Research ArticleDistributed Cooperative Backpressure-Based Traffic LightControl Method

Shenxue Hao 12 Licai Yang 1 Li Ding 2 and Yajuan Guo 1

1School of Control Science and Engineering Shandong University Jinan 250061 China2School of Information and Engineering Shandong Yingcai University Jinan 250104 China

Correspondence should be addressed to Licai Yang yanglcsdueducn

Received 6 November 2018 Accepted 18 February 2019 Published 5 March 2019

Academic Editor Eneko Osaba

Copyright copy 2019 Shenxue Hao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

On the foundation of the original backpressure-based traffic light control algorithm a distributed cooperative backpressure-basedtraffic light control method is proposed in this paper The urban traffic network is modeled as a smart agent-controlled queuingnetwork inwhich the intersection agents exchange the queue length information and the selected activating light phase informationof neighboring intersections through communications and determine the activating light phase at each time slot according tolocal traffic informationThe improved phase pressure computation method considers the phase state of downstream intersectionsinstead of only the queue length of the local intersections Light phase switching coordination among adjacent intersections isachieved using the consensus-based bundle algorithm in which the cooperative light phase switching problem is viewed as atask assignment issue among adjacent intersections Simulation results illustrated that the proposed cooperative backpressure-based traffic light control method obtained better performance than the original backpressure-based and fixed-time traffic controlmethods

1 Introduction

With the increase of vehicles travelling on roads in citiescongestion has become a major problem for urban trafficsystems Optimal traffic control strategy can improve traf-fic efficiency and balance the traffic load which plays animportant role in urban traffic systems There are three typesof conventional traffic control methods fixed-time trafficcontrol method actuated traffic control and adaptive trafficcontrol In the 1960s TRANSYT [1] and MAXBAND [2]were proposed based on offline historical traffic data whichselected optimal timing plans at different times of the dayHowever these fixed-time traffic control methods cannotdeal with the fluctuation of traffic demand due to predefinedtraffic parameters Next an actuated traffic control methodwas proposed to extend the light phase duration accordingto the detected traffic flow data in real time using detectorsinstalled at upstream segments However the detection ofsparse traffic has a considerable influence on delay time[3] and it was applied mainly at isolated intersections [4]Adaptive traffic control methods (SCOOT [5] and SCATS

[6]) were proposed to adjust signal timing plans based ononline traffic information for responding to real-time trafficdemand These methods can respond to traffic fluctuationusing detector input historical trends and predictive models[7] Although adaptive traffic control systems have beenimplied in real urban traffic networks the cooperation ofmultiple intersections requires a control center which hasobstructed further development In addition the centralizedsignal timing scheme is always calculated according to thecurrent traffic conditions but is implied in the next cyclewhen the traffic situation may have already changed [4]

To fix these deficiencies there is an agreement amongmany researchers that a distributed traffic control strategy isan ideal alternative In a distributed traffic control systemsignalized intersections are viewed as smart agents andeach agent determines the traffic control parameters accord-ing to current local information For real-time dynamictraffic conditions this kind of traffic control method canachieve better performance and adaptation [8 9] Manyagent-based traffic control approaches have been studied inthe past decades and different theories and methods have

HindawiJournal of Advanced TransportationVolume 2019 Article ID 7481489 14 pageshttpsdoiorg10115520197481489

2 Journal of Advanced Transportation

been utilized to optimize traffic control among intersectionsaccording to different traffic parameters Researchers haveutilized intelligent computation methods to obtain the cyclelength and splits by minimizing traffic parameters such astotal travel time etc These types of methods avoid hugecomputations such as the colony optimization approach [10]and reinforcement learning algorithm [11] However it is stilldifficult for these types of traffic control methods to achieveonline traffic timing decision making since the traffic signaltiming problem usually is NP hard it may take long time tofind an optimal solution for simple transportation systems[12 13] Therefore to simplify the traffic control problema better way of achieving distributed traffic system controlwith lower computation is needed In 2012 Wongpiromsarnet al first introduced a backpressure algorithm to solve trafficcontrol problems and develop a traffic signal control strategyby viewing the traffic network as a queuing network [14]Thebackpressure algorithm is usually used in wireless multihopnetworks as an optimal strategy for resource allocation It hasmany features including throughput optimality achievableadaptive resource allocation and simplicity [15]

Since the backpressure-based traffic signal control algo-rithm can maximize the throughput of the traffic network ina completely distributed manner researchers have paid moreattention to this algorithm and have obtained many achieve-ments In 2013 Varaiya presented amaximumpressure trafficcontrol strategy based on the backpressure algorithmAt eachintersection the active phase is selected depending on thelocal queue lengthmean turn ratios and saturation rates [16]Considering the influence of the routing rate of the queuedvehicle on links Gregoire et al presented a backpressure-based traffic signal control algorithm with unknown rout-ing rates and an estimated aggregated queue length Thevehicle routing information can be detected using detectorson dedicated lanes [17] In their later research a routingmodel of the traffic network was established with partialcontrollable vehicles to be used for pressure computation [18]With the development of communication technology moreinformation could be obtained from a vehicular network anda multicommodity backpressure algorithm for traffic lightcontrol was proposed where it is assumed that all vehiclesrsquoroutes are known [19] To further improve the efficiency ofthe traffic network Taale et al integrated route guidancetechnology with traffic signal control based on the backpres-sure algorithm [20] and Le et al proposed a cyclic phasebackpressure control policy with online estimation of turningfraction and measurement of queue size [21] Howeverthese backpressure-based traffic light control methods onlydetermine the activating light phase according to the phasepressure computed using local information and neglect thepossible coordination with adjacent intersections for phasepressure computation

In this paper we propose a cooperative backpressure-based traffic control method in which the phase pressureis computed by considering the phase state of downstreamintersections Actually the phase pressure is influenced by thequeue length on the downstream segmentWhen the vehiclesqueued in front of the downstream intersections obtain theright of way the corresponding phase of the downstream

intersection is activated and the current phase pressure willincrease due to the decrease of queued vehicles on down-stream segmentThese phase pressure changes may affect thechoice of the activating light phase Therefore we propose amodified phase pressure computation method which con-siders the phase state of downstream intersections to achievecooperative light phase switching among intersections Inaddition for the cooperative light phase switching decisionamong intersections the traffic light switching problem isviewed as a task assignment issue Moreover the consensus-based bundle algorithm (CBBA) which is usually utilized fordecentralized task selection for a multiagent system [22] isintroduced to achieve coordination among intersections

To summarize there are two main innovations in thispaper (1) A modified phase pressure computation methodis proposed considering the phase state of downstreamintersections this method is appropriate for cooperative lightphase switching among intersections (2)CBBA is introducedto solve the conflicts in cooperative light phase switchingdecision by viewing the cooperative traffic light switching asa cooperative task assignment problem

The remainder of this paper is organized as follows InSection 2 the urban traffic network is modeled as an agent-controlled queuing networkThe phase pressure computationmethod is presented in Section 3 In Section 4 the CBBAbased traffic light cooperative control algorithm is describedIn Section 5 the stability of the proposed algorithm isanalyzed Simulations are carried out and the results arediscussed in Section 6 This paper is concluded in Section 7

2 Traffic Network Modeling

The urban traffic network is modeled as an agent-controlledqueuing network Road segments are considered link nodesThere are 3 types of link nodes ingress node internal nodeand exit node Both ingress nodes and internal nodes havea downstream intersection Vehicle flows entering a nodegenerate vehicle queues when the corresponding light phaseis red The exit nodes from which vehicles are leavingthe traffic network have no downstream intersection andtherefore do not generate queued vehicles It assumes thatvehicle flows entering internal nodes are coming completelyfrom an upstream intersection

The intersections controlled by Smart Traffic Light Con-trol Agents (STLCA) are viewed as junctions that connectlink nodes Although it is unnecessary to explain the lightphase of each intersection under backpressure-based trafficcontrol method used in this paper to describe the topologyof vehicle flows in traffic network clearly it assumes thatthere are 4 light phases at each intersection north-southstraight phase north-south left-turn phase west-east straightphase and west-east left-turn phase as shown in Figure 1There are 3 dedicated lanes on each road segment ie a left-turn lane a straight lane and a right-turn lane It assumesthat the vehicles entering a road segment will drive into thededicated lane immediately and the right-turn vehicle flowsare free up to the traffic light Vehicle flow on the straightlane or left-turn lane is controlled by the correspondinglight phases For example as shown in Figure 2 there are

Journal of Advanced Transportation 3

Φ1 Φ2

Φ3 Φ4

Figure 1 The light phases of one intersection

Exit node Ingress nodeInternal node Internal node

au2

au1

au3 b

fl3

fl2

fl1

fl4

fs3

fs2

fs1

fs4fr3

fr2

fr1

fr4

ai

b

b

i-1

qab

qab

qab

c

c

c

i+1

qbc

qbc

qbc

Figure 2 The topology of queuing traffic network

12 vehicle flows passing through intersection i The right-turn flows 1198911199031 1198911199032 1198911199033 1198911199034 are not controlled by the trafficlight of STLCA i the straight flows 1198911199041 1198911199043 and 1198911199042 1198911199044are controlled by west-east straight phase and north-southstraight phase of intersection i and the left-turn flows1198911198971 1198911198973 1198911198972 1198911198974 are controlled by west-east left-turn phaseand north-south left-turn phase respectively

Time is slotted for the queuing network control LetAa(t)denote the exogenous arrivals from upstream intersectionDab(t) represent the number of vehicles moving from seg-ment a to b and Tab(t) is the turn ratio of exogenous arrivalsthat is there are Tab(t)Aa(t) vehicles add to queued vehiclesqab(t) during time slot t Then queue length qab on a waitingformoving to b can be computed using (1) Shown as Figure 2the exogenous arrival Aa(t) of a comes from the three vehicleflows 1198911198974 1198911199041 1198911199032 of intersection i-1

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119879119886119887119860119886 (119905) minus 119863119886119887 (119905) (1)

In the agent-controlled queuing network STLCA i selects alight phase to activate with a given green time or extendsthe current activating light phase with a short green timeaccording to the backpressure algorithm at each time slot tIn this paper we only consider the cooperative light phaseswitching strategy The activating phase extension strategywill be researched in the future work

3 Pressure Computation Method ofLight Phase

In the traffic network vehicle flows on segment a waiting topass through the intersection will generate traffic pressureto the downstream intersection According to the original

4 Journal of Advanced Transportation

p

p

d2 c1

b2 p2

c2 d1

a2

a1 b1

Figure 3 The case with similar queue length for downstream and upstream links

backpressure algorithm the pressure of each light phasepressurep is defined as the sum of the pressure associated withall vehicle flows controlled by light phase p computed using(2) and (3) The pressure associated with vehicle flow f ab isinpis defined as the current flow rate weighted by wab(t) (thedifference between qa and qb) where f ab isinp indicates that f abobtains the right of way when p is activated [14 23]

119908119886119887 (119905) = 119902119886 (119905) minus 119902119887 (119905) (2)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) 119908119886119887 (119905) (3)

where qa(t) is the total number of vehicles waiting on segmenta at the beginning of slot t and 120583ab(p) is the number ofvehicles passing through intersection per unit time when pis activated If p is not activated then 120583ab(p) = 0

Only these vehicles queued on segment a waiting formoving to node b generate pressure to the light phase pinstead of all the vehicles on segment aTherefore inVaraiyarsquospaper [17] the weight 119908119886119887(119905) was modified as

119908119886119887 (119905) = 119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905) (4)

where rbc denotes the proportion of vehicle flow leaving b andentering c and sum119888 119903119887119888119902119887119888(119905) indicates the average queue lengthof b

According to the backpressure-based traffic control algo-rithm [15] STLCA i chooses a phase with maximum pressureto activate in the next time slot using (5) to achievemaximumthroughput of a single intersection

119901lowast = arg max119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (5)

where Pi is the light phase set of intersection i and 119901lowast is theselected phase to activate in the next time slot

However when the vehicle queue on the downstreamsegments of an intersection is longer than the vehicle queueon an upstream segment or when the two queue lengths are

close (as shown in Figure 3) according to the backpressurealgorithm phase 1199011015840 will be activated instead of phase p in thenext time slot In this situation the two pairs of queue lengths(qa1 qb1) and (qa2 qb2) both have a small difference and thetwo pairs of (qc1 qd1) and (qc2 qd2) have a larger differenceThis results in the pressure of phase p being smaller thanphase1199011015840 and phase1199011015840 with a smaller volume is to be activatedin the next time slot It is obvious that in this circumstance thebackpressure-based phase switching strategy described abovecannot achieve an ideal effect

Considering this situation the cooperative traffic controlstrategy may be a good choice for some subareas of the trafficnetwork Based on this view in this paper the phase pressureis calculated considering the phase state of downstreamintersections to achieve a cooperative backpressure-basedphase switching algorithm Shown in Figure 4(a) for thestraight phase 119901119894119895 of intersection i there is 119891119886119887 isin 119901119894119895 Forthe downstream intersection i+1 there are 1198911198871198881015840 isin 119901119894+1119896 and11989111988711988810158401015840 isin 119901119894+1119896+1 119901119894119895 denotes the jth light phase of intersection i1198891198871198881015840 and 11988911988711988810158401015840 denote the number of vehicles departing fromb to 1198881015840 or 11988810158401015840 respectively If 119901119895119896 or 119901119895

119896+1is activated then the

queued vehicles on b will decrease and the pressure of 119901119894119895will increaseThen we can conclude that the pressure of119901119894119895 ofintersection i is affected by the phase state of the downstreamintersection i+1 A similar situation of the left-turn lightphase of intersection i is shown in Figure 4(b) Accordingto the above-stated analysis there are two cases for thephase switching coordination of adjacent intersections Case1 occurs when intersection i is determining the activatingphase of the next time slot the corresponding downstreamphase 119901119894+1119896 or 119901119894+1119896+1 is still activating Case 2 occurs when thetwo intersections i and i+1 are determining the activatingphase at the same time For these two cases the phasepressure computation method of 119901119894119895 should be modified as(6)-(9)

119908119886119887 (119905) = 119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905)) (6)

Journal of Advanced Transportation 5

a b

iqabpij

dbc

c

c

pi+1kdbc

i + 1qbc

qbcpi+1k+1

(a)

a

qbc

dbc

c

c

pi+1k+1 pi+1

k

dbc

qabi

i+1

b

qbc

pij

(b)

Figure 4 Pressure of straight phase and left-turn phase of intersection i affected by downstream intersection i+1 (a) Straight phase ofintersection i affected by downstream light phase (b) left-turn phase of intersection i affected by downstream light phase

119901119903119890119904119904119906119903119890119901= sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))) (7)

where dbc(t) is the number of vehicles departing from b to cdefined as (8) (9)

119889119887119888 (119905)=

119904 (119901119896 (119905)) 119902119887119888 (119905) for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) lt 119891max

119904 (119901119896 (119905)) 119891max for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) gt= 119891max

(8)

where

119904 (119901119896 (119905)) = 1 if 119901119896 (119905) is activated0 if 119901119896 (119905) is not activated (9)

where 119891max is the maximum number of vehicles passingthrough the intersection when the downstream light phasepk is activated

Using the proposed method of phase pressure computa-tion by considering the phase state of downstream intersec-tions the light phase is to be switched in cooperation modeThe light phase switching of adjacent intersections shouldbe able to make a cooperative decision without conflictsHowever it is difficult to obtain the global optimal trafficcontrol strategy amongmultiple intersections with a complextraffic state CBBA which is usually used for task assignmentproblems [22] is introduced to solve this problem in the nextsection

4 Cooperative Light Phase Switching DecisionUsing CBBA

According to the idea of the backpressure traffic light controlalgorithm the phase withmaximum pressure will be selectedto activate in the next time slot However the phase pressure

computed using the proposedmethod is affected by the phasestate of the downstream intersection The selected phase 119901119894119895of intersection i may be in conflict with the phase 119901119894+1119896 ofdownstream intersection i+1 This is because the pressureof 119901119894119895 may be computed by assuming that phase 119901119894+1119896+1 isactivated To solve the conflicts of phase switching amongintersections the CBBA is introduced in this section

CBBA is a market-based distributed task assignmentalgorithm and has been shown to produce a conflict freesolution [24 25] There are two stages in CBBA [22] ie theauction stage and the consensus stage In the auction stageeach agent bids on a task and calculates a score based on thelocal information to decide whether it assigns a new task tothe task bundle or not The task bundle of agent i is definedas the possible task set with task score If agent i does notwin any tasks in the auction stage the task bundle of agenti is empty In the consensus stage agents exchange the taskinformation through communication If an agent is outbidfor a task then it is released from the task bundle For atask the agent with the highest score is the winner Based onthe idea of CBBA in the next section the cooperative phaseswitching problem in urban traffic network is modeled as atask assignment problem

41 Cooperative Light Phase SwitchingModeled as TaskAssign-ment There are four light phases for the light controllerof each intersection which are managed by a smart agentSTLCA The task of STLCA is to determine which phaseis to be activated in the next time slot For each intersec-tion there are two vehicle flows with opposite directionscontrolled by each light phase For each vehicle flow thereis one downstream intersection at most If there is onedownstream intersection for one vehicle flow then thereare three cooperative phase switching possibilities Shown inFigure 5 vehicle flows f 1 and f 2 are controlled by phase 119901119894119895When 119901119894119895 is activated the vehicles waiting on segments a1and a2 obtain the right of way to pass through intersectioni The pressure of 119901119894119895 is consists of two parts the pressure

6 Journal of Advanced Transportation

c21

c22

cp21

cp22

b2

a1 f1c11

c12

cp11

cp12

b1

a2f2

pij

Intersection i

DownstreamIntersection2

DownstreamIntersection1

Figure 5 Coordination relationships of light phases among adjacent intersections

(1)

(3)

(5)

(6)

(8) (9)

(7)

(4)

(2)

Figure 6 Nine cooperative phase switching possibilities of one light phase

from f 1 and the pressure from f 2 The pressure generatedfrom vehicle flow f 1 is associated with the cooperative phasescp11 and cp12 If cp11 or cp12 are activated vehicles onsegment b1 can drive into segments c11 or c12 In this casethe number of vehicles departing from segment b1 shouldbe considered If cp11 and cp12 are not to be activated orthe DownstreamIntersection1 does not exist the number ofvehicles departing from segment b1 is zero Similarly thecomputation of pressure generated from vehicle flow f 2should consider the downstream phase states cp21 or cp22Therefore when considering the downstream phase statethere are 9 possible cooperative phase switching strategies forphase 119901119894119895 as shown in Figure 6 For each intersection thereare at most 36 possible cooperative phase switching

In order to solve the cooperative phase switching problemusing CBBA these possible cooperative phase switchingstrategies are viewed as the task bundle of STLCA i Eachpossible phase switching strategy is viewed as a possibletask of intersection i The task score is represented by thephase pressure calculated using (6)-(9) according to thedownstream phase state

42 Cooperative Backpressure-Based Light Phase SwitchingAlgorithm In this section the CBBA is utilized to solve thecooperative phase switching problem among intersectionsAccording to the CCBA the cooperative backpressure-basedlight phase switching algorithm consists of three steps asfollows

(a) Task Bundle Construction STLCA i constructs thetask bundle according to its topology in the traffic network

The possible switching strategies of intersection i are consid-ered as the possible tasks of STLCA i The amount of possibleswitching strategies for a particular intersection is fixed sincethe topology of an intersection in urban traffic network isfixed The task bundle of each intersection is constructedbefore phase switching decision making can improve thealgorithm performance Shown in Table 1 there are 9 possiblephase switching strategies for one phase of intersection iTherefore there are 36 possible switching strategies in oneintersection at most

(b) Phase Pressure Computation STLCA i collects thequeue length information of each segment and broadcaststhe information to the adjacent intersectionsThe pressure ofeach strategy is calculated using (6)-(9) according to the localqueue length information andqueue length information fromadjacent intersections The possible strategies of intersectioni are ordered by the phase pressure and all the possiblestrategies are set to be available at beginning After thepressure computation for each intersection the strategy withmaximum pressure is selected to be the candidate strategyand the responding phase to be the candidate light phase foractivating in the next time slot

(c) Activating Phase Decision and Conflicts ResolutionIn this step STLCA i determines the candidate phase asthe activating phase of intersection i at time t+1 directlyif the candidate strategy need not coordinate with theneighboring downstream intersections These intersectionswhose light phase has been determined are called activating-phase-determined intersections If STLCA i has determinedthe activating phase STLCA i broadcasts this information

Journal of Advanced Transportation 7

If SelectedStrategyDownstreamIntersection1 = 0 And SelectedStrategyDownstreamIntersection2 = 0ThenIntersectioniDeterminedFlag = TrueIntersectioniSelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfIf IntersectioniDeterminedFlag = False Then

If STLCAi receives the decisions from adjacent intersections ThenFor each AdjacentIntersection of IntersectioniIf AdjacentIntersectionjDetermineFlag = TrueThenRelease the strategies of Intersectioni conflicting with AdjacentIntersectionjSelectedPhase

End IfNextReselect the IntersectioniSelectedStrategy with maximum pressure from the available strategiesBroadcast IntersectioniSelectedStrategy to the adjacent intersections

End IfElect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithmIf IntersectioniSelectedStrategy is the elected strategy with maximum pressure ThenIntersectioni DeterminedFlag = TrueIntersectioni SelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfEnd If

Algorithm 1 The algorithm of intersection i for activating phase decision and conflicts resolution

Table 1 The possible cooperative phase switching strategies of 119901119894119895Strategy Number Light phase DownstreamIntersection1 CooperativePhase1 DownstreamIntersection2 CooperativePhase21 119901119894119895 0 times 0 times2 119901119894119895 1 cp11 0 times3 119901119894119895 1 cp12 0 times4 119901119894119895 0 times 1 cp215 119901119894119895 0 times 1 cp226 119901119894119895 1 cp11 1 cp217 119901119894119895 1 cp11 1 cp228 119901119894119895 1 cp12 1 cp219 119901119894119895 1 cp12 1 cp22Note times indicates that the cooperative phase is not to be activated 0 indicates that the downstream intersection does not exist or does not coordinate with 119901119894119895cp11 cp12 cp21 and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2

to its adjacent intersections Then the adjacent STLCAs ofintersection i release the strategies conflict with intersectioni For example if intersection i has determined the activatinglight phase to be west-east straight phase the strategies ofneighboring intersections (if the neighboring intersectionsare not activating-phase-determined intersections) which areconflict with the activating light phase of intersection i will bereleased that is the invalid flag of these strategies are set to betrue For these activating-phase-undermined intersectionsSTLCAs elect a strategy with maximum pressure in dis-tributed mode by communication The elected intersectiondetermines the activating phase and broadcasts it to adjacentintersections Step (a) and (b) are repeated until all STLCAs

have determined the activating phase A detailed descriptionof the algorithm is given in Algorithm 1

Based on the three steps the cooperative backpressure-based traffic control method (simplified as CBP method) canobtain a conflict free phase switching strategy with max-imum phase pressure The backpressure-based traffic lightcontrol method (simplified as BP method) only considersthe queue length of the current intersection and neglectsthe decrease of queued vehicles on the downstream segmentwhen the corresponding downstream phase is activatedThe CBP method fixes this deficiency and considers thephase state of downstream intersections to achieve coordi-nation among intersections Furthermore all the switching

8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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Journal of Advanced Transportation 3

Φ1 Φ2

Φ3 Φ4

Figure 1 The light phases of one intersection

Exit node Ingress nodeInternal node Internal node

au2

au1

au3 b

fl3

fl2

fl1

fl4

fs3

fs2

fs1

fs4fr3

fr2

fr1

fr4

ai

b

b

i-1

qab

qab

qab

c

c

c

i+1

qbc

qbc

qbc

Figure 2 The topology of queuing traffic network

12 vehicle flows passing through intersection i The right-turn flows 1198911199031 1198911199032 1198911199033 1198911199034 are not controlled by the trafficlight of STLCA i the straight flows 1198911199041 1198911199043 and 1198911199042 1198911199044are controlled by west-east straight phase and north-southstraight phase of intersection i and the left-turn flows1198911198971 1198911198973 1198911198972 1198911198974 are controlled by west-east left-turn phaseand north-south left-turn phase respectively

Time is slotted for the queuing network control LetAa(t)denote the exogenous arrivals from upstream intersectionDab(t) represent the number of vehicles moving from seg-ment a to b and Tab(t) is the turn ratio of exogenous arrivalsthat is there are Tab(t)Aa(t) vehicles add to queued vehiclesqab(t) during time slot t Then queue length qab on a waitingformoving to b can be computed using (1) Shown as Figure 2the exogenous arrival Aa(t) of a comes from the three vehicleflows 1198911198974 1198911199041 1198911199032 of intersection i-1

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119879119886119887119860119886 (119905) minus 119863119886119887 (119905) (1)

In the agent-controlled queuing network STLCA i selects alight phase to activate with a given green time or extendsthe current activating light phase with a short green timeaccording to the backpressure algorithm at each time slot tIn this paper we only consider the cooperative light phaseswitching strategy The activating phase extension strategywill be researched in the future work

3 Pressure Computation Method ofLight Phase

In the traffic network vehicle flows on segment a waiting topass through the intersection will generate traffic pressureto the downstream intersection According to the original

4 Journal of Advanced Transportation

p

p

d2 c1

b2 p2

c2 d1

a2

a1 b1

Figure 3 The case with similar queue length for downstream and upstream links

backpressure algorithm the pressure of each light phasepressurep is defined as the sum of the pressure associated withall vehicle flows controlled by light phase p computed using(2) and (3) The pressure associated with vehicle flow f ab isinpis defined as the current flow rate weighted by wab(t) (thedifference between qa and qb) where f ab isinp indicates that f abobtains the right of way when p is activated [14 23]

119908119886119887 (119905) = 119902119886 (119905) minus 119902119887 (119905) (2)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) 119908119886119887 (119905) (3)

where qa(t) is the total number of vehicles waiting on segmenta at the beginning of slot t and 120583ab(p) is the number ofvehicles passing through intersection per unit time when pis activated If p is not activated then 120583ab(p) = 0

Only these vehicles queued on segment a waiting formoving to node b generate pressure to the light phase pinstead of all the vehicles on segment aTherefore inVaraiyarsquospaper [17] the weight 119908119886119887(119905) was modified as

119908119886119887 (119905) = 119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905) (4)

where rbc denotes the proportion of vehicle flow leaving b andentering c and sum119888 119903119887119888119902119887119888(119905) indicates the average queue lengthof b

According to the backpressure-based traffic control algo-rithm [15] STLCA i chooses a phase with maximum pressureto activate in the next time slot using (5) to achievemaximumthroughput of a single intersection

119901lowast = arg max119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (5)

where Pi is the light phase set of intersection i and 119901lowast is theselected phase to activate in the next time slot

However when the vehicle queue on the downstreamsegments of an intersection is longer than the vehicle queueon an upstream segment or when the two queue lengths are

close (as shown in Figure 3) according to the backpressurealgorithm phase 1199011015840 will be activated instead of phase p in thenext time slot In this situation the two pairs of queue lengths(qa1 qb1) and (qa2 qb2) both have a small difference and thetwo pairs of (qc1 qd1) and (qc2 qd2) have a larger differenceThis results in the pressure of phase p being smaller thanphase1199011015840 and phase1199011015840 with a smaller volume is to be activatedin the next time slot It is obvious that in this circumstance thebackpressure-based phase switching strategy described abovecannot achieve an ideal effect

Considering this situation the cooperative traffic controlstrategy may be a good choice for some subareas of the trafficnetwork Based on this view in this paper the phase pressureis calculated considering the phase state of downstreamintersections to achieve a cooperative backpressure-basedphase switching algorithm Shown in Figure 4(a) for thestraight phase 119901119894119895 of intersection i there is 119891119886119887 isin 119901119894119895 Forthe downstream intersection i+1 there are 1198911198871198881015840 isin 119901119894+1119896 and11989111988711988810158401015840 isin 119901119894+1119896+1 119901119894119895 denotes the jth light phase of intersection i1198891198871198881015840 and 11988911988711988810158401015840 denote the number of vehicles departing fromb to 1198881015840 or 11988810158401015840 respectively If 119901119895119896 or 119901119895

119896+1is activated then the

queued vehicles on b will decrease and the pressure of 119901119894119895will increaseThen we can conclude that the pressure of119901119894119895 ofintersection i is affected by the phase state of the downstreamintersection i+1 A similar situation of the left-turn lightphase of intersection i is shown in Figure 4(b) Accordingto the above-stated analysis there are two cases for thephase switching coordination of adjacent intersections Case1 occurs when intersection i is determining the activatingphase of the next time slot the corresponding downstreamphase 119901119894+1119896 or 119901119894+1119896+1 is still activating Case 2 occurs when thetwo intersections i and i+1 are determining the activatingphase at the same time For these two cases the phasepressure computation method of 119901119894119895 should be modified as(6)-(9)

119908119886119887 (119905) = 119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905)) (6)

Journal of Advanced Transportation 5

a b

iqabpij

dbc

c

c

pi+1kdbc

i + 1qbc

qbcpi+1k+1

(a)

a

qbc

dbc

c

c

pi+1k+1 pi+1

k

dbc

qabi

i+1

b

qbc

pij

(b)

Figure 4 Pressure of straight phase and left-turn phase of intersection i affected by downstream intersection i+1 (a) Straight phase ofintersection i affected by downstream light phase (b) left-turn phase of intersection i affected by downstream light phase

119901119903119890119904119904119906119903119890119901= sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))) (7)

where dbc(t) is the number of vehicles departing from b to cdefined as (8) (9)

119889119887119888 (119905)=

119904 (119901119896 (119905)) 119902119887119888 (119905) for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) lt 119891max

119904 (119901119896 (119905)) 119891max for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) gt= 119891max

(8)

where

119904 (119901119896 (119905)) = 1 if 119901119896 (119905) is activated0 if 119901119896 (119905) is not activated (9)

where 119891max is the maximum number of vehicles passingthrough the intersection when the downstream light phasepk is activated

Using the proposed method of phase pressure computa-tion by considering the phase state of downstream intersec-tions the light phase is to be switched in cooperation modeThe light phase switching of adjacent intersections shouldbe able to make a cooperative decision without conflictsHowever it is difficult to obtain the global optimal trafficcontrol strategy amongmultiple intersections with a complextraffic state CBBA which is usually used for task assignmentproblems [22] is introduced to solve this problem in the nextsection

4 Cooperative Light Phase Switching DecisionUsing CBBA

According to the idea of the backpressure traffic light controlalgorithm the phase withmaximum pressure will be selectedto activate in the next time slot However the phase pressure

computed using the proposedmethod is affected by the phasestate of the downstream intersection The selected phase 119901119894119895of intersection i may be in conflict with the phase 119901119894+1119896 ofdownstream intersection i+1 This is because the pressureof 119901119894119895 may be computed by assuming that phase 119901119894+1119896+1 isactivated To solve the conflicts of phase switching amongintersections the CBBA is introduced in this section

CBBA is a market-based distributed task assignmentalgorithm and has been shown to produce a conflict freesolution [24 25] There are two stages in CBBA [22] ie theauction stage and the consensus stage In the auction stageeach agent bids on a task and calculates a score based on thelocal information to decide whether it assigns a new task tothe task bundle or not The task bundle of agent i is definedas the possible task set with task score If agent i does notwin any tasks in the auction stage the task bundle of agenti is empty In the consensus stage agents exchange the taskinformation through communication If an agent is outbidfor a task then it is released from the task bundle For atask the agent with the highest score is the winner Based onthe idea of CBBA in the next section the cooperative phaseswitching problem in urban traffic network is modeled as atask assignment problem

41 Cooperative Light Phase SwitchingModeled as TaskAssign-ment There are four light phases for the light controllerof each intersection which are managed by a smart agentSTLCA The task of STLCA is to determine which phaseis to be activated in the next time slot For each intersec-tion there are two vehicle flows with opposite directionscontrolled by each light phase For each vehicle flow thereis one downstream intersection at most If there is onedownstream intersection for one vehicle flow then thereare three cooperative phase switching possibilities Shown inFigure 5 vehicle flows f 1 and f 2 are controlled by phase 119901119894119895When 119901119894119895 is activated the vehicles waiting on segments a1and a2 obtain the right of way to pass through intersectioni The pressure of 119901119894119895 is consists of two parts the pressure

6 Journal of Advanced Transportation

c21

c22

cp21

cp22

b2

a1 f1c11

c12

cp11

cp12

b1

a2f2

pij

Intersection i

DownstreamIntersection2

DownstreamIntersection1

Figure 5 Coordination relationships of light phases among adjacent intersections

(1)

(3)

(5)

(6)

(8) (9)

(7)

(4)

(2)

Figure 6 Nine cooperative phase switching possibilities of one light phase

from f 1 and the pressure from f 2 The pressure generatedfrom vehicle flow f 1 is associated with the cooperative phasescp11 and cp12 If cp11 or cp12 are activated vehicles onsegment b1 can drive into segments c11 or c12 In this casethe number of vehicles departing from segment b1 shouldbe considered If cp11 and cp12 are not to be activated orthe DownstreamIntersection1 does not exist the number ofvehicles departing from segment b1 is zero Similarly thecomputation of pressure generated from vehicle flow f 2should consider the downstream phase states cp21 or cp22Therefore when considering the downstream phase statethere are 9 possible cooperative phase switching strategies forphase 119901119894119895 as shown in Figure 6 For each intersection thereare at most 36 possible cooperative phase switching

In order to solve the cooperative phase switching problemusing CBBA these possible cooperative phase switchingstrategies are viewed as the task bundle of STLCA i Eachpossible phase switching strategy is viewed as a possibletask of intersection i The task score is represented by thephase pressure calculated using (6)-(9) according to thedownstream phase state

42 Cooperative Backpressure-Based Light Phase SwitchingAlgorithm In this section the CBBA is utilized to solve thecooperative phase switching problem among intersectionsAccording to the CCBA the cooperative backpressure-basedlight phase switching algorithm consists of three steps asfollows

(a) Task Bundle Construction STLCA i constructs thetask bundle according to its topology in the traffic network

The possible switching strategies of intersection i are consid-ered as the possible tasks of STLCA i The amount of possibleswitching strategies for a particular intersection is fixed sincethe topology of an intersection in urban traffic network isfixed The task bundle of each intersection is constructedbefore phase switching decision making can improve thealgorithm performance Shown in Table 1 there are 9 possiblephase switching strategies for one phase of intersection iTherefore there are 36 possible switching strategies in oneintersection at most

(b) Phase Pressure Computation STLCA i collects thequeue length information of each segment and broadcaststhe information to the adjacent intersectionsThe pressure ofeach strategy is calculated using (6)-(9) according to the localqueue length information andqueue length information fromadjacent intersections The possible strategies of intersectioni are ordered by the phase pressure and all the possiblestrategies are set to be available at beginning After thepressure computation for each intersection the strategy withmaximum pressure is selected to be the candidate strategyand the responding phase to be the candidate light phase foractivating in the next time slot

(c) Activating Phase Decision and Conflicts ResolutionIn this step STLCA i determines the candidate phase asthe activating phase of intersection i at time t+1 directlyif the candidate strategy need not coordinate with theneighboring downstream intersections These intersectionswhose light phase has been determined are called activating-phase-determined intersections If STLCA i has determinedthe activating phase STLCA i broadcasts this information

Journal of Advanced Transportation 7

If SelectedStrategyDownstreamIntersection1 = 0 And SelectedStrategyDownstreamIntersection2 = 0ThenIntersectioniDeterminedFlag = TrueIntersectioniSelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfIf IntersectioniDeterminedFlag = False Then

If STLCAi receives the decisions from adjacent intersections ThenFor each AdjacentIntersection of IntersectioniIf AdjacentIntersectionjDetermineFlag = TrueThenRelease the strategies of Intersectioni conflicting with AdjacentIntersectionjSelectedPhase

End IfNextReselect the IntersectioniSelectedStrategy with maximum pressure from the available strategiesBroadcast IntersectioniSelectedStrategy to the adjacent intersections

End IfElect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithmIf IntersectioniSelectedStrategy is the elected strategy with maximum pressure ThenIntersectioni DeterminedFlag = TrueIntersectioni SelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfEnd If

Algorithm 1 The algorithm of intersection i for activating phase decision and conflicts resolution

Table 1 The possible cooperative phase switching strategies of 119901119894119895Strategy Number Light phase DownstreamIntersection1 CooperativePhase1 DownstreamIntersection2 CooperativePhase21 119901119894119895 0 times 0 times2 119901119894119895 1 cp11 0 times3 119901119894119895 1 cp12 0 times4 119901119894119895 0 times 1 cp215 119901119894119895 0 times 1 cp226 119901119894119895 1 cp11 1 cp217 119901119894119895 1 cp11 1 cp228 119901119894119895 1 cp12 1 cp219 119901119894119895 1 cp12 1 cp22Note times indicates that the cooperative phase is not to be activated 0 indicates that the downstream intersection does not exist or does not coordinate with 119901119894119895cp11 cp12 cp21 and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2

to its adjacent intersections Then the adjacent STLCAs ofintersection i release the strategies conflict with intersectioni For example if intersection i has determined the activatinglight phase to be west-east straight phase the strategies ofneighboring intersections (if the neighboring intersectionsare not activating-phase-determined intersections) which areconflict with the activating light phase of intersection i will bereleased that is the invalid flag of these strategies are set to betrue For these activating-phase-undermined intersectionsSTLCAs elect a strategy with maximum pressure in dis-tributed mode by communication The elected intersectiondetermines the activating phase and broadcasts it to adjacentintersections Step (a) and (b) are repeated until all STLCAs

have determined the activating phase A detailed descriptionof the algorithm is given in Algorithm 1

Based on the three steps the cooperative backpressure-based traffic control method (simplified as CBP method) canobtain a conflict free phase switching strategy with max-imum phase pressure The backpressure-based traffic lightcontrol method (simplified as BP method) only considersthe queue length of the current intersection and neglectsthe decrease of queued vehicles on the downstream segmentwhen the corresponding downstream phase is activatedThe CBP method fixes this deficiency and considers thephase state of downstream intersections to achieve coordi-nation among intersections Furthermore all the switching

8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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4 Journal of Advanced Transportation

p

p

d2 c1

b2 p2

c2 d1

a2

a1 b1

Figure 3 The case with similar queue length for downstream and upstream links

backpressure algorithm the pressure of each light phasepressurep is defined as the sum of the pressure associated withall vehicle flows controlled by light phase p computed using(2) and (3) The pressure associated with vehicle flow f ab isinpis defined as the current flow rate weighted by wab(t) (thedifference between qa and qb) where f ab isinp indicates that f abobtains the right of way when p is activated [14 23]

119908119886119887 (119905) = 119902119886 (119905) minus 119902119887 (119905) (2)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) 119908119886119887 (119905) (3)

where qa(t) is the total number of vehicles waiting on segmenta at the beginning of slot t and 120583ab(p) is the number ofvehicles passing through intersection per unit time when pis activated If p is not activated then 120583ab(p) = 0

Only these vehicles queued on segment a waiting formoving to node b generate pressure to the light phase pinstead of all the vehicles on segment aTherefore inVaraiyarsquospaper [17] the weight 119908119886119887(119905) was modified as

119908119886119887 (119905) = 119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905) (4)

where rbc denotes the proportion of vehicle flow leaving b andentering c and sum119888 119903119887119888119902119887119888(119905) indicates the average queue lengthof b

According to the backpressure-based traffic control algo-rithm [15] STLCA i chooses a phase with maximum pressureto activate in the next time slot using (5) to achievemaximumthroughput of a single intersection

119901lowast = arg max119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (5)

where Pi is the light phase set of intersection i and 119901lowast is theselected phase to activate in the next time slot

However when the vehicle queue on the downstreamsegments of an intersection is longer than the vehicle queueon an upstream segment or when the two queue lengths are

close (as shown in Figure 3) according to the backpressurealgorithm phase 1199011015840 will be activated instead of phase p in thenext time slot In this situation the two pairs of queue lengths(qa1 qb1) and (qa2 qb2) both have a small difference and thetwo pairs of (qc1 qd1) and (qc2 qd2) have a larger differenceThis results in the pressure of phase p being smaller thanphase1199011015840 and phase1199011015840 with a smaller volume is to be activatedin the next time slot It is obvious that in this circumstance thebackpressure-based phase switching strategy described abovecannot achieve an ideal effect

Considering this situation the cooperative traffic controlstrategy may be a good choice for some subareas of the trafficnetwork Based on this view in this paper the phase pressureis calculated considering the phase state of downstreamintersections to achieve a cooperative backpressure-basedphase switching algorithm Shown in Figure 4(a) for thestraight phase 119901119894119895 of intersection i there is 119891119886119887 isin 119901119894119895 Forthe downstream intersection i+1 there are 1198911198871198881015840 isin 119901119894+1119896 and11989111988711988810158401015840 isin 119901119894+1119896+1 119901119894119895 denotes the jth light phase of intersection i1198891198871198881015840 and 11988911988711988810158401015840 denote the number of vehicles departing fromb to 1198881015840 or 11988810158401015840 respectively If 119901119895119896 or 119901119895

119896+1is activated then the

queued vehicles on b will decrease and the pressure of 119901119894119895will increaseThen we can conclude that the pressure of119901119894119895 ofintersection i is affected by the phase state of the downstreamintersection i+1 A similar situation of the left-turn lightphase of intersection i is shown in Figure 4(b) Accordingto the above-stated analysis there are two cases for thephase switching coordination of adjacent intersections Case1 occurs when intersection i is determining the activatingphase of the next time slot the corresponding downstreamphase 119901119894+1119896 or 119901119894+1119896+1 is still activating Case 2 occurs when thetwo intersections i and i+1 are determining the activatingphase at the same time For these two cases the phasepressure computation method of 119901119894119895 should be modified as(6)-(9)

119908119886119887 (119905) = 119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905)) (6)

Journal of Advanced Transportation 5

a b

iqabpij

dbc

c

c

pi+1kdbc

i + 1qbc

qbcpi+1k+1

(a)

a

qbc

dbc

c

c

pi+1k+1 pi+1

k

dbc

qabi

i+1

b

qbc

pij

(b)

Figure 4 Pressure of straight phase and left-turn phase of intersection i affected by downstream intersection i+1 (a) Straight phase ofintersection i affected by downstream light phase (b) left-turn phase of intersection i affected by downstream light phase

119901119903119890119904119904119906119903119890119901= sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))) (7)

where dbc(t) is the number of vehicles departing from b to cdefined as (8) (9)

119889119887119888 (119905)=

119904 (119901119896 (119905)) 119902119887119888 (119905) for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) lt 119891max

119904 (119901119896 (119905)) 119891max for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) gt= 119891max

(8)

where

119904 (119901119896 (119905)) = 1 if 119901119896 (119905) is activated0 if 119901119896 (119905) is not activated (9)

where 119891max is the maximum number of vehicles passingthrough the intersection when the downstream light phasepk is activated

Using the proposed method of phase pressure computa-tion by considering the phase state of downstream intersec-tions the light phase is to be switched in cooperation modeThe light phase switching of adjacent intersections shouldbe able to make a cooperative decision without conflictsHowever it is difficult to obtain the global optimal trafficcontrol strategy amongmultiple intersections with a complextraffic state CBBA which is usually used for task assignmentproblems [22] is introduced to solve this problem in the nextsection

4 Cooperative Light Phase Switching DecisionUsing CBBA

According to the idea of the backpressure traffic light controlalgorithm the phase withmaximum pressure will be selectedto activate in the next time slot However the phase pressure

computed using the proposedmethod is affected by the phasestate of the downstream intersection The selected phase 119901119894119895of intersection i may be in conflict with the phase 119901119894+1119896 ofdownstream intersection i+1 This is because the pressureof 119901119894119895 may be computed by assuming that phase 119901119894+1119896+1 isactivated To solve the conflicts of phase switching amongintersections the CBBA is introduced in this section

CBBA is a market-based distributed task assignmentalgorithm and has been shown to produce a conflict freesolution [24 25] There are two stages in CBBA [22] ie theauction stage and the consensus stage In the auction stageeach agent bids on a task and calculates a score based on thelocal information to decide whether it assigns a new task tothe task bundle or not The task bundle of agent i is definedas the possible task set with task score If agent i does notwin any tasks in the auction stage the task bundle of agenti is empty In the consensus stage agents exchange the taskinformation through communication If an agent is outbidfor a task then it is released from the task bundle For atask the agent with the highest score is the winner Based onthe idea of CBBA in the next section the cooperative phaseswitching problem in urban traffic network is modeled as atask assignment problem

41 Cooperative Light Phase SwitchingModeled as TaskAssign-ment There are four light phases for the light controllerof each intersection which are managed by a smart agentSTLCA The task of STLCA is to determine which phaseis to be activated in the next time slot For each intersec-tion there are two vehicle flows with opposite directionscontrolled by each light phase For each vehicle flow thereis one downstream intersection at most If there is onedownstream intersection for one vehicle flow then thereare three cooperative phase switching possibilities Shown inFigure 5 vehicle flows f 1 and f 2 are controlled by phase 119901119894119895When 119901119894119895 is activated the vehicles waiting on segments a1and a2 obtain the right of way to pass through intersectioni The pressure of 119901119894119895 is consists of two parts the pressure

6 Journal of Advanced Transportation

c21

c22

cp21

cp22

b2

a1 f1c11

c12

cp11

cp12

b1

a2f2

pij

Intersection i

DownstreamIntersection2

DownstreamIntersection1

Figure 5 Coordination relationships of light phases among adjacent intersections

(1)

(3)

(5)

(6)

(8) (9)

(7)

(4)

(2)

Figure 6 Nine cooperative phase switching possibilities of one light phase

from f 1 and the pressure from f 2 The pressure generatedfrom vehicle flow f 1 is associated with the cooperative phasescp11 and cp12 If cp11 or cp12 are activated vehicles onsegment b1 can drive into segments c11 or c12 In this casethe number of vehicles departing from segment b1 shouldbe considered If cp11 and cp12 are not to be activated orthe DownstreamIntersection1 does not exist the number ofvehicles departing from segment b1 is zero Similarly thecomputation of pressure generated from vehicle flow f 2should consider the downstream phase states cp21 or cp22Therefore when considering the downstream phase statethere are 9 possible cooperative phase switching strategies forphase 119901119894119895 as shown in Figure 6 For each intersection thereare at most 36 possible cooperative phase switching

In order to solve the cooperative phase switching problemusing CBBA these possible cooperative phase switchingstrategies are viewed as the task bundle of STLCA i Eachpossible phase switching strategy is viewed as a possibletask of intersection i The task score is represented by thephase pressure calculated using (6)-(9) according to thedownstream phase state

42 Cooperative Backpressure-Based Light Phase SwitchingAlgorithm In this section the CBBA is utilized to solve thecooperative phase switching problem among intersectionsAccording to the CCBA the cooperative backpressure-basedlight phase switching algorithm consists of three steps asfollows

(a) Task Bundle Construction STLCA i constructs thetask bundle according to its topology in the traffic network

The possible switching strategies of intersection i are consid-ered as the possible tasks of STLCA i The amount of possibleswitching strategies for a particular intersection is fixed sincethe topology of an intersection in urban traffic network isfixed The task bundle of each intersection is constructedbefore phase switching decision making can improve thealgorithm performance Shown in Table 1 there are 9 possiblephase switching strategies for one phase of intersection iTherefore there are 36 possible switching strategies in oneintersection at most

(b) Phase Pressure Computation STLCA i collects thequeue length information of each segment and broadcaststhe information to the adjacent intersectionsThe pressure ofeach strategy is calculated using (6)-(9) according to the localqueue length information andqueue length information fromadjacent intersections The possible strategies of intersectioni are ordered by the phase pressure and all the possiblestrategies are set to be available at beginning After thepressure computation for each intersection the strategy withmaximum pressure is selected to be the candidate strategyand the responding phase to be the candidate light phase foractivating in the next time slot

(c) Activating Phase Decision and Conflicts ResolutionIn this step STLCA i determines the candidate phase asthe activating phase of intersection i at time t+1 directlyif the candidate strategy need not coordinate with theneighboring downstream intersections These intersectionswhose light phase has been determined are called activating-phase-determined intersections If STLCA i has determinedthe activating phase STLCA i broadcasts this information

Journal of Advanced Transportation 7

If SelectedStrategyDownstreamIntersection1 = 0 And SelectedStrategyDownstreamIntersection2 = 0ThenIntersectioniDeterminedFlag = TrueIntersectioniSelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfIf IntersectioniDeterminedFlag = False Then

If STLCAi receives the decisions from adjacent intersections ThenFor each AdjacentIntersection of IntersectioniIf AdjacentIntersectionjDetermineFlag = TrueThenRelease the strategies of Intersectioni conflicting with AdjacentIntersectionjSelectedPhase

End IfNextReselect the IntersectioniSelectedStrategy with maximum pressure from the available strategiesBroadcast IntersectioniSelectedStrategy to the adjacent intersections

End IfElect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithmIf IntersectioniSelectedStrategy is the elected strategy with maximum pressure ThenIntersectioni DeterminedFlag = TrueIntersectioni SelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfEnd If

Algorithm 1 The algorithm of intersection i for activating phase decision and conflicts resolution

Table 1 The possible cooperative phase switching strategies of 119901119894119895Strategy Number Light phase DownstreamIntersection1 CooperativePhase1 DownstreamIntersection2 CooperativePhase21 119901119894119895 0 times 0 times2 119901119894119895 1 cp11 0 times3 119901119894119895 1 cp12 0 times4 119901119894119895 0 times 1 cp215 119901119894119895 0 times 1 cp226 119901119894119895 1 cp11 1 cp217 119901119894119895 1 cp11 1 cp228 119901119894119895 1 cp12 1 cp219 119901119894119895 1 cp12 1 cp22Note times indicates that the cooperative phase is not to be activated 0 indicates that the downstream intersection does not exist or does not coordinate with 119901119894119895cp11 cp12 cp21 and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2

to its adjacent intersections Then the adjacent STLCAs ofintersection i release the strategies conflict with intersectioni For example if intersection i has determined the activatinglight phase to be west-east straight phase the strategies ofneighboring intersections (if the neighboring intersectionsare not activating-phase-determined intersections) which areconflict with the activating light phase of intersection i will bereleased that is the invalid flag of these strategies are set to betrue For these activating-phase-undermined intersectionsSTLCAs elect a strategy with maximum pressure in dis-tributed mode by communication The elected intersectiondetermines the activating phase and broadcasts it to adjacentintersections Step (a) and (b) are repeated until all STLCAs

have determined the activating phase A detailed descriptionof the algorithm is given in Algorithm 1

Based on the three steps the cooperative backpressure-based traffic control method (simplified as CBP method) canobtain a conflict free phase switching strategy with max-imum phase pressure The backpressure-based traffic lightcontrol method (simplified as BP method) only considersthe queue length of the current intersection and neglectsthe decrease of queued vehicles on the downstream segmentwhen the corresponding downstream phase is activatedThe CBP method fixes this deficiency and considers thephase state of downstream intersections to achieve coordi-nation among intersections Furthermore all the switching

8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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Journal of Advanced Transportation 5

a b

iqabpij

dbc

c

c

pi+1kdbc

i + 1qbc

qbcpi+1k+1

(a)

a

qbc

dbc

c

c

pi+1k+1 pi+1

k

dbc

qabi

i+1

b

qbc

pij

(b)

Figure 4 Pressure of straight phase and left-turn phase of intersection i affected by downstream intersection i+1 (a) Straight phase ofintersection i affected by downstream light phase (b) left-turn phase of intersection i affected by downstream light phase

119901119903119890119904119904119906119903119890119901= sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888isin1198881015840 11988810158401015840

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))) (7)

where dbc(t) is the number of vehicles departing from b to cdefined as (8) (9)

119889119887119888 (119905)=

119904 (119901119896 (119905)) 119902119887119888 (119905) for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) lt 119891max

119904 (119901119896 (119905)) 119891max for 119891119887119888 isin 119901119896 (119905) if 119902119887119888 (119905) gt= 119891max

(8)

where

119904 (119901119896 (119905)) = 1 if 119901119896 (119905) is activated0 if 119901119896 (119905) is not activated (9)

where 119891max is the maximum number of vehicles passingthrough the intersection when the downstream light phasepk is activated

Using the proposed method of phase pressure computa-tion by considering the phase state of downstream intersec-tions the light phase is to be switched in cooperation modeThe light phase switching of adjacent intersections shouldbe able to make a cooperative decision without conflictsHowever it is difficult to obtain the global optimal trafficcontrol strategy amongmultiple intersections with a complextraffic state CBBA which is usually used for task assignmentproblems [22] is introduced to solve this problem in the nextsection

4 Cooperative Light Phase Switching DecisionUsing CBBA

According to the idea of the backpressure traffic light controlalgorithm the phase withmaximum pressure will be selectedto activate in the next time slot However the phase pressure

computed using the proposedmethod is affected by the phasestate of the downstream intersection The selected phase 119901119894119895of intersection i may be in conflict with the phase 119901119894+1119896 ofdownstream intersection i+1 This is because the pressureof 119901119894119895 may be computed by assuming that phase 119901119894+1119896+1 isactivated To solve the conflicts of phase switching amongintersections the CBBA is introduced in this section

CBBA is a market-based distributed task assignmentalgorithm and has been shown to produce a conflict freesolution [24 25] There are two stages in CBBA [22] ie theauction stage and the consensus stage In the auction stageeach agent bids on a task and calculates a score based on thelocal information to decide whether it assigns a new task tothe task bundle or not The task bundle of agent i is definedas the possible task set with task score If agent i does notwin any tasks in the auction stage the task bundle of agenti is empty In the consensus stage agents exchange the taskinformation through communication If an agent is outbidfor a task then it is released from the task bundle For atask the agent with the highest score is the winner Based onthe idea of CBBA in the next section the cooperative phaseswitching problem in urban traffic network is modeled as atask assignment problem

41 Cooperative Light Phase SwitchingModeled as TaskAssign-ment There are four light phases for the light controllerof each intersection which are managed by a smart agentSTLCA The task of STLCA is to determine which phaseis to be activated in the next time slot For each intersec-tion there are two vehicle flows with opposite directionscontrolled by each light phase For each vehicle flow thereis one downstream intersection at most If there is onedownstream intersection for one vehicle flow then thereare three cooperative phase switching possibilities Shown inFigure 5 vehicle flows f 1 and f 2 are controlled by phase 119901119894119895When 119901119894119895 is activated the vehicles waiting on segments a1and a2 obtain the right of way to pass through intersectioni The pressure of 119901119894119895 is consists of two parts the pressure

6 Journal of Advanced Transportation

c21

c22

cp21

cp22

b2

a1 f1c11

c12

cp11

cp12

b1

a2f2

pij

Intersection i

DownstreamIntersection2

DownstreamIntersection1

Figure 5 Coordination relationships of light phases among adjacent intersections

(1)

(3)

(5)

(6)

(8) (9)

(7)

(4)

(2)

Figure 6 Nine cooperative phase switching possibilities of one light phase

from f 1 and the pressure from f 2 The pressure generatedfrom vehicle flow f 1 is associated with the cooperative phasescp11 and cp12 If cp11 or cp12 are activated vehicles onsegment b1 can drive into segments c11 or c12 In this casethe number of vehicles departing from segment b1 shouldbe considered If cp11 and cp12 are not to be activated orthe DownstreamIntersection1 does not exist the number ofvehicles departing from segment b1 is zero Similarly thecomputation of pressure generated from vehicle flow f 2should consider the downstream phase states cp21 or cp22Therefore when considering the downstream phase statethere are 9 possible cooperative phase switching strategies forphase 119901119894119895 as shown in Figure 6 For each intersection thereare at most 36 possible cooperative phase switching

In order to solve the cooperative phase switching problemusing CBBA these possible cooperative phase switchingstrategies are viewed as the task bundle of STLCA i Eachpossible phase switching strategy is viewed as a possibletask of intersection i The task score is represented by thephase pressure calculated using (6)-(9) according to thedownstream phase state

42 Cooperative Backpressure-Based Light Phase SwitchingAlgorithm In this section the CBBA is utilized to solve thecooperative phase switching problem among intersectionsAccording to the CCBA the cooperative backpressure-basedlight phase switching algorithm consists of three steps asfollows

(a) Task Bundle Construction STLCA i constructs thetask bundle according to its topology in the traffic network

The possible switching strategies of intersection i are consid-ered as the possible tasks of STLCA i The amount of possibleswitching strategies for a particular intersection is fixed sincethe topology of an intersection in urban traffic network isfixed The task bundle of each intersection is constructedbefore phase switching decision making can improve thealgorithm performance Shown in Table 1 there are 9 possiblephase switching strategies for one phase of intersection iTherefore there are 36 possible switching strategies in oneintersection at most

(b) Phase Pressure Computation STLCA i collects thequeue length information of each segment and broadcaststhe information to the adjacent intersectionsThe pressure ofeach strategy is calculated using (6)-(9) according to the localqueue length information andqueue length information fromadjacent intersections The possible strategies of intersectioni are ordered by the phase pressure and all the possiblestrategies are set to be available at beginning After thepressure computation for each intersection the strategy withmaximum pressure is selected to be the candidate strategyand the responding phase to be the candidate light phase foractivating in the next time slot

(c) Activating Phase Decision and Conflicts ResolutionIn this step STLCA i determines the candidate phase asthe activating phase of intersection i at time t+1 directlyif the candidate strategy need not coordinate with theneighboring downstream intersections These intersectionswhose light phase has been determined are called activating-phase-determined intersections If STLCA i has determinedthe activating phase STLCA i broadcasts this information

Journal of Advanced Transportation 7

If SelectedStrategyDownstreamIntersection1 = 0 And SelectedStrategyDownstreamIntersection2 = 0ThenIntersectioniDeterminedFlag = TrueIntersectioniSelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfIf IntersectioniDeterminedFlag = False Then

If STLCAi receives the decisions from adjacent intersections ThenFor each AdjacentIntersection of IntersectioniIf AdjacentIntersectionjDetermineFlag = TrueThenRelease the strategies of Intersectioni conflicting with AdjacentIntersectionjSelectedPhase

End IfNextReselect the IntersectioniSelectedStrategy with maximum pressure from the available strategiesBroadcast IntersectioniSelectedStrategy to the adjacent intersections

End IfElect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithmIf IntersectioniSelectedStrategy is the elected strategy with maximum pressure ThenIntersectioni DeterminedFlag = TrueIntersectioni SelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfEnd If

Algorithm 1 The algorithm of intersection i for activating phase decision and conflicts resolution

Table 1 The possible cooperative phase switching strategies of 119901119894119895Strategy Number Light phase DownstreamIntersection1 CooperativePhase1 DownstreamIntersection2 CooperativePhase21 119901119894119895 0 times 0 times2 119901119894119895 1 cp11 0 times3 119901119894119895 1 cp12 0 times4 119901119894119895 0 times 1 cp215 119901119894119895 0 times 1 cp226 119901119894119895 1 cp11 1 cp217 119901119894119895 1 cp11 1 cp228 119901119894119895 1 cp12 1 cp219 119901119894119895 1 cp12 1 cp22Note times indicates that the cooperative phase is not to be activated 0 indicates that the downstream intersection does not exist or does not coordinate with 119901119894119895cp11 cp12 cp21 and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2

to its adjacent intersections Then the adjacent STLCAs ofintersection i release the strategies conflict with intersectioni For example if intersection i has determined the activatinglight phase to be west-east straight phase the strategies ofneighboring intersections (if the neighboring intersectionsare not activating-phase-determined intersections) which areconflict with the activating light phase of intersection i will bereleased that is the invalid flag of these strategies are set to betrue For these activating-phase-undermined intersectionsSTLCAs elect a strategy with maximum pressure in dis-tributed mode by communication The elected intersectiondetermines the activating phase and broadcasts it to adjacentintersections Step (a) and (b) are repeated until all STLCAs

have determined the activating phase A detailed descriptionof the algorithm is given in Algorithm 1

Based on the three steps the cooperative backpressure-based traffic control method (simplified as CBP method) canobtain a conflict free phase switching strategy with max-imum phase pressure The backpressure-based traffic lightcontrol method (simplified as BP method) only considersthe queue length of the current intersection and neglectsthe decrease of queued vehicles on the downstream segmentwhen the corresponding downstream phase is activatedThe CBP method fixes this deficiency and considers thephase state of downstream intersections to achieve coordi-nation among intersections Furthermore all the switching

8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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6 Journal of Advanced Transportation

c21

c22

cp21

cp22

b2

a1 f1c11

c12

cp11

cp12

b1

a2f2

pij

Intersection i

DownstreamIntersection2

DownstreamIntersection1

Figure 5 Coordination relationships of light phases among adjacent intersections

(1)

(3)

(5)

(6)

(8) (9)

(7)

(4)

(2)

Figure 6 Nine cooperative phase switching possibilities of one light phase

from f 1 and the pressure from f 2 The pressure generatedfrom vehicle flow f 1 is associated with the cooperative phasescp11 and cp12 If cp11 or cp12 are activated vehicles onsegment b1 can drive into segments c11 or c12 In this casethe number of vehicles departing from segment b1 shouldbe considered If cp11 and cp12 are not to be activated orthe DownstreamIntersection1 does not exist the number ofvehicles departing from segment b1 is zero Similarly thecomputation of pressure generated from vehicle flow f 2should consider the downstream phase states cp21 or cp22Therefore when considering the downstream phase statethere are 9 possible cooperative phase switching strategies forphase 119901119894119895 as shown in Figure 6 For each intersection thereare at most 36 possible cooperative phase switching

In order to solve the cooperative phase switching problemusing CBBA these possible cooperative phase switchingstrategies are viewed as the task bundle of STLCA i Eachpossible phase switching strategy is viewed as a possibletask of intersection i The task score is represented by thephase pressure calculated using (6)-(9) according to thedownstream phase state

42 Cooperative Backpressure-Based Light Phase SwitchingAlgorithm In this section the CBBA is utilized to solve thecooperative phase switching problem among intersectionsAccording to the CCBA the cooperative backpressure-basedlight phase switching algorithm consists of three steps asfollows

(a) Task Bundle Construction STLCA i constructs thetask bundle according to its topology in the traffic network

The possible switching strategies of intersection i are consid-ered as the possible tasks of STLCA i The amount of possibleswitching strategies for a particular intersection is fixed sincethe topology of an intersection in urban traffic network isfixed The task bundle of each intersection is constructedbefore phase switching decision making can improve thealgorithm performance Shown in Table 1 there are 9 possiblephase switching strategies for one phase of intersection iTherefore there are 36 possible switching strategies in oneintersection at most

(b) Phase Pressure Computation STLCA i collects thequeue length information of each segment and broadcaststhe information to the adjacent intersectionsThe pressure ofeach strategy is calculated using (6)-(9) according to the localqueue length information andqueue length information fromadjacent intersections The possible strategies of intersectioni are ordered by the phase pressure and all the possiblestrategies are set to be available at beginning After thepressure computation for each intersection the strategy withmaximum pressure is selected to be the candidate strategyand the responding phase to be the candidate light phase foractivating in the next time slot

(c) Activating Phase Decision and Conflicts ResolutionIn this step STLCA i determines the candidate phase asthe activating phase of intersection i at time t+1 directlyif the candidate strategy need not coordinate with theneighboring downstream intersections These intersectionswhose light phase has been determined are called activating-phase-determined intersections If STLCA i has determinedthe activating phase STLCA i broadcasts this information

Journal of Advanced Transportation 7

If SelectedStrategyDownstreamIntersection1 = 0 And SelectedStrategyDownstreamIntersection2 = 0ThenIntersectioniDeterminedFlag = TrueIntersectioniSelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfIf IntersectioniDeterminedFlag = False Then

If STLCAi receives the decisions from adjacent intersections ThenFor each AdjacentIntersection of IntersectioniIf AdjacentIntersectionjDetermineFlag = TrueThenRelease the strategies of Intersectioni conflicting with AdjacentIntersectionjSelectedPhase

End IfNextReselect the IntersectioniSelectedStrategy with maximum pressure from the available strategiesBroadcast IntersectioniSelectedStrategy to the adjacent intersections

End IfElect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithmIf IntersectioniSelectedStrategy is the elected strategy with maximum pressure ThenIntersectioni DeterminedFlag = TrueIntersectioni SelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfEnd If

Algorithm 1 The algorithm of intersection i for activating phase decision and conflicts resolution

Table 1 The possible cooperative phase switching strategies of 119901119894119895Strategy Number Light phase DownstreamIntersection1 CooperativePhase1 DownstreamIntersection2 CooperativePhase21 119901119894119895 0 times 0 times2 119901119894119895 1 cp11 0 times3 119901119894119895 1 cp12 0 times4 119901119894119895 0 times 1 cp215 119901119894119895 0 times 1 cp226 119901119894119895 1 cp11 1 cp217 119901119894119895 1 cp11 1 cp228 119901119894119895 1 cp12 1 cp219 119901119894119895 1 cp12 1 cp22Note times indicates that the cooperative phase is not to be activated 0 indicates that the downstream intersection does not exist or does not coordinate with 119901119894119895cp11 cp12 cp21 and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2

to its adjacent intersections Then the adjacent STLCAs ofintersection i release the strategies conflict with intersectioni For example if intersection i has determined the activatinglight phase to be west-east straight phase the strategies ofneighboring intersections (if the neighboring intersectionsare not activating-phase-determined intersections) which areconflict with the activating light phase of intersection i will bereleased that is the invalid flag of these strategies are set to betrue For these activating-phase-undermined intersectionsSTLCAs elect a strategy with maximum pressure in dis-tributed mode by communication The elected intersectiondetermines the activating phase and broadcasts it to adjacentintersections Step (a) and (b) are repeated until all STLCAs

have determined the activating phase A detailed descriptionof the algorithm is given in Algorithm 1

Based on the three steps the cooperative backpressure-based traffic control method (simplified as CBP method) canobtain a conflict free phase switching strategy with max-imum phase pressure The backpressure-based traffic lightcontrol method (simplified as BP method) only considersthe queue length of the current intersection and neglectsthe decrease of queued vehicles on the downstream segmentwhen the corresponding downstream phase is activatedThe CBP method fixes this deficiency and considers thephase state of downstream intersections to achieve coordi-nation among intersections Furthermore all the switching

8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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Journal of Advanced Transportation 7

If SelectedStrategyDownstreamIntersection1 = 0 And SelectedStrategyDownstreamIntersection2 = 0ThenIntersectioniDeterminedFlag = TrueIntersectioniSelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfIf IntersectioniDeterminedFlag = False Then

If STLCAi receives the decisions from adjacent intersections ThenFor each AdjacentIntersection of IntersectioniIf AdjacentIntersectionjDetermineFlag = TrueThenRelease the strategies of Intersectioni conflicting with AdjacentIntersectionjSelectedPhase

End IfNextReselect the IntersectioniSelectedStrategy with maximum pressure from the available strategiesBroadcast IntersectioniSelectedStrategy to the adjacent intersections

End IfElect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithmIf IntersectioniSelectedStrategy is the elected strategy with maximum pressure ThenIntersectioni DeterminedFlag = TrueIntersectioni SelectedPhase = SelectedStrategyphaseLet the selected phase be the activating phase in the next time slotBroadcast the decision of Intersectioni to the adjacent intersections

End IfEnd If

Algorithm 1 The algorithm of intersection i for activating phase decision and conflicts resolution

Table 1 The possible cooperative phase switching strategies of 119901119894119895Strategy Number Light phase DownstreamIntersection1 CooperativePhase1 DownstreamIntersection2 CooperativePhase21 119901119894119895 0 times 0 times2 119901119894119895 1 cp11 0 times3 119901119894119895 1 cp12 0 times4 119901119894119895 0 times 1 cp215 119901119894119895 0 times 1 cp226 119901119894119895 1 cp11 1 cp217 119901119894119895 1 cp11 1 cp228 119901119894119895 1 cp12 1 cp219 119901119894119895 1 cp12 1 cp22Note times indicates that the cooperative phase is not to be activated 0 indicates that the downstream intersection does not exist or does not coordinate with 119901119894119895cp11 cp12 cp21 and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2

to its adjacent intersections Then the adjacent STLCAs ofintersection i release the strategies conflict with intersectioni For example if intersection i has determined the activatinglight phase to be west-east straight phase the strategies ofneighboring intersections (if the neighboring intersectionsare not activating-phase-determined intersections) which areconflict with the activating light phase of intersection i will bereleased that is the invalid flag of these strategies are set to betrue For these activating-phase-undermined intersectionsSTLCAs elect a strategy with maximum pressure in dis-tributed mode by communication The elected intersectiondetermines the activating phase and broadcasts it to adjacentintersections Step (a) and (b) are repeated until all STLCAs

have determined the activating phase A detailed descriptionof the algorithm is given in Algorithm 1

Based on the three steps the cooperative backpressure-based traffic control method (simplified as CBP method) canobtain a conflict free phase switching strategy with max-imum phase pressure The backpressure-based traffic lightcontrol method (simplified as BP method) only considersthe queue length of the current intersection and neglectsthe decrease of queued vehicles on the downstream segmentwhen the corresponding downstream phase is activatedThe CBP method fixes this deficiency and considers thephase state of downstream intersections to achieve coordi-nation among intersections Furthermore all the switching

8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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8 Journal of Advanced Transportation

possibilities based on BP method (that is the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 =0) are contained in the task bundle of each intersection inCBP method In other words the CBP method consideringdownstream phase state can obtain equal or greater trafficperformance compared to the BP method To illustrate thefeasibility of the proposed CBP method the stability isdiscussed in the next section

5 Stability Analysis

In this section the stability of the proposed CBP methodis analyzed As described in references [26] and [14] for anetwork with queue vector U = 1198801 1198802 119880119873 a sufficientcondition for stability can be provided using Lyapunov driftwhich is given as below

Lemma 1 Suppose E119880119894(119905) lt infin for all 119894 isin 1 2 119873 andthere exist constants B gt0 and 120576gt0 which satisfies

E 119871 (U (119905 + 1) minus 119871 (U (119905)) | U (119905) le 119861 minus 120576 119873sum119894=1

119880119894 (119905) (10)

then the network is stable where the Lyapunov function isdefined as

119871 (119880) = 119873sum119894=1

1198802119894 (11)

To describe simplicity define the function 119881119894119899119886119887(119901(119905)) and119881119900119906119905119886119887 (119901(119905)) anddenote the vehicles entering qab(t) and vehiclesdeparting from qab(t) under the current light phase switchingstrategy P(t) during slot t

119881119900119906119905119886119887 (119875 (119905)) = sum119891119886119887isin119901119894(119905)

120583119886119887 (119901119894 (119905)) (12)

where pi(t) is the activated light phase of intersection i andvehicle flow f ab is controlled by pi(t) b is the downstreamnode of a

119881119894119899119886119887 (119875 (119905)) = sum119891119888119886isin119901119894minus1(119905)

120583119888119886 (119901119894minus1 (119905)) (13)

where pi-1(t) is the activated light phase of intersection i-1 andvehicle flow f ca is controlled by pi-1(t) c is the upstream nodeof a

Then (1) is rewritten as

119902119886119887 (119905 + 1) = 119902119886119887 (119905) + 119881119894119899119886119887 (119875 (119905)) minus 119881119900119906119905119886119887 (119875 (119905)) (14)

According to the Lyapunov function and (14) 119881119894119899119886119887(119875(119905)) and119881119900119906119905119886119887 (119875(119905)) are simplified as 119881119894119899119886119887 and 119881119900119906119905119886119887 we obtain119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) minus 2sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2 (sum

119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887)

(15)

For queue network in this paper it assumes that the arrivalvehicles of ingress node are all come from the upstreamnodes For example shown in Figure 7 there are two vehiclequeues 1199021198871198881 1199021198871198882 on node b the arrival vehicle 1198811198941198991198871198881 iscomputed using 1198811198941198991198871198881 = 1199031198871198881119881119900119906119905119886119887 where rbc1 is the proportionof vehicles entering qbc1 from qab when the correspondinglight phase is activated Similarly there is 1198811198941198991198871198882 = 1199031198871198882119881119900119906119905119886119887 Furthermore for the node b under a given light phasestrategy P(t) one of 1198811199001199061199051198871198881 and 1198811199001199061199051198871198882 is zero at least

Based on these properties of the traffic network the rightterm of (15) is expanded as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887= (119902119886119887 (119905) 119881119900119906119905119886119887 minus 119902119886119887 (119905) 119881119894119899119886119887)

+ (1199021198871198881 (119905) 1198811199001199061199051198871198881 minus 1199021198871198881 (119905) 1198811198941198991198871198881)+ (1199021198871198882 (119905) 1198811199001199061199051198871198882 minus 11990211988711988821 (119905) 1198811198941198991198871198882) + + 119881119894119899119886119887119881119900119906119905119886119887+ 11988111989411989911988711988811198811199001199061199051198871198881 + 11988111989411989911988711988821198811199001199061199051198871198882 +

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887minus 119902119886119887 (119905) 119881119894119899119886119887 + + (119881119894119899119886119887119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 + )

= 119902119886119887 (119905) 119881119900119906119905119886119887 minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887+ 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881 + 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882+ (minus119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887 ) +

(16)

For the term underline-marked in the above equationminus119902119886119887(119905)119881119894119899119886119887+119881119894119899119886119887119881119900119906119905119886119887 a is an ingress node It assumes that nodex is a virtual node with qx=0119881119900119906119905119909 is the vehicle input on nodea and rab is the proportion of 119881119900119906119905119909 for vehicle entering qabduring time slot tThen this term can be rewritten as follows

minus 119902119886119887 (119905) 119881119894119899119886119887 + 119881119894119899119886119887119881119900119906119905119886119887= 119902119909 (119905) 119881119900119906119905119909 minus 119902119886119887 (119905) 119903119886119887119881119900119906119905119909 + 119903119886119887119881119900119906119905119909 119881119900119906119905119886119887 (17)

Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

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Journal of Advanced Transportation 9

a

qabVoutab

Vinbc1

Vinbc2 Vout

bc2

Voutbc1

b

qbc1

qbc2

e1 e2

c1qc1e1 qc1e2

Vinc1e1 Vin

c1e2

d1

d2

c2

qc2d1

qc2d2

Vinc2d1

Vinc2d2

Figure 7 The relationship of arrival and departing vehicle flow in traffic network

Similarly the queues in the exit nodes also can be expressedas the similar forms Therefore the right term of (15) can berewritten as

sum119886119887

(119902119886119887 (119905) [119881119900119906119905119886119887 minus 119881119894119899119886119887]) + sum119886119887

119881119900119906119905119886119887 119881119894119899119886119887 = sum119886119887

119902119886119887 (119905) 119881119900119906119905119886119887minus 1199021198871198881 (119905) 1199031198871198881119881119900119906119905119886119887 minus 1199021198871198882 (119905) 1199031198871198882119881119900119906119905119886119887 + 1199031198871198881119881119900119906119905119886119887 1198811199001199061199051198871198881+ 1199031198871198882119881119900119906119905119886119887 1198811199001199061199051198871198882 = sum

119886119887

(119902119886119887 (119905) minus 1199021198871198881 (119905) 1199031198871198881 minus 1199021198871198882 (119905)sdot 1199031198871198882 + 11990311988711988811198811199001199061199051198871198881 + 11990311988711988821198811199001199061199051198871198882 ) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus (1199021198871198881 (119905) 1199031198871198881 + 1199021198871198882 (119905) 1199031198871198882 minus 11990311988711988811198811199001199061199051198871198881minus 11990311988711988821198811199001199061199051198871198882 )) 119881119900119906119905119886119887 = sum

119886119887

(119902119886119887 (119905)minus sum119888

(119902119887119888 (119905) 119903119887119888 minus 119903119887119888119881119900119906119905119887119888 )) 119881119900119906119905119886119887 = sum119886119887

(119902119886119887 (119905)minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887

(18)

Then L(U(t+1))-L(U(t)) can be expressed as

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))= sum119886119887

((119881119894119899119886119887)2 + (119881119900119906119905119886119887 )2)minus 2sum119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887(19)

Let 119861 = sum119886119887((sup(119881119894119899119886119887))2 + (sup(119881119900119906119905119886119887 ))2) then there is

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

(119902119886119887 (119905) minus sum119888

119903119887119888 (119902119887119888 (119905) minus 119881119900119906119905119887119888 )) 119881119900119906119905119886119887 (20)

In the above equation119881119900119906119905119887119888 is the departing vehicles from qbcand if the corresponding downstream light phase of node a

is not activated 119881119900119906119905119887119888 will be zero Here 119881119900119906119905119887119888 is same to dbc in(6) Inject (6) (12) into (17) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905) (21)

Since the queuing networkmodeled in this paper is similar tothe model in reference [14 16] there are similar properties

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)](22)

From (22) the following equation can be deduced

sum119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905))]= sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) [119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905))sdot [119902119886119887 (119905) minus sum

119888

119903119887119888 (119902119887119888 (119905) minus 119889119887119888 (119905))]le sum119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)

(23)

It means that under the same given light phase switchingstrategy P(t) if the downstream phase is coordinated withthe phase of current intersection it achieves equal or betterthroughput If dbc(t) is zero the CBP method is degeneratedto the BP method When the downstream light phase iscoordinated with the current light phase it achieves betterthroughput

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Journal of Advanced Transportation

(a)

(b) (c) (d)

Figure 8 Topology and settings of simulation traffic network (a) The topology of simulation traffic network (b) route decisions setting (c)travel time sections setting (d) queue Counters setting

According to (21) and (23) we obtain

119871 (119880 (119905 + 1)) minus 119871 (119880 (119905))le 119861 minus 2sum

119886119887

sum119891119886119887isin119901(119905)

120583119886119887 (119901 (119905)) 119908119886119887 (119905)le 119861 minus 2sum

119886119887

119902119886119887 (119905) [119881119900119906119905119886119887 (119901) minus 119881119894119899119886119887 (119901)](24)

In additional in [14] it has been proved that for theadmissible arrival rate 120582 in capacity region Λ and 120576 gt0 thereis

E 119881119900119906119905119886119887 (119875 (119905)) minus 119881119894119899119886119887 (119875 (119905)) | 119880 (119905) = 120582119886119887 + 120576 (25)

Then we obtain

E 119871 (U (119905 + 1)) minus 119871 (U (119905)) | U (119905) le 119861 minus 2120576sum119886119887

119902119886119887 (119905) (26)

According to Lemma 1 the network controlled by the pro-posed CBP method is stable under the admissible arrivalrate 120582 in capacity region Λ To illustrate the effectiveness ofthe proposed CBP method simulations are conducted and adescription is given in the following section

6 Simulation and Discussion

Three traffic light control methods are implemented includ-ing fixed-time control BP control and CBP control TheBP method implemented in simulations is proposed byVaraiya [16] Although there are several improved versionsof backpressure-based traffic controlmethods thesemethodsdo not consider the coordination of light phase switchingamong neighboring intersections The CBP control methodconsidering the downstream light phase state is proposed

on the foundation of basic backpressure-based traffic controlmethod proposed by Varaiya Therefore in this section wecompare the three traffic control methods In the furtherresearch work based on these improved versions the coor-dination should be considered

The simulation traffic network consists of 15 intersectionsand 76 links constructed in Vissim The network includes 16ingress links and 16 exit links as shown in Figure 8(a) Thevehicle input of ingress links is 800vehh There are 15 signalcontrollers and 4 light phases for each controller ie north-south straight north-south left-turn west-east straight andwest-east left-turn There are 3 lanes on each link The left-turn vehicle flow drives on Lane 3 and the straight vehicleflow drives on Lane 2 The two vehicle flows are controlledby a traffic light The right-turn vehicle flow drives on Lane 1and is free of the traffic light To obtain the traffic parametersduring simulation 60 queue counters 60 travel time sectionsand 60 routing decisions are laid on the simulation trafficnetwork as shown in Figures 8(b) 8(c) and 8(d) During thesimulation the green light time of each light phase is assumedto be 21s which is the pedestrian clearance time (the roadwidth is about 21m and the pedestrians speed is about 1ms)[4] The yellow light time is assumed to be 3s

The three traffic light controlmethods are implemented inVisual Studio 2010 The simulation programs communicatewith Vissim through the Vissim COM programming inter-face to obtain the traffic parameters and decide the trafficlight signal at each time slot The simulation runs for 7200sand the traffic network performance is evaluated from 1000sto 7200s using Vissim This is because in the first 1000s ofsimulation time there are not enough vehicles entering thesimulation traffic network

The simulation results of traffic network performance aregiven in Table 2 The results show that under similar trafficvolume condition the fixed-time control algorithm resultsin a higher delay time and travel time due to the lower

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 11

Table 2 Traffic network performance comparison

Parameters Fixed-Time Method BP Method CBP MethodTotal travel time (h) 149411 1440434 1332647Total delay time (h) 824652 761403 648577Number of Stops 120850 70359 68554Total stopped delay (h) 593744 587938 482059Average delay time (s) 133379 123199 105175Average number of stops 543 3162 3088Average stopped delay (s) 96032 95131 78172Average speed (kmh) 23549 24761 26965Number of vehicles in network 938 880 806Number of vehicles left network 21320 21369 21394Total Number of vehicles 22258 22249 22200

Fixed-time MethodBP MethodCBP Method

010203040506070

Aver

age d

elay

(s)

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

Figure 9 Average vehicle delay during simulation

average speed Vehicles stopping in front of intersections haveto wait for the next cycle to get the right of way to passthrough the intersection which results in an increase in thevehicle delay time Although the BP method can switch lightsignal according to the pressure of each light phase it cannotdeal well with the situation described in Figure 3 This kindof situation results in an increase of vehicle delay becausethe vehicles on a1 and a2 cannot be released in time TheCBP method solves this problem well From Table 2 wecan conclude that the CBP control can obtain better trafficnetwork performance than the other two methods

To further illustrate the effectiveness of the CBP methodthe average delay average stop delay average queue lengthand the maximum queue length during the simulation areextracted from the Vissim evaluation files and listed inFigures 9 10 11 and 12

Figure 9 displays the average delay of the three methodsduring simulation timeThe delay time of fixed-time methodis larger than the other two methods since each light phaseis activated in same time interval in the fixed-time controland the segments with longer queue length cannot get theright of way in priorityThe BP andCBPmethods have a closeaverage vehicle delay because these twomethods have similarlight phase switching strategy by selecting the light phasewith the maximum pressure for activation The CBP method

considers the phase state of downstream intersections andachieved coordination among intersections It can effectivelyspeed up the release of queued vehicles Therefore the CBPmethod obtains a smaller average delay during simulation

For the average stop delay shown in Figure 10 theCBP method shows a smaller average stop delay duringsimulation than the other two methods By considering thephase state of downstream intersections the CBP methodachieves coordination among intersections in a distributedpattern which releases more vehicles due to the cooperativelight phase switching among adjacent intersections

During the simulation there is an average of three queuelengths increase in the first 1000s with the increase ofsimulation time as shown in Figure 11 Under the samevehicle input the average queue length of the CBP methodgenerates a smaller average queue length in the traffic net-work Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queuedvehicle releasing

From Figure 12 according to the curve of the fixed-time traffic light control method queue lengths of somesegments in the traffic network reaches the maximum valueat about 2600s The queue length of the BP method reachesthe maximum value at about 6600s The queue length ofthe CBP method does not reach the maximum value in the

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Journal of Advanced Transportation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

010203040506070

Aver

age s

top

delay

(s)

Figure 10 Average stop delay during simulation

Fixed-time MethodBP MethodCBP Method

600 120018002400300036004200480054006000660072000Time (s)

05

101520253035404550

Aver

age q

ueue

leng

th (m

)

Figure 11 Average queue length during simulation

Fixed-time MethodBP MethodCBP Method

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 72000Time (s)

0255075

100125150175200225250

Max

imum

que

ue le

ngth

(m)

Figure 12 Maximum queue length during simulation

entire simulation time This simulation result indicates thatthe fixed-time control method is more likely to lead to queuespillover in the segments whose queue length reaches themaximum value The queue spillover of some segments mayresult in traffic congestion propagation or a traffic collapse

This is a seriously negative effect to urban traffic Althoughthe time of the queue length for reaching the maximumvalue of the BP method is later than the fixed-time controlmethod it still reaches the maximum queue length after allAs shown in Figure 12 the queue lengths of all the segments

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 13

do not reach the maximum value during the CBP methodsimulation This result indicates that the CBP method canobtain better performance in the traffic network

Based on the above discussion the proposed trafficcontrol method performs better than the other two methodsin the simulations However the CBP method requirescommunication among intersections due to the distributedphase switching decision This results in an increase of thecommunication cost With the development of an intelli-gent transport system the communication problem in theproposed method can be solved completely Therefore thisproposed cooperative traffic light control method may be anuseful attempt for improving the efficiency of the urban trafficsystem

7 Conclusion

In this paper a cooperative backpressure-based traffic lightcontrol method is proposed to improve the efficiency ofurban traffic network The traffic network is modeled as aqueuing network in which the light phases of each inter-section are controlled by STLCA The light phase pressureis computed using the proposed light phase pressure com-putation method considering the phase state of downstreamintersections which is the basis for cooperative light phaseswitching decision among intersections In the cooperativelight phase switching decision the CBBA is introduced tosolve the conflicts in the switching strategies of adjacentintersections in a distributed mode Simulations show thatthe proposed method obtains better performance than theoriginal backpressure-based traffic light control method andfixed-time traffic light control method

However there are further works to be done in the futureFirstly the pressure computation should be improved toconsider other factors such as the capacity of the downstreamsegment the priority of queued vehicles and the number ofpassengers in queued vehicles These factors may influencethe light phase switching decision in different scenariosSecondly for simplicity the cooperative backpressure-basedtraffic light control method only considers situations ofsynchronous traffic light switching In reality traffic lightsdo not switch at the same time and there are also differentgreen times at different intersections Actually some group-based traffic control methods have considered this problemin these methods the current light phase can be decided toextend the green time or finish it according to the max-pressure signal control policy [27 28] In the future work onthe foundation of these methods the cooperative light phaseswitching method can be further improved for better trafficnetwork performance Thirdly the physical traffic modelshould be designed in the future research work to achieve theactual application in the urban traffic network

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial Natural Sci-ence Foundation China [Grant no ZR2018FM027] Projectof Shandong Province Higher Educational Science and Tech-nology Program [Grant no J17KB181] Key Research andDevelopment Project in Shandong Province China [Grantsnos 2015GGX101048 2016GSF120009] and National Natu-ral Science Foundation of Youth Fund Project [Grant no61501284]

References

[1] D I Robertson ldquoTRANSYT a traffic network study toolrdquo TechRep LR-253 Road Res Lab Berkshire UK 1969

[2] J D C Little M D Kelson and N H Gartner ldquoMAXBAND aprogram for setting signal on arteries and triangular networkrdquoTransportation Research Record no 795 pp 40ndash46 1981

[3] S Araghi A Khosravi and D Creighton ldquoA review on com-putational intelligence methods for controlling traffic signaltimingrdquo Expert Systems with Applications vol 42 no 3 pp1538ndash1550 2015

[4] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

[5] P B Hunt D I Robertson R D Bretherton R Winton andL Scoot ldquoA traffic responsive method of coordinating signalsrdquoTech Rep LR1014 TRRL Crowthorne UK 1981

[6] A G Sims ldquoThe Sydney coordinated adaptive traffic systemrdquoin Proceedings of the Engineering Foundation Conference onResearch Directions in Computer Control of Urban Traffic Sys-tems pp 12ndash27 Calif USA 1979

[7] H M Abdelghaffar H Yang and H A Rakha ldquoIsolated trafficsignal control using a game theoretic frameworkrdquo inProceedingsof the 19th IEEE International Conference on Intelligent Trans-portation Systems ITSC 2016 pp 1496ndash1501 Rio de JaneiroBrazil November 2016

[8] Q Wu B Li and K Chen ldquoA multi-agent traffic control modelbased on distributed systemrdquo Sensors amp Transducers vol 173pp 60ndash67 2014

[9] D McKenney and T White ldquoDistributed and adaptive trafficsignal control within a realistic traffic simulationrdquo EngineeringApplications of Artificial Intelligence vol 26 no 1 pp 574ndash5832013

[10] A Jovanovic M Nikolic and D Teodorovic ldquoArea-wide urbantraffic control a bee colony optimization approachrdquoTransporta-tion Research Part C Emerging Technologies vol 77 pp 329ndash350 2017

[11] J Jin and X Ma ldquoHierarchical multi-agent control of trafficlights based on collective learningrdquo Engineering Applications ofArtificial Intelligence vol 68 pp 236ndash248 2018

[12] P Shao L Wang W Qian Q-G Wang and X-H Yang ldquoAdistributed traffic control strategy based on cell-transmissionmodelrdquo IEEE Access vol 6 pp 10771ndash10778 2018

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

14 Journal of Advanced Transportation

[13] L Wang C Li M Z Q Chen Q-G Wang and F TaoldquoConnectivity-based accessibility for public bicycle sharingsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 99 no 4 pp 1ndash12 2018

[14] T Wongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 Anchorage Alaska USA September2012

[15] Z Jiao B Zhang C Li and H T Mouftah ldquoBackpressure-based routing and scheduling protocols for wireless multihopnetworks a surveyrdquo IEEE Wireless Communications Magazinevol 23 no 1 pp 102ndash110 2016

[16] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[17] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo in Proceedings of the 19th IFACWorld Congress onInternational Federation of Automatic Control IFAC 2014 vol47 pp 11332ndash11337 South Africa August 2014

[18] J Gregoire S Samaranayake and E Frazzoli ldquoBack-pressuretraffic signal control with partial routing controlrdquo inProceedingsof the 55th IEEE Conference on Decision and Control CDC 2016pp 6753ndash6758 Las Vegas Nev USA December 2016

[19] A A Zaidi B Kulcsar and H Wymeersch ldquoBack-pressuretraffic signal control with fixed and adaptive routing for urbanvehicular networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 17 no 8 pp 2134ndash2143 2016

[20] H Taale J Van Kampen and S Hoogendoorn ldquoIntegratedsignal control and route guidance based on back-pressureprinciplesrdquo Transportation Research Procedia vol 10 pp 226ndash235 2015

[21] T Le P Kovacs N Walton H L Vu L L H Andrew andS S P Hoogendoorn ldquoDecentralized signal control for urbanroad networksrdquo Transportation Research Part C EmergingTechnologies vol 58 pp 431ndash450 2015

[22] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[23] J Gregoire X Qian E Frazzoli A De La Fortelle and TWongpiromsarn ldquoCapacity-aware backpressure traffic signalcontrolrdquo IEEE Transactions on Control of Network Systems vol2 no 2 pp 164ndash173 2015

[24] X Hu J Cheng and H Luo ldquoTask assignment for multi-uav under severe uncertainty by using stochastic multicriteriaacceptability analysisrdquo Mathematical Problems in Engineeringvol 2015 Article ID 249825 10 pages 2015

[25] W Zhao Q Meng and PW H Chung ldquoA heuristic distributedtask allocationmethod formultivehicle multitask problems andits application to search and rescue scenariordquo IEEE Transactionson Cybernetics vol 46 no 4 pp 902ndash915 2016

[26] L GeorgiadisM J Neely and L Tassiulas ldquoResource allocationand cross-layer control in wireless networksrdquo Foundations andTrends in Networking vol 1 no 1 pp 1ndash144 2006

[27] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control part I formulationrdquo Transporta-tion Research Part B Methodological vol 104 pp 1ndash18 2017

[28] S Lee S C Wong and P Varaiya ldquoGroup-based hierarchicaladaptive traffic-signal control Part II implementationrdquo Trans-portation Research Part B Methodological vol 105 pp 376ndash3972017

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

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