Research Article Simulation-Based Sensor Location Model ...€¦ · Research Article Simulation-Based Sensor Location Model for Arterial Street QinxiaoYu,NingZhu,GengLi,andShoufengMa

Research ArticleSimulation-Based Sensor Location Model for Arterial Street

Qinxiao Yu Ning Zhu Geng Li and Shoufeng Ma

Institute of Systems Engineering College of Management amp Economics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Geng Li ligeng345126com

Received 25 December 2014 Accepted 26 May 2015

Academic Editor Essam Radwan

Copyright copy 2015 Qinxiao Yu 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

Traffic sensors serve as an important way to a number of intelligent transportation system applications which rely heavily on real-time data However traffic sensors are costly Therefore it is necessary to optimize sensor placement to maximize various benefitsArterial street traffic is highly dynamic and the movement of vehicles is disturbed by signals and irregular vehicle maneuver It ischallenging to estimate the arterial street travel time with limited sensors In order to solve the problem the paper presents traveltime estimation models that rely on speed data collected by sensorThe relationship between sensor position and vehicle trajectoryin single link is investigated A sensor location model in signalized arterial is proposed to find the optimal sensor placement withthe minimum estimation error of arterial travel time Numerical experiments are conducted in 3 conditions synchronized trafficsignals green wave traffic signals and vehicle-actuated signals The results indicate that the sensors should not be placed in vehiclequeuing area Intersection stop line is an ideal sensor position There is not any fixed sensor position that can cope with all trafficconditions

1 Introduction

Traffic sensors (eg magnetic detectors cameras and blue-tooth detectors) are widely used in transportation systemfor systematic surveillance Various traffic applications needdifferent sensor data OD estimation may require the trafficcounting information on links Travel time estimation asksfor information about link travel time or path travel timeThese information can be obtained either from float vehiclesuch as taxi or cameras which are able to read plate licensenumber One of the most intuitive pieces of informationfor advanced traveler information system is travel timeTherefore a simple and implementation-wise easy method isneeded to estimate travel time

Attempt to estimate arterial street travel time is very chal-lenging Arterial street traffic is highly dynamic and themovement of vehicles is disturbed by signals and irregularvehicle maneuver Zhang [1] developed a travel time andjourney speed estimation method for freeways by the uti-lization of volume to capacity ratio volume and occupancySkabardonis and Geroliminis [2] employed Kinematic wavetheory to model the spatial-temporal queueing at the signalsLiu and Ma [3] proposed that loop detector data and signalphase changes information which in a high-resolution data

context are required to estimate travel time Without usingtraditional traffic flow theory Takaba et al [4] developedsome heuristic models for travel time estimation by usingloop detector and license plate reader A neural networkframework is developed to fuse probe vehicle data and loopdetector data by Cheu et al [5] Dailey and Cathey [6]employed probe sensor to define vehicle speed functionCurrently Li et al [7] studied that most of the travel timeestimation methods or algorithms require many sources ofreal time data Investment in these transportation surveil-lance devices is costly and thus providing these pieces ofreal time information is expensive Among all these trafficsensors loop detectors are relatively cheap and popular Inaddition loop detectors are also widely used in travel timeestimation field [8 9] Due to the high cost of transportationinfrastructure it is necessary to find away to save investment

Sensor location problem works for this purpose It aimsto find an optimal sensor placement pattern either in trans-portation network or freeway The purpose of sensor place-ment is mainly for various flow estimations which are ODtrips estimation link flows estimation path flows estimationand its related application Yang and Zhou [10] proposed foursensor location rules in transportation network mainly forOD estimation This paper can be seen the seminal paper in

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 854089 13 pageshttpdxdoiorg1011552015854089

2 Discrete Dynamics in Nature and Society

the sensor location literature Bianco et al [11] introduceda linear system approach for sensor location problem mod-eling Gentili and Mirchandani [12] extended linear systemapproach by introducing active and passive sensors Ehlertet al [13] presented several models to cover as many flowsas possible OD estimation using generalized least-squaremethod is applied to seek for the optimal sensor placementpattern by Fei et al [14] and Eisenman et al [15] In additiontoOD estimation path estimation or identification is anotherhot topic Normally license plate reader is employed torecognize route or estimate route travel time by Castillo etal [16] andMınguez et al [17] Modeling techniques adoptedare integer program Commercial software or heuristics isused for solving these problems Hu et al [18] studied the linksensor placement problem to infer all link flow informationof the network of interests Viti et al [19] investigated partialobservation problems and gave a simple metric for quantifythe quality of a sensor placement pattern Other sensorlocation problems include mobile sensor routing problem(Zhu et al [20]) bottleneck identification oriented sensorlocation problem (Liu andDanczyk [21]) and sensor locationproblem considering time-spatial correlation (Liu et al [22])

Sensor location problem on freeway is relatively limitedparticularly for travel time estimation Kim et al [23] adoptedgenetic algorithm to find an optimal sensor placement loca-tion on freeway with the minimization of mean absoluterelative error Kianfar and Edara [24] used clustering tech-nology for optimizing freeway traffic sensors Other freewaysensor location problems employ empirical study methods(Kwon et al [25]) simulation (Thomas [26]) and dynamicprogramming (Ban et al [27])

Due to the complexity of urban transportation systemnone of current studies consider combining travel timeestimation method with a sensor location pattern to seekan optimal sensor placement pattern Another importantcharacteristic of urban transportation system is traffic signalcontrol This paper attempts to find optimal sensor locationpattern with traffic signals A simulation tool is used to gen-erate basic traffic flow dataThe rest of this paper is organizedas follows Section 2 offers a description of travel time esti-mation method and partition rule in arterial street Section 3presents optimal sensor location pattern on a single linkSection 4 gives results for multiple links and also with trafficsignal control Section 5 concludes the whole paper

2 Sensor Location Model Description

21 Section Partition Rule for Arterial Street In our modela signalized arterial street is partitioned into sections Eachsection is associated with a sensor and the speed of sectionis represented by the average instantaneous speed (normallyit is collected from the 30s time interval) at the sensor spotThe estimated travel time of each section is calculated by thelength and speed of the section By summing up estimatedtravel time across all sections the estimated arterial traveltime can be obtained

In our study the boundary of section is determinedby three sensors which are located at the section and the

adjacent upstream and downstream section respectivelyThetotal length of signalized arterial is set to 119871 the total numberof sensors is 119899 the location of the sensor 119894 is 119909

119894 and the length

of section 119894 is 119897119894which is calculated as follows

119897119894

=

1199091 +1199092 minus 1199091

2 119894 = 1

119909119894+1 minus 119909

119894minus12

1 lt 119894 lt 119899

119871 minus 119909119899

+119909119899

minus 119909119899minus1

2 119894 = 119899

(1)

22 Travel Time Estimation Model There are many traveltime estimationmodels among which the speed-based traveltime estimation model is easy-to-operate and widely applied[6ndash9 28ndash30]These three models proposed by Li et al [7] areadopted in our studyThe principle of all these threemodels isto calculate travel time according to the spot-speed obtainedby sensors

The first model is instantaneous model A vehicle issupposed to enter the arterial street at time 119896 The detectedspeed at time 119896 is considered as the average speed of vehiclesat that sectionThe travel time of vehicle at section 119894 is denotedas 119905(119894 119896)

119905 (119894 119896) =119897119894

V (119894 119896) (2)

where 119897119894is the length of the section 119894 and V(119894 119896) is the

measured speed of section 119894 at time 119896 The travel time 119879(119896)

of vehicle passing through the entire signalized arterial streetis the sum of all sectionsrsquo travel times

119879 (119896) =

119899

sum

119894=1119905 (119894 119896) (3)

In the instantaneous model speeds from only one pointon each section are used to estimate travel time whileignoring the speed variations within a section This does notmeet the authentic traveling situation of vehicles Howeverfrom the perspective of calculation when the travel timeis fixed its corresponding average speed is fixed Thereforethe sensor location pattern is very critical A suitable senorlocation pattern can accurately capture the average travel timeof its spatial influence area On the other hand all speedsare collected at the time of vehicle entering the arterial Thespeed associatedwith the downstream sectionwill not changedramaticallywhen the vehicle traverses the arterialThese tworeasons result in travel time estimation error The other twospeed estimation models are Time Slice Model and DynamicTime Slice Model

The difference among the three travel time estimationmodels lies in the calculation method for vehicle speed oneach section which is the main factor that affects errorThesethree models all use the speed measured by sensors so thetravel time estimation error is caused by not only inevitablecalculation error but also the error arising from the locationof sensor in arterial street Different combinations of sensorlocations generate different section partitions thus resultingin different estimation errors

Discrete Dynamics in Nature and Society 3

Midpoint

l l l l l

S1 SnSnminus1

1km800 vph

middot middot middotmiddot middot middotmiddot middot middot

Figure 1 Partitioned arterial street with traffic signal

To compare the differences among these three modelsnumerical experiments are conducted for the identical sensorlocation pattern In our study a traffic simulation tool isused for a street with length 1 km as shown in Figure 1Intersection signal cycle is 100 s green light duration is 60 sand the vehicle enters the street with a speed of 36 kmhThe maximum speed is 576 kmh The vehiclersquos arrival rateis 800 vph and obeys Poisson distribution Sensors are evenlydistributed on the road at equal interval of 119897 The influentialarea of each sensor is shown in Figure 1 The number ofsensors is increased from 1 to 10 Travel time is estimated withabove-mentioned three methods respectively

In order to compare the resulted obtained by these threemethods we use three measures for evaluation which are themean absolute error (MAE) rootmean square error (RMSE)and mean absolute relative error (MARE) respectively

MAE =1119899

119899

sum

119894=1

1003816100381610038161003816ett119894 minus gttt119894

1003816100381610038161003816

RMSE = radic1119899

119899

sum

119894=1(ett119894

minus gttt119894)2

MARE =1119899

119899

sum

119894=1

1003816100381610038161003816ett119894 minus gttt119894

1003816100381610038161003816

gttt119894

(4)

ett119894refers to the estimated travel time gttt

119894refers to the

ground-truth travel time and 119899 refers to the number ofvehicles Through 10 traffic simulations 10 groups of traveltime 119878

119903| 119878119903

= (gttt1199031 gttt

1199032 gttt119903119899

) 119903 = 1 2 10 areobtained These three travel time estimation methods wereused to calculate the corresponding estimated travel time andtheMAE

119903MARE

119903 and RMSE

119903respectively Finally 10 sets of

three measures are obtained for each travel time estimationmethod The mean values and standard deviations are cal-culated according to these 10 sets of data The experimentalresults are shown in Figure 2

As shown in Figure 2 the three estimation models havelittle differences in most cases Particularly the time slicemodel and dynamic time slice model almost obtain the sameresults The instantaneous model outperforms other twoestimation models although instantaneous model (IM) onlymakes estimation according to the traffic condition whenvehicle enters the street But under the traffic signal controlthe traffic flow has certain reproducibility The vehiclersquostravel time can be accurately estimated when the vehiclestraverse the arterial street smoothlyTherefore in subsequentexperiments IM method is adopted to estimate the traveltime

3 Sensor Location Setting in Single Link

In the urban transportation network the movement ofvehicles is with some regularity due to the traffic signalcontrol Generally it can be summarized as after passingthrough previous intersection vehicles enter the street atlow speed or original speed Then the vehicles accelerateto the maximum allowable speed and keep moving Whenapproaching the next intersection it determines whether toslow down or keep moving at the original speed according totraffic signalsThe speed contour profile is shown in Figure 3

The key point of estimating vehiclersquos travel time on acertain link is to find the vehiclersquos average speed on that linkVehicle moves with different speed at different positions ofthe link In order to accurately estimate travel time we needto find an appropriate location for the sensor making surethat the instantaneous speed is close to the average speedTherefore the location of sensor has great impact on thetravel time estimation error Figure 4 shows the relationshipbetween travel time estimation error (MARE) and differentsensor locations when a sensor is placed on a 1 km link underdifferent arrival rates

As can be seen from Figure 4 the smallest travel timeestimation error is obtained when the sensor is placed at the50m and the mean travel time estimation error of the fourarrival rates is about 8 In the link from 100m to 850m thesensor location has little impact on travel time estimation andthe error is about 29 In the 100m of downstream sectionthe travel time estimation error increases about 120

Sensor location pattern is closely related to the vehiclersquostrajectory on the link (i) If a vehicle enters the intersectionat a slow speed and the sensor is placed at the inlet (0m)position where the average speed is small the estimationerror will be large (ii) If the vehicle accelerates to themaximum speed after entering the link the vehicle speedvaries greatly within this distance (from 36 to 566 kmh)so it is easy to find a position that can represent the averagespeed of the link (iii) The vehicle runs at the limited speedin the middle section of the link and the speed fluctuationis small Therefore no matter where the sensor is placed thedetected speed is almost equal to the maximum speed Thusit cannot reflect the average speed on the link (iv)Within the100m in the downstream vehicles enter the queuing areaThelength of queue increases as the arrival rate increases Whenthe sensor is closer to the intersection the sensor is likelyto be occupied by vehicles When the sensor is occupied itcannot detect speed The estimation error is large Howeverwhen the arrival rate is less than the saturation volume thenumber of queuing vehicles is small All the vehicles can passthrough the intersection in a signal cycle The proportionof sensorrsquos occupied time is small Therefore the travel timecan be estimated When the arrival rate is greater than thesaturated volume vehicles at the tail of the queue need towait for two or more signal cycles before passing throughthe intersectionThe sensor is more likely to be occupied andcannot obtain vehicle speed during a long period of time


1012141618202224262830

0 1 2 3 4 5 6 7 8 9 10

Mea

n ab

solu

te er

ror (

s)M

ean

abso

lute

relat

ive e

rror

(s) (

)

Number of sensors

0

02

04

06

08

1

12

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18

The s

td o

f MA

E (s

)

0

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35

0 1 2 3 4 5 6 7 8 9 10Number of sensors

00

02

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18

The s

td o

f MA

RE (s

) (

)

10

15

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25

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35

0 1 2 3 4 5 6 7 8 9 10

Root

mea

n sq

uare

erro

r (s)

Number of sensors

IMTSMDTSM

002040608

112141618

2

0 1 2 3 4 5 6 7 8 9 10

The s

td o

f roo

t RM

SE (s

)

Number of sensors



IMTSMDTSM

Figure 2 Errors and their standard deviations versus number of sensors

4 Sensor Location in Arterial Street

41 Sensor Location Model Description A signalized arterialstreet is usually composed of many links and there is atraffic signal between every two links In order to studysensor placement pattern in such a signalized arterial streetwe divide arterial street into equal-length cells as shown in

Figure 5 If a cell is equipped with a sensor then the sensorwill be placed on the cellrsquos right boundary Section is definedas the influential area of the sensor The partition method isgiven in previous section

Assume that the entire horizon of the study is119879 and a totalof 119872 vehicles pass through the entire street 119870 sensors will beplaced on the arterial street that is there will be 119870 sections


02468

1012141618

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Spee

d (m

s)

Travel time (s)

Figure 3 Speed contour profile in a signal cycle

020406080

100120140160180200

0 50 100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

MA

RE (

)

Position (m)Arrival rate is 400 vphArrival rate is 800 vph

Arrival rate is 1200 vphArrival rate is 1600 vph

Figure 4 Estimate error versus different sensor locations

The actual travel time of the 119898th vehicle that passes throughthe arterial is GTTT

119898 which is obtained by simulation The

estimated travel time is ETT119898 which is the total sum of 119870

sectionsrsquo estimated travel time ETT119898119896 The goal of sensor

location model is to minimize the error between estimatedtravel time and actual travel timeThe decision variable of themodel is 119909

119894isin 0 1 which indicates whether the sensor is

placed on 119894th cellThemodel is shown as follows and is calledM1

M1

min 1119872

119872

sum

119898=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(sum119870

119896=1 ETT119898119896) minus GTTT119898

GTTT119898

10038161003816100381610038161003816100381610038161003816100381610038161003816

(5a)

subject to119873

sum

119894=1119909119894

= 119870 (5b)

119909119894

isin 0 1 (5c)

119884 is set of the index of 119909119894

where 119909119894

= 1

119910119896is the 119896th element in 119884

(5d)

Link

SectionCell Section

Sensor

Figure 5 Partitioned arterial street

119904119896

=

119910119896+1 minus 119910

119896minus12

119896 = 2 119870 minus 1

1199101 +1199102 minus 1199101

2 119896 = 1

119910119870

+119873 minus 119910

119870

2 119896 = 119870

(5e)

ETT119896

=119904119896

V119896

sdot 119897 (5f)

In M1 119872 is the number of total vehicles 119873 is thenumber of cells and 119870 is the number of sensors namely thebudget constraint 119909

119894is the decision variable and 119909

119894= 1

indicates that the 119894th cell has a sensor otherwise 119909119894

= 0119910 represents the set of cells that are installed with sensorswherein the total number of elements is 119870 For instanceif the entire arterial street is divided into 10 cells budgetconstraints are three sensors 119884 = 3 6 8 means that sensorsare placed in the 3th 6th and 8th cells 119910

119896refers to the 119896th

element of set 119884 and 119878119896is the coverage area of 119896th sensor

namely the number of cells contained in the 119896th sectionwhich is determined by the position of adjacent upstreamand downstream sensors When the positions of a group ofsensors are given the length of 119870 sections can be calculatedAlso taking 119884 = 3 6 8 as an example 119904

1covers 45 cells

1199042covers 25 cells and 119904

3covers 3 cells 119881

119896is the average

speed of section 119896 which is calculated according to the speeddetected by corresponding sensor After solving this modelthe optimal sensor placement pattern can be obtained Due tothe complexity of the combinational optimization problemexact algorithm is very hard Thus in our study geneticalgorithm is employed

In this section a mathematic model is proposed to solvethe sensor placement problem in arterial street It is a 0-1 pro-gramming model The objective function is to minimize therelative travel time estimation error between estimated andactual travel time The input data of the model is travel timeinformation of all vehicles in computer simulation whichis treated as ground-truth travel time Once the ground-truth travel time information is given and our proposedmathematical programming model can decide the optimalsensor placement pattern that minimizes the relative traveltime estimation errorTherefore ourmodel is amathematicalmodel In addition the model is deterministic

42 Case Study In this study we only consider the influenceof vehicle arrival rate and traffic signal strategy on sensorplacement pattern which are two major factors that affecttravel time on urban network In the simulation the signal-ized arterial street is set as 3 km long and it is composed of6 links Each link is 05 km long Each link is divided into


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

(a) Arrival rate is 400 vph

Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

(b) Arrival rate is 800 vph

Time (s)

Posit

ion

(m)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

1000

1500

2000

2500

0

2

4

6

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16

(c) Arrival rate is 1200 vph

Posit

ion

(m)

500

1000

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2000

2500

0

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4

6

8

10

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14

16

Time (s)

500

1000

1500

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3000

3500

4000

4500

5000

(d) Arrival rate is 1600 vph

Figure 6 Speed contour plot in synchronized traffic signals

five cells and each cell is 01 km long The limited speedis 576 kmh Vehicle arrival rate at the entrance is set as400 vph 800 vph 1200 vph and 1600 vph respectively andarrival rate obeys Poisson distribution When the arrival rateexceeds the intersection capacity there will be congestionIn each simulation the arrival rate is fixed In the urbantransportation network traffic signals strategy can greatlyaffect the capacity of each intersection and thereby affectthe trajectory of vehicle after entering the arterial streetTherefore we need to consider different traffic signals strate-gies In this study we consider three common traffic signalsstrategies synchronized traffic signals green wave trafficsignals and vehicle-actuated signals In the experiment weanalyzed three strategies separately under different vehiclearrival rates

421 Synchronized Traffic Signals Synchronized controlrefers that all intersections of the arterial street use the samesignal configuration and display the same traffic signal atthe same time In the simulation the signal cycle is set as100 s and green ratio is 50 Vehiclersquos speed trajectoriesunder different arrival rates are shown in Figure 6 The 119909-axis represents time and the 119910-axis represents the vehicleposition Speed is represented by different colors Colorchange from red to blue refers to the gradual decrease of

speed Each row represents the speed variation of the sameposition at different time points Each column represents thespeed of each position at the same time

According to Figure 6(a) the arrival rate is 400 vphwhich is far less than the intersection capacity In the wholesimulation horizon the number of vehicles that arrive atthe intersection is small and all vehicles can pass throughthe intersection within one signal cycle so the queue lengthof each intersection is short In Figure 6(b) the vehiclearrival rate is approximately equal to the intersection capacityBecause the arrival rate obeys Poisson distribution vehicleshave different actual arrival rates in each signal cycle Somearrival rates are greater than the intersection capacity andsome are less than the intersection capacity When the arrivalrate is greater than the intersection capacity in a signal cyclethere will be multiple queues While the queuing at thedownstream intersection spreads to the upstream it reducesthe actual capacity of upstream intersection making thequeuing at upstream section much longer as the first andsecond intersections after 2000 s shown in Figure 6(b)Whenthe arrival rate is much greater than the intersection capacitysome queuing vehicles may not be able to pass through theintersection within one signal cycle Vehicle release rate willbe exactly equal to the intersection capacity that is thearrival rate of the next intersection is equal to the intersection


Table 1 Comparison among different arrival rates

Arrival rate(vph)

Number ofsensors Sensor location pattern

40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900

lowastThe sensor is located in the queuing area

capacity This intersection will not form a secondary queueAs shown in the yellow rectangle of Figures 6(b) 6(c) and6(d) if the queue of vehicles at the upstream intersectionis too long then the queue of vehicles at the downstreamintersection is short

Table 1 shows the optimal sensor placement pattern ofcertain number of sensors at different arrival rates Someobservations can be summarized as follows (i) among the12 optimal sensor placement patterns only three patterns setthe sensor in the queuing area (blue area in Figure 6) Thebold numbers in Table 1 represent the queuing area and thecorresponding influence areas of the 3 sensors are very shortwhich are 300m 400m and 200m respectively In thesethree areas blue area takes up a large proportion that isthe average speed of vehicles is small The queuing is fairlysevere Thus when sensor is placed in the queuing area thecorresponding influence area is with long queuing lengthThis is because the average speed in queuing area is smallIt has great impact on the estimated travel time The sensorshould not be placed in the queuing area When sensor isplaced in the queuing area its corresponding detection rangeshould not be too large and it should be able to accuratelycapture the length of the queuing (ii) Among the 12 optimalplacement patterns 9 patterns set sensors at one or morestop line positions (positions 500 1000 1500 2000 2500 and3000) Compared with other locations the average speed ofvehicles at stop lines has the largest variations and the speedprofile at stop line is repetitive with regard to cycle timeWhen the red traffic light is on the speed is 0 When thegreen light is turned on the speed gradually increases to themaximum speed of 576 kmh When the arrival rate is smallthe number of queuing vehicles is small and then the averagespeed is high When the arrival rate is large the number ofqueuing vehicles is large and then the average speed is low Inaddition speed variation is determined by both the upstreamqueuing and downstream traffic congestion situation Thuswhen there are a lot of vehicles queuing at both the upstreamand downstream links the sensor at stop line can timely andflexibly detect the variation of traffic situation (iii) Whenthe number of sensors is fixed the corresponding sensor

0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)

4 sensors5 sensors6 sensors

Figure 7 Comparison amongMARE in different number of sensorsfor synchronized traffic signals

placement patterns of different arrival rates vary greatly Thisis because different arrival rates leads to different queuelengths so the position of sensor should be determined bythe road conditions

In Figure 7 each point represents the travel time esti-mation error under the optimal sensor placement patternThis pattern is obtained under certain arrival rate and certainnumber of sensors When the arrival rate is 800 vph 4sensors are better than 5 or 6 sensors When the arrival rateis 1600 vph 5 sensors are better than 6 sensors Thereforeoptimizing the sensor placement pattern can reduce errorsand number of sensors When the number of sensors is4 MARE is in proportional with the arrival rate that isin this case the estimation error cannot be reduced bychanging sensor location However the estimation error canbe reduced by increasing the number of sensors

422 Influence of Green Ratio Green ratio is an importantparameter in traffic signals strategy Congestion occurs whentraffic volume is greater than the intersection capacity Inorder to timely dissipate queuing vehicles green ratio shouldbe lengthened When traffic volume is less than the inter-section capacity the green ratio should be carefully reducedGreen ratiowill directly affect intersection capacityWhen thearrival rate is a constant vehicle queuing will change thusaffecting the position of sensor

In order to study the influence of sensor location onestimated travel time under different green ratios we setthree green ratios which are 40 50 and 60 respectivelyOther parameters are the same as previous section By usinggenetic algorithm the optimal sensor placement pattern andthe travel time estimation errors are obtained As shown inFigure 8 each point represents the travel time estimationerror of the optimal sensor placement pattern It can be seenfrom Figure 8 that regardless of the number of sensors smallgreen ratio causes larger travel time estimation error Thereason could be explained as that traffic condition becomes


0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors

Green ratio is 40Green ratio is 50Green ratio is 60

5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

Green rate is 40Green rate is 50Green rate is 60


Figure 8 Comparison among different green ratios

congested and complicated as the green ratio reducedThere-fore it is hard to predict the travel time

423 Green Wave Traffic Signals Green wave control meansthat if a vehicle passes through one intersection at a givenspeed with a green light phase it will pass through all the

downstream intersections at green light phase [31]The offsettime of adjacent intersectionrsquos traffic signal equals to thelength of the section divided by the given speed If the vehiclemeets a red light phase at the intersection it will stop andwaitWhen the light turns green the vehicle accelerates from zeroto the limited speed and moves to the next intersection Theaverage speed of vehicle on the link is slightly less than the



Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000

limited speed and travel time is slightly larger than the offsettime Therefore during the green light period if the numberof queuing vehicles exceeds the intersection capacity vehiclesin the front of queue can successfully pass through the nextintersection However vehicles in the tail part of the queuewill meet red light at the next intersection

In our simulation the signal cycle is set as 100 s Greenratio is 50 Each link is 05 km long The limited vehiclespeed is 576 kmh The relative offset time between adjacentintersections is 32 sTherefore there will be a band alongwiththe arterial street As long as the vehicle arrives within theband and keeps moving at the limited speed of 576 kmh itcan travel smoothly through the all intersections Figure 11shows the speed trajectories under four different arrival ratesGenerally speaking vehicles can move at the maximumspeed on the road most of the time and can smoothly passthrough each intersection If the arrival rate is greater than theintersection capacity the first intersection will have a smallnumber of queuing vehicles this is because some vehiclesenter the arterial street at a smaller speed their travel time onthe link is greater than the travel time of vehicle which movesat the limited speed (offset time) They will meet red light atthe first intersection

Table 2 shows the corresponding optimal sensor place-ment patterns of fixed number of sensors for different arrivalrates All sensors are set after the first intersection This isbecause the first link has stable traffic condition without anyfluctuations under different arrival rates This can be seenfrom the four subgraphs of Figure 9 In the section from0m to 500m the vehicle accelerates to maximum speed andmoves to the first intersection after entering the arterial Inthis section the speed trajectory is very similar Besideslike the synchronized traffic signal strategy 10 out of the 12patterns set the sensor on one or more stop lines

Figure 10 shows the estimation error under the optimalsensor placement pattern for greenwave traffic signal controlCompared with other two signals strategies it is easier toestimate the travel time under green wave control Theestimation error is less than 2 Regardless of the arrivalrate the travel time estimation error is minimized whenthe number of detectors is 4 This is because that the traffic

condition is simple under the green wave traffic signalsWhen the traffic is not complicated less sensors should beplaced When traffic condition is very complicated moresensors should be placed

424 Vehicle-Actuated Traffic Signals The above two trafficsignals strategies are fixed traffic signal control methodTheyare developed based on historical data The disadvantage ofthese strategies is that it is unable to meet real-time changesof traffic flow In order to overcome this deficiency vehicle-actuated signals [32] are adopted This strategy changes thegreen time adaptively according to real-time traffic volume

In the simulation of vehicle-actuated control we set eachintersectionrsquos signal cycle as 100 s The minimum green timeis 50 s The maximum green time is 70 s The unit extensioninterval is 3 s When the arterial street gets the access rightthe signal system will first give the signal phase a minimumgreen time of 50 s enabling the vehicle that has arrived at theintersection to pass through the intersection If there is novehicle after this the access right will be transferred to thesubsequent link If a vehicle is detected within the green timethe green time will be extended a unit time interval of 3 sThemaximum extension can be 70 s

Figure 11 is the speed trajectory for 4 different arrivalrates Comparedwith the synchronized traffic signals controlthe queuing length at the intersection is shorter and large-scale long queue appears only once when the arrival rate is1600 vph When the traffic volume is large the signal systemcan detect the changes of volume increasing the green time ina timelymanner As a result intersection capacity is improvedand vehicle queue length is reduced By comparing Figures 6and 11 it can be seen that under the vehicle traffic signalscontrol the average vehicle queue length is short

The optimal sensor location pattern under vehicle-actuated traffic signals control is shown in Table 3 Thecorresponding travel time estimation error of each optimalsensor mode is shown in Figure 12 Similar to synchronizedtraffic signals control among the 12 optimum placementpatterns only three patterns set the sensor in the queuingarea (see the bold numbers in the table) Two of these threesensors have small detection coverage area which are 250mand 400m respectively 12 sensor location patterns all chooseto place the sensor on a stop line As can be seen fromFigures 11(c) and 11(d) blue area takes up a large proportionin these two areas that is the average speed of vehicles issmall and the queuing situation is severe Thus when thesensor is located in queuing area its corresponding influencearea is the area with long queues This is because the averagespeed of queuing area is small which has great impact on theestimation of travel timeThe sensors should be avoided frombeing placed in queuing area When the sensor is placed inthe queue area its corresponding influence area should notbe too large It should accurately correspond to the lengthof the queue In addition it can be seen from Figure 12 that(i) the travel time estimation error calculated by differentnumber of sensors differs slightly for different arrival rates(ii) As for a fixed number of sensors travel time estimationerror increases as the arrival rate increases (iii) Regardless of


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Figure 9 Speed contour plot in green wave traffic signals

10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)


Figure 10 Comparison among MARE in different number ofsensors for green wave traffic signals

arrival rate the estimated travel time predicted by 6 sensorsis the most accurate The number of arrived vehicles withinone signal cycle changes the green time length adaptively

resulting in the change of intersection capacity The trafficcondition of each link becomes more dynamic and randomwhich differs greatly from the cyclical and repetitive trafficcondition under synchronized control Therefore it requiresmore sensors to estimate travel time

Through experimental analyses of these three trafficsignals strategies observations are stated as follows (i) whenthe sensor is located in the queuing area its correspondinginfluence area is short Its length should be close to thequeuing length Therefore when a sensor is located in thequeuing area some sensors should be set in the adjacentupstream and downstream areas in a cooperative manner(ii) Stop line is an ideal sensor position place Comparedwith other location places the average speed of vehicles atstop lines has the largest variations Small arrival rate maycause fewer queuing vehicles The average speed is high Alarge arrival rate may cause more queuing vehicles where theaverage speed is lowThe sensor can timely and flexibly reflectthe traffic conditions (iii) Under simple traffic situationwhere vehicle speed is stable with small speed fluctuation fewsensors can accurately estimate travel time Otherwise moresensors are needed


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Figure 11 Speed contour plot in vehicle-actuated traffic signals


Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100

lowastThe sensor is located in queuing area

5 Conclusion

The paper studies the sensor location problem in urbanarterial street for travel time estimation and proposes optimal

00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)


Figure 12 Comparison among MARE in different number ofsensors for vehicle-actuated traffic signals

sensor location model (M1) to obtain the minimum traveltime estimation error Based on this model the influence oftraffic signals strategies on sensor location is also discussed


By comparing the synchronized traffic signals green wavetraffic signals and vehicle-actuated signals it is found thatsensor should not be placed in vehicle queuing area If thesensor is located in the queuing area its associated coveragearea should include the vehicle queuing area as precise aspossible Intersection stop line is an ideal sensor positionThere is not any fixed sensor position that can cope withall traffic conditions and the sensor location should bedetermined according to the characteristics of traffic flowon the road Under simple traffic situation where vehiclespeed is stable and speed fluctuation is small few sensorscan accurately estimate travel time In case of complex trafficconditions with large fluctuations of vehicle speed moresensors are required to estimate travel time

The future research direction can be considered as fol-lows (i) our study only takes into account the traffic conditionof single lane with fixed traffic volume and the futureresearch can consider more complex traffic situations suchas the dynamic changes of multilane road with dynamictraffic volume and other real phenomena that match withactual road conditions (ii) The future research can take thisstudy on urban arterial street as background taking intoaccount the vehicle turnings and study the sensor locationunder the road network layout (iii) A more efficient modelfor algorithm should be explored in the future researchalthough the genetic algorithm used in the paper is aneffective solutionmodel (iv)The study only seeks the optimalsensor location for travel time estimation Future research canfocus on optimized combination of more traffic informationapplications or more information applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation Council of China under Projects 7130111571431005 71271150 and 71101102 and Specialized ResearchFund for theDoctoral Program ofHigher Education of China(SRFDP) under Grant no 20130032120009

References

[1] H M Zhang ldquoLink-journey-speed model for arterial trafficrdquoTransportation Research Record vol 1676 no 1 pp 109ndash1151999

[2] A Skabardonis and N Geroliminis ldquoReal-time estimation oftravel times on signalized arterialsrdquo Transportation and TrafficTheory no LUTS-ARTICLE-2009-003 pp 387ndash406 2005

[3] H X Liu and W Ma ldquoA virtual vehicle probe model for time-dependent travel time estimation on signalized arterialsrdquoTrans-portation Research Part C Emerging Technologies vol 17 no 1pp 11ndash26 2009

[4] S Takaba T Morita T Hada T Usami and M YamaguchildquoEstimation and measurement of travel time by vehicle detec-tors and license plate readersrdquo in Proceedings of the Vehicle

Navigation and Information Systems Conference vol 2 pp 257ndash267 IEEE October 1991

[5] R L Cheu D H Lee and C Xie ldquoAn arterial speed estimationmodel fusing data from stationary and mobile sensorsrdquo inProceedings of the IEEE Intelligent Transportation Systems Pro-ceedings pp 573ndash578 August 2001

[6] D J Dailey and F W Cathey ldquoAVL-equipped vehicles as trafficprobe sensorsrdquo Tech Rep WA-RD 5341 Washington StateDepartment of Transportation 2002

[7] R Li G Rose and M Sarvi ldquoEvaluation of speed-based traveltime estimation modelsrdquo Journal of Transportation Engineeringvol 132 no 7 pp 540ndash547 2006

[8] J W C Van Lint ldquoEmpirical evaluation of new robust traveltime estimation algorithmsrdquo Transportation Research Recordvol 2160 no 1 pp 50ndash59 2010

[9] D Ni and H Wang ldquoTrajectory reconstruction for travel timeestimationrdquo Journal of Intelligent Transportation Systems vol 12no 3 pp 113ndash125 2008

[10] H Yang and J Zhou ldquoOptimal traffic counting locations fororigin-destination matrix estimationrdquo Transportation ResearchPart B Methodological vol 32B no 2 pp 109ndash126 1997

[11] L Bianco G Confessore and P Reverberi ldquoA network basedmodel for traffic sensor location with implications on ODmatrix estimatesrdquo Transportation Science vol 35 no 1 pp 50ndash60 2001

[12] M Gentili and P B Mirchandani ldquoLocating active sensors ontraffic networksrdquo Annals of Operations Research vol 136 no 1pp 229ndash257 2005

[13] A EhlertM GH Bell and S Grosso ldquoThe optimisation of tra-ffic count locations in road networksrdquo Transportation ResearchPart B Methodological vol 40 no 6 pp 460ndash479 2006

[14] X Fei H SMahmassani and SM Eisenman ldquoSensor coverageand location for real-time traffic prediction in large-scalenetworksrdquoTransportation Research Record Journal of the Trans-portation Research Board vol 2039 no 1 pp 1ndash15 2007

[15] S M Eisenman X Fei X Zhou and H S Mahmassani ldquoNum-ber and location of sensors for real-time network traffic esti-mation and prediction sensitivity analysisrdquo TransportationResearch Record vol 1964 no 1 pp 253ndash259 2006

[16] E Castillo J M Menendez and P Jimenez ldquoTrip matrix andpath flow reconstruction and estimation based on plate scan-ning and link observationsrdquo Transportation Research Part BMethodological vol 42 no 5 pp 455ndash481 2008

[17] R Mınguez S Sanchez-Cambronero E Castillo and PJimenez ldquoOptimal traffic plate scanning location for OD tripmatrix and route estimation in road networksrdquo TransportationResearch Part B Methodological vol 44 no 2 pp 282ndash2982010

[18] S-R Hu S Peeta and C-H Chu ldquoIdentification of vehiclesensor locations for link-based network traffic applicationsrdquoTransportation Research Part B Methodological vol 43 no 8-9 pp 873ndash894 2009

[19] F Viti M Rinaldi F Corman and C M Tampere ldquoAssessingpartial observability in network sensor location problemsrdquoTransportation Research Part B Methodological vol 70 pp 65ndash89 2014

[20] N Zhu Y Liu S Ma and Z He ldquoMobile traffic sensor routingin dynamic transportation systemsrdquo IEEE Transactions on Intel-ligent Transportation Systems Magazine vol 15 no 5 pp 2273ndash2285 2014


[21] H X Liu and A Danczyk ldquoOptimal sensor locations for free-way bottleneck identificationrdquoComputer-AidedCivil and Infras-tructure Engineering vol 24 no 8 pp 535ndash550 2009

[22] Y Liu N Zhu S Ma and N Jia ldquoTraffic sensor locationapproach for flow inferencerdquo IET Intelligent Transport Systemsvol 9 no 2 pp 184ndash192 2015

[23] J Kim B Park J Lee and JWon ldquoDetermining optimal sensorlocations in freeway using genetic algorithm-based optimiza-tionrdquo Engineering Applications of Artificial Intelligence vol 24no 2 pp 318ndash324 2011

[24] J Kianfar and P Edara ldquoOptimizing freeway traffic sensorlocations by clustering global-positioning-system-derivedspeed patternsrdquo IEEE Transactions on Intelligent TransportationSystems vol 11 no 3 pp 738ndash747 2010

[25] J Kwon K Petty and P Varaiya ldquoProbe vehicle runs or loopdetectors Effect of detector spacing and sample size on accu-racy of freeway congestion monitoringrdquo TransportationResearch Record vol 2012 pp 57ndash63 2007

[26] G B Thomas ldquoThe relationship between detector location andtravel characteristics on arterial streetsrdquo ITE Journal vol 69 no10 pp 36ndash42 1999

[27] X Ban L Chu R Herring and J D Margulici ldquoSequentialmodeling framework for optimal sensor placement for multipleintelligent transportation system applicationsrdquo Journal of Trans-portation Engineering vol 137 no 2 pp 112ndash120 2010

[28] J W C van Lint ldquoOnline learning solutions for freeway traveltime predictionrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 9 no 1 pp 38ndash47 2008

[29] L D Vanajakshi B M Williams and L R Rilett ldquoImprovedflow-based travel time estimation method from point detectordata for freewaysrdquo Journal of Transportation Engineering vol135 no 1 pp 26ndash36 2009

[30] L Li X Chen Z Li and L Zhang ldquoFreeway travel-time estima-tion based on temporal-spatial queueing modelrdquo IEEE Trans-actions on Intelligent Transportation Systems vol 14 no 3 pp1536ndash1541 2013

[31] J T Morgan and J D C Little ldquoSynchronizing traffic signalsfor maximal bandwidthrdquoOperations Research vol 12 no 6 pp896ndash912 1964

[32] R Hall Handbook of Transportation Science Springer BerlinGermany 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the sensor location literature Bianco et al [11] introduceda linear system approach for sensor location problem mod-eling Gentili and Mirchandani [12] extended linear systemapproach by introducing active and passive sensors Ehlertet al [13] presented several models to cover as many flowsas possible OD estimation using generalized least-squaremethod is applied to seek for the optimal sensor placementpattern by Fei et al [14] and Eisenman et al [15] In additiontoOD estimation path estimation or identification is anotherhot topic Normally license plate reader is employed torecognize route or estimate route travel time by Castillo etal [16] andMınguez et al [17] Modeling techniques adoptedare integer program Commercial software or heuristics isused for solving these problems Hu et al [18] studied the linksensor placement problem to infer all link flow informationof the network of interests Viti et al [19] investigated partialobservation problems and gave a simple metric for quantifythe quality of a sensor placement pattern Other sensorlocation problems include mobile sensor routing problem(Zhu et al [20]) bottleneck identification oriented sensorlocation problem (Liu andDanczyk [21]) and sensor locationproblem considering time-spatial correlation (Liu et al [22])

Sensor location problem on freeway is relatively limitedparticularly for travel time estimation Kim et al [23] adoptedgenetic algorithm to find an optimal sensor placement loca-tion on freeway with the minimization of mean absoluterelative error Kianfar and Edara [24] used clustering tech-nology for optimizing freeway traffic sensors Other freewaysensor location problems employ empirical study methods(Kwon et al [25]) simulation (Thomas [26]) and dynamicprogramming (Ban et al [27])

Due to the complexity of urban transportation systemnone of current studies consider combining travel timeestimation method with a sensor location pattern to seekan optimal sensor placement pattern Another importantcharacteristic of urban transportation system is traffic signalcontrol This paper attempts to find optimal sensor locationpattern with traffic signals A simulation tool is used to gen-erate basic traffic flow dataThe rest of this paper is organizedas follows Section 2 offers a description of travel time esti-mation method and partition rule in arterial street Section 3presents optimal sensor location pattern on a single linkSection 4 gives results for multiple links and also with trafficsignal control Section 5 concludes the whole paper

2 Sensor Location Model Description

21 Section Partition Rule for Arterial Street In our modela signalized arterial street is partitioned into sections Eachsection is associated with a sensor and the speed of sectionis represented by the average instantaneous speed (normallyit is collected from the 30s time interval) at the sensor spotThe estimated travel time of each section is calculated by thelength and speed of the section By summing up estimatedtravel time across all sections the estimated arterial traveltime can be obtained

In our study the boundary of section is determinedby three sensors which are located at the section and the

adjacent upstream and downstream section respectivelyThetotal length of signalized arterial is set to 119871 the total numberof sensors is 119899 the location of the sensor 119894 is 119909

119894 and the length

of section 119894 is 119897119894which is calculated as follows

119897119894

=

1199091 +1199092 minus 1199091

2 119894 = 1

119909119894+1 minus 119909

119894minus12

1 lt 119894 lt 119899

119871 minus 119909119899

+119909119899

minus 119909119899minus1

2 119894 = 119899

(1)

22 Travel Time Estimation Model There are many traveltime estimationmodels among which the speed-based traveltime estimation model is easy-to-operate and widely applied[6ndash9 28ndash30]These three models proposed by Li et al [7] areadopted in our studyThe principle of all these threemodels isto calculate travel time according to the spot-speed obtainedby sensors

The first model is instantaneous model A vehicle issupposed to enter the arterial street at time 119896 The detectedspeed at time 119896 is considered as the average speed of vehiclesat that sectionThe travel time of vehicle at section 119894 is denotedas 119905(119894 119896)

119905 (119894 119896) =119897119894

V (119894 119896) (2)

where 119897119894is the length of the section 119894 and V(119894 119896) is the

measured speed of section 119894 at time 119896 The travel time 119879(119896)

of vehicle passing through the entire signalized arterial streetis the sum of all sectionsrsquo travel times

119879 (119896) =

119899

sum

119894=1119905 (119894 119896) (3)

In the instantaneous model speeds from only one pointon each section are used to estimate travel time whileignoring the speed variations within a section This does notmeet the authentic traveling situation of vehicles Howeverfrom the perspective of calculation when the travel timeis fixed its corresponding average speed is fixed Thereforethe sensor location pattern is very critical A suitable senorlocation pattern can accurately capture the average travel timeof its spatial influence area On the other hand all speedsare collected at the time of vehicle entering the arterial Thespeed associatedwith the downstream sectionwill not changedramaticallywhen the vehicle traverses the arterialThese tworeasons result in travel time estimation error The other twospeed estimation models are Time Slice Model and DynamicTime Slice Model

The difference among the three travel time estimationmodels lies in the calculation method for vehicle speed oneach section which is the main factor that affects errorThesethree models all use the speed measured by sensors so thetravel time estimation error is caused by not only inevitablecalculation error but also the error arising from the locationof sensor in arterial street Different combinations of sensorlocations generate different section partitions thus resultingin different estimation errors


Midpoint

l l l l l

S1 SnSnminus1

1km800 vph





MAE =1119899

119899

sum

119894=1

1003816100381610038161003816ett119894 minus gttt119894

1003816100381610038161003816

RMSE = radic1119899

119899

sum

119894=1(ett119894

minus gttt119894)2

MARE =1119899

119899

sum

119894=1

1003816100381610038161003816ett119894 minus gttt119894

1003816100381610038161003816

gttt119894

(4)


119894refers to the


119903| 119878119903

= (gttt1199031 gttt

1199032 gttt119903119899


119903MARE

119903 and RMSE










1012141618202224262830

0 1 2 3 4 5 6 7 8 9 10

Mea

n ab

solu

te er

ror (

s)M

ean

abso

lute

relat

ive e

rror

(s) (

)

Number of sensors

0

02

04

06

08

1

12

14

16

18

The s

td o

f MA

E (s

)

0

5

10

15

20

25

30

35


00

02

04

06

08

10

12

14

16

18

The s

td o

f MA

RE (s

) (

)

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10

Root

mea

n sq

uare

erro

r (s)

Number of sensors

IMTSMDTSM

002040608

112141618

2

0 1 2 3 4 5 6 7 8 9 10

The s

td o

f roo

t RM

SE (s

)

Number of sensors



IMTSMDTSM







02468

1012141618

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Spee

d (m

s)

Travel time (s)


020406080

100120140160180200

0 50 100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

MA

RE (

)











M1

min 1119872

119872

sum

119898=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(sum119870

119896=1 ETT119898119896) minus GTTT119898

GTTT119898

10038161003816100381610038161003816100381610038161003816100381610038161003816

(5a)

subject to119873

sum

119894=1119909119894

= 119870 (5b)

119909119894

isin 0 1 (5c)


where 119909119894

= 1


(5d)

Link

SectionCell Section

Sensor


119904119896

=

119910119896+1 minus 119910

119896minus12

119896 = 2 119870 minus 1

1199101 +1199102 minus 1199101

2 119896 = 1

119910119870

+119873 minus 119910

119870

2 119896 = 119870

(5e)

ETT119896

=119904119896

V119896

sdot 119897 (5f)



119894= 1






1covers 45 cells








Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Time (s)

Posit

ion

(m)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000









Arrival rate(vph)


40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900




0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)








0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Midpoint

l l l l l

S1 SnSnminus1

1km800 vph





MAE =1119899

119899

sum

119894=1

1003816100381610038161003816ett119894 minus gttt119894

1003816100381610038161003816

RMSE = radic1119899

119899

sum

119894=1(ett119894

minus gttt119894)2

MARE =1119899

119899

sum

119894=1

1003816100381610038161003816ett119894 minus gttt119894

1003816100381610038161003816

gttt119894

(4)


119894refers to the


119903| 119878119903

= (gttt1199031 gttt

1199032 gttt119903119899


119903MARE

119903 and RMSE










1012141618202224262830

0 1 2 3 4 5 6 7 8 9 10

Mea

n ab

solu

te er

ror (

s)M

ean

abso

lute

relat

ive e

rror

(s) (

)

Number of sensors

0

02

04

06

08

1

12

14

16

18

The s

td o

f MA

E (s

)

0

5

10

15

20

25

30

35


00

02

04

06

08

10

12

14

16

18

The s

td o

f MA

RE (s

) (

)

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10

Root

mea

n sq

uare

erro

r (s)

Number of sensors

IMTSMDTSM

002040608

112141618

2

0 1 2 3 4 5 6 7 8 9 10

The s

td o

f roo

t RM

SE (s

)

Number of sensors



IMTSMDTSM







02468

1012141618

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Spee

d (m

s)

Travel time (s)


020406080

100120140160180200

0 50 100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

MA

RE (

)











M1

min 1119872

119872

sum

119898=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(sum119870

119896=1 ETT119898119896) minus GTTT119898

GTTT119898

10038161003816100381610038161003816100381610038161003816100381610038161003816

(5a)

subject to119873

sum

119894=1119909119894

= 119870 (5b)

119909119894

isin 0 1 (5c)


where 119909119894

= 1


(5d)

Link

SectionCell Section

Sensor


119904119896

=

119910119896+1 minus 119910

119896minus12

119896 = 2 119870 minus 1

1199101 +1199102 minus 1199101

2 119896 = 1

119910119870

+119873 minus 119910

119870

2 119896 = 119870

(5e)

ETT119896

=119904119896

V119896

sdot 119897 (5f)



119894= 1






1covers 45 cells








Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Time (s)

Posit

ion

(m)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000









Arrival rate(vph)


40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900




0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)








0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1012141618202224262830

0 1 2 3 4 5 6 7 8 9 10

Mea

n ab

solu

te er

ror (

s)M

ean

abso

lute

relat

ive e

rror

(s) (

)

Number of sensors

0

02

04

06

08

1

12

14

16

18

The s

td o

f MA

E (s

)

0

5

10

15

20

25

30

35


00

02

04

06

08

10

12

14

16

18

The s

td o

f MA

RE (s

) (

)

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10

Root

mea

n sq

uare

erro

r (s)

Number of sensors

IMTSMDTSM

002040608

112141618

2

0 1 2 3 4 5 6 7 8 9 10

The s

td o

f roo

t RM

SE (s

)

Number of sensors



IMTSMDTSM







02468

1012141618

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Spee

d (m

s)

Travel time (s)


020406080

100120140160180200

0 50 100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

MA

RE (

)











M1

min 1119872

119872

sum

119898=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(sum119870

119896=1 ETT119898119896) minus GTTT119898

GTTT119898

10038161003816100381610038161003816100381610038161003816100381610038161003816

(5a)

subject to119873

sum

119894=1119909119894

= 119870 (5b)

119909119894

isin 0 1 (5c)


where 119909119894

= 1


(5d)

Link

SectionCell Section

Sensor


119904119896

=

119910119896+1 minus 119910

119896minus12

119896 = 2 119870 minus 1

1199101 +1199102 minus 1199101

2 119896 = 1

119910119870

+119873 minus 119910

119870

2 119896 = 119870

(5e)

ETT119896

=119904119896

V119896

sdot 119897 (5f)



119894= 1






1covers 45 cells








Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Time (s)

Posit

ion

(m)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000









Arrival rate(vph)


40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900




0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)








0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02468

1012141618

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Spee

d (m

s)

Travel time (s)


020406080

100120140160180200

0 50 100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

MA

RE (

)











M1

min 1119872

119872

sum

119898=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(sum119870

119896=1 ETT119898119896) minus GTTT119898

GTTT119898

10038161003816100381610038161003816100381610038161003816100381610038161003816

(5a)

subject to119873

sum

119894=1119909119894

= 119870 (5b)

119909119894

isin 0 1 (5c)


where 119909119894

= 1


(5d)

Link

SectionCell Section

Sensor


119904119896

=

119910119896+1 minus 119910

119896minus12

119896 = 2 119870 minus 1

1199101 +1199102 minus 1199101

2 119896 = 1

119910119870

+119873 minus 119910

119870

2 119896 = 119870

(5e)

ETT119896

=119904119896

V119896

sdot 119897 (5f)



119894= 1






1covers 45 cells








Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Time (s)

Posit

ion

(m)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000









Arrival rate(vph)


40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900




0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)








0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Time (s)

Posit

ion

(m)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000









Arrival rate(vph)


40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900




0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)








0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Arrival rate(vph)


40080012001600

6

500 600 1100 1700 2900 3000500 600 2000 2300 2500 30001000 1100 1300 2100 2500 2900lowast200 500 800 1600 2500 2600

40080012001600

5

1100 1200 1300 1500 1700200 600 1100 1800 2200

1000 1200 2000 2300 2900lowast100 600 1600 2200 2800

40080012001600

4

500 800 1900 2300500 2100 2700 3000200 500 1100 2800200 400lowast 600 2900




0

5

10

15

20

25

30

0 400 800 1200 1600 2000

MA

RE (

)

Arrival rate (vph)








0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

20

25

30

35

40

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)

4 sensors


5 sensors

6 sensors

0

5

10

15

20

25

30

35

40

45

400 800 1200 1600M

ARE

()

Arrival rate (vph)

0

5

10

15

20

25

30

35

40

50

45

400 800 1200 1600

MA

RE (

)

Arrival rate (vph)









Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Arrival rate(vph)


40080012001600

6

600 700 900 1500 2600 3000700 1000 1300 1400 2300 27001500 1700 1800 2100 2600 2800800 1200 1400 2300 2400 2800

40080012001600

5

1400 1900 2200 2500 3000900 1300 2700 2800 3000900 1300 1400 2500 2900700 800 1600 2400 3000

40080012001600

4

1200 1500 2200 2800600 800 1100 1300

1200 1500 2700 28001300 1500 2600 3000











Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000



10111213141516171819

0 400 800 1200 1600

MA

RE (

)

Arrival rate (vph)







Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Posit

ion

(m)

500

1000

1500

2000

2500

0

2

4

6

8

10

12

14

16

Time (s)

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Arrival rate(vph)


40080012001600

6

200 300 500 1100 1400 1800400 500 1200 1400 1900 2800500 600 1900 2200 2600 2900lowast500 900lowast 1300 1400 2100 2600

40080012001600

5

600 700 1300 1500 2200600 900 1100 1500 1900100 500 1800 2100 2400900lowast 1000 1900 2300 3000

40080012001600

4

600 1000 1100 1700400 500 1400 2600100 500 1300 1900300 700 1000 2100


5 Conclusion


00

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800

MA

RE (

)

Arrival rate (vph)









Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Simulation-Based Sensor Location Model ...€¦ · Research Article Simulation-Based Sensor Location Model for Arterial Street QinxiaoYu,NingZhu,GengLi,andShoufengMa