Traffic Reliability Model of Linked Intersections Based on Demand/Capacity

Runlin CAI1, 2 and Xiaoguang YANG1 1 School of Transportation Engineering; Tongji University; Shanghai, 200092; PH

13811458606; FAX (010) 58323100; email: cairunlin@vip.tom.com 2 China Academy of Urban Planning & Design; Beijing 100037 ABSTRACT

Increasing traffic congestion arouses concern with the reliability of transportation systems. In particular, the interference between adjacent intersections often results in the devastation of local road networks. This paper first defines the category of linked intersections and concludes the influencing factors and determination standard on the nature of linking. The paper proposes the concept of linked-intersection reliability, and advances the research category due to the construction of linked intersections. Furthermore, according to the methodology that functional failure should be determined by demand-capacity, this paper develops the model for intersections based on D/C, the model for road section based on congestion level, and combined model based on series system. Last, the applicability of the model is analyzed.

KEY WORDS

Traffic reliability, Linked intersections, Traffic congestion, Demand/Capacity, Probability

INTRODUCTION In the recent years, traffic congestion in major cities of China becomes more

and more serious. In addition to the expanding traffic flow, the traffic system reliability, the lack of flexibility in handling traffic flow changes and incidents are also responsible for the deterioration of the traffic situation. Actually, the poor traffic situation at one intersection could affect adjacent road sections and other intersections, which could result in the inefficiency of local road network. Currently, the urban road network is intensifying and the intersection spacing is growing shorter and shorter. Take Siping Road as an example, which is one of main roads in Shanghai. Its average distance between intersections is only 180 meters. Under such conditions the relevance between intersections cannot be ignored. And it is impossible to resolve the congestion if one intersection is studied and optimized separately. Therefore, it is necessary to combine linked intersections as a whole system and study the reliability so as to improve the overall traffic efficiency of local road system.

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ANALYSIS OF INTERSECTION RELEVANCE Definition

Linked intersections refer to two intersections, which are closely linked. The relevance lies in that the distribution of the arrived vehicles would take on a grouping state at the downstream intersection, when the traffic volume of the main road is overly large because of the short intersection spacing. And the traffic situation at the upstream would be thus probably influenced by the vehicles queuing at the downstream.

The intersection spacing differs extremely in current Chinese urban road networks. In certain road sections the spacing is only tens of meters, which cannot even meet the requirements of arrived vehicles in one cycle to queue. During time intervals of large traffic flow (e.g. peak hours) the traffic at the downstream intersection is seriously oversaturated, which also makes a great impact on the upstream traffic, thereby leading to congestion and even devastation in local road network.

Influencing Factors of Intersection Relevance

The following main factors should be taken into consideration when analyzing the degree of the intersection relevance.

The queuing space for the connecting road section. Studies on the discrete state of traffic flow suggest that the growing driving distance can increase the degree of traffic dispersion. Because of the growth of driving time between intersections and the influences from the branch roads out of the controls by the traffic signal and the traffic flow pulling into and out of the exit or entrance, the vehicles arrive at the downstream intersection will always show a random distributing state, which gradually weakens the relevance between upstream and downstream intersections. Besides, the downstream queuing vehicles left by the last signal cycle are another factor that influences the queuing space in road section. It changes dynamically according to different coordination phrase separation. Because the upstream vehicles will regard the waiting vehicles as the “stop line” before queuing, the actual spacing will be thus gradually shortened by the growing queue length of waiting vehicles. The spacing between intersections and the initial queuing length at the downstream jointly determine the capacity of the connecting road section to accommodate the traffic flow from upstream.

Demand and capacity ratio. It is determined by the demand of the upstream traffic flow and the supply of roads. When the traffic of main roads is heavily burdened, the randomness of arrived vehicles will reduce greatly because of the growing traffic flow. The traffic flow released at the upstream intersection will take on a pulse saturation flow, which seriously impacts the downstream intersection and connecting road sections.

Coordination of the phases. When the common cycle at upstream and downstream intersections are decided, the offset will become the key factor in coordinating the arriving and releasing of traffic flow. Proper offset will enable the connecting road section to maintain enough space and reduce queuing vehicles.

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It should be pointed out that due to the directionality of traffic flow and controlling conditions the two-dimensional relevance of the same road section is different and in most cases it could differs greatly, which can be given consideration respectively.

Determination of Intersection Relevance

There are two methods to determine the intersection-group relevance, namely volume method which doesn’t take the speed into account and travel time method which takes the speed of traffic flow between upstream and downstream intersections into consideration.

The volume method introduces the concept of coupling index. Coupling index I refers to the simple ratio of the main road’s traffic volume Q to the intersection spacing L .

The travel time method is an analysis based on the influencing factors of relevance, which takes the intersection spacing and the flow volume of the main road as independent variable to determine the coupling correlation between traffic relevance of intersection-group, distance and traffic volume. The specific functional form is as follows:

11

5.0

1

maxn

iiq

nq

tI

(1)

In the function I refer to the relevance index between intersections; n is the

quantity of branches of the traffic flow putting in from the upstream intersections.

maxq is the traffic volume of straight running vehicle from the upstream

intersection in the direction of the main line and is the maximum value in iq .

n

iiq

1

is the total traffic volume at downstream intersection and for cross road

intersection 3n . t is the travel time between two intersections, equaling to the intersection

spacing L divided by the average velocity V . Regardless of the queuing length of vehicles and its dynamic influence on the

speed of traffic flow, the intersection relevance degree is reflected by the traffic density between upstream and downstream intersections in the volume method. And the travel time method takes the influence of traffic flow on the driving speed and queuing length of vehicles into consideration, which is of great significance to the practical function of linked intersections.

The critical value of relevance determination can be evaluated according to the actual operation of intersection group. Generally considering 5.0I means high traffic relevance and 5.0I means low traffic relevance. (Hang, 2002)

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RESEARCH CATEGORY OF LINKED INTERSECTIONS RELIABILITY According to the traditional definition in systematic reliability theory,

reliability is the ability with which products can actualize its function under the given conditions and within the given time. The probabilistic measurement unit of reliability is called degree of reliability. Referring to the systematic definition, the intersection reliability can be defined as the probability of the condition that the single-intersection traffic operation satisfies the normal operation under required conditions and within the required time.

And in the study on reliability there are only two the working conditions of system or unit, i.e. normal condition and disabled condition. (Guo and Wu, 2002) Given that the probabilistic of normal condition is wP and that of disabled condition

is , that is to say the reliability of system is fw PPAP 1)( .

Factors that influence the traffic reliability are complicated. In order to simplify the research, the paper defines the content of research in the following three aspects. Firstly, the subject of research is intersection in microcosmic level. In the early researches, the studies on reliability are set about mostly from the category of road network and O-D pair and simplify the nodal points and road section contrarily. The content of this paper can be considered as a complement to the former researches. Secondly, the disabled condition of traffic function is defined. To the traffic system of urban roads, traffic congestion is the most prominent representation of the failure of part road network. The results caused by traffic congestion may be traffic accidents, delay of going out, the increasing of fuel oil consumption and the deterioration of exhaust emissions etc. The basic function of traffic system is not to be actualized. Therefore, it is appropriate to define the traffic congestion in some degree as the disabled condition of traffic system. Thirdly, the research is positioned on the problems of traffic reliability caused by recurrent congestion. Recurrent congestion is the traffic jams and congestion that is caused by the change of traffic demand and capacity in the daily life, the most obvious example of which is the traffic congestion raised by concentrated going out at morning and evening rush hour.

According to the above-mentioned definition the traffic reliability of linked intersections can be defined as the capability of that the traffic system, with the linked intersection as its basic unit, meets the normal operation needs under normal traffic conditions and within given time.

Linked intersection is composed of two intersections and the connecting road section. Linked intersection is regarded as directional and for this reason there is only one connection road section between intersections, namely the one that has the same direction as the linked intersection. Thus we can divide the linked intersection system into three units: two intersections and the directional connecting road section between them.

TRAFFIC RELIABILITY MODEL BASED ON D/C

Determination Standard of Functional Failure

Functional failure determination standard of linked intersection system is the key to establish the reliability model. Traffic demand and capacity are always the principal contradictions and this paper regards the relation of demand and capacity as

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the basic standard of functional failure determination. Regardless of the interference of un-normal factors such as traffic accidents,

road construction and disasters, the entry lane congestion of intersection can be regarded as the disabled condition of the entry lane function. Studies on the congestion condition emphasize how to quantize and determine the congestion condition, i.e. to what extent the congestion can be considered as the traffic disabled condition. Nowadays there have been a number of studies on quantifying the congestion level and its space-time expansion, such as congestion level index and K-Factor. (Liang, 2002) The congestion level in microcosmic view is usually reflected by the space-time degree of saturation of the road network and the severity of its consequence.

(1) Determination of functional failure on single intersection

Intersection is the joint point of two roads, whose traffic function depends on its entry and exit lanes. Under normal traffic conditions congestion of intersections always takes place in the entry lane.

And the traffic capacity does not depend on road conditions only. With the growing traffic volume, the random disturbance of the flow, the mutual influence between automobile, non-motorized vehicle and pedestrian will increase too. Especially when the traffic demand approximates to or exceeds the traffic capacity, this disturbance to the practical capacity can not be neglected.

Former research suggests that the intersection should be congested when CDD 9.00 (Lomax, 1988). We will use to VISSIM Simulation to verify the

feasibility of this standard. We establish an abstract model of an intersection and add different traffic

demand to its far enough upstream. After we take different signal cycles at the downstream and set up appropriate detector to obtain the benefit values such as delay, queue length, average cut-off interval, we get the following variation chart and compare the variation tendency.

From the Figure 1 we can see obviously that when D/C is over 0.9, delay, and average queue length will increase sharply and the traffic condition will also deteriorate obviously. Furthermore it is to point out that the disturbance from pedestrian and bicycles are not taken into account under the simulation conditions, which has more important effects on the capacity and traffic condition. That is to say, when D/C increases to about 0.9 in practice, the congestion at intersection will be worse.

Figure 1 Variation of average straight-going delay and queue in different cycles

0

10

20

30

40

50

60

70

80

90

100

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

D/C

Del

ay（

s）

C 40s

C 60s

C 80s

C 100s

C 120s

C 140s

C 160s

C 180s

0

50

100

150

200

250

300

350

400

450

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

D/C

Queue(m)

C 40s

C 60s

C 80s

C 100s

C 120s

C 140s

C 160s

C 180s

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Thus, we can define: When CDD 0.1~85.00 , the intersection will be

congested. 0D can be called as the volume limit which can guarantee the normal

operation of road sections and intersections. According to the extent of mutual interference between the practical management conditions of road sections and intersections and the vehicle flow, it can be evaluated within this limit.

(2) Determination of functional failure on connecting road sections

Congestion is always related to the feature of volume and traffic volume and capacity are the two main features. Moreover, the consequences of congestion are related to the operation characteristics, which include speed, density, travel time and delay. Considering all the factors and the level of reliable service on the key road section, we regard traffic volume as the main factor of the former and speed as that of the later. We establish therefore the congestion model that combines the two main factors which evaluates the urban roads with a certain traffic volume. The specific form of the model is as follows:

1001

b

LV V

VCG

(2) b

f

LL S

SVV

1

)1(1

(3) VCG －degree of congestion，b－ratio of speed reduction

V－traffic volume， LV －actual traffic volume

fS－maximum speed of free flow，

LS －actual flow speed

iimpb (4)

ip is the proportion of the traffic volume i and coefficient im means the

weight of vehicles I in reducing the whole traffic flow speed. The value of parameter b and can be corrected according to the data obtained under conditions of different types of vehicles and road conditions.

The exponential term b＋1 in the model reflects the influence of traffic volume movement under normal road and controlling conditions. LV and LS reflect respectively the maximum actual traffic volume and speed. The level of congestion is a quantitative description of the free movement loss of traffic volume and the latter explains the variation of traffic volume speed with the growing traffic flow under normal road and controlling conditions.

Research shows that the road section can be regarded as congested when the index of the congestion level reaches more than 80 (Maitra ,1999).

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Establishment of Reliability Model

(1) Reliability model of single intersection

Suppose that the function condition of the road operation depends on two elements, i.e. traffic capacity C and demand flow D, we can establish the function Z of road unit as

D

kCDCgZ ),( (5)

In this formula, k is the coefficient of congestion, which can be evaluated according to the value interval of the congestion judgment in last chapter and the practical traffic flow condition.

We can determine on basis of the value of Z whether the road condition satisfies the demand of some definite function:

When 1Z , the road section condition is normal； When 1Z , the road section is congested; When 1Z , the road section is in a limit condition. Given that the probability density function )(ZfZ

of road unit function Z is

known, the reliability of entry lane lR can be calculated through the following formula:

1

)()1( dZZfZPR Zl (6)

If the probability of the congestion condition at entry lane is named as

congestion probability fP , then

1

0)()1( dZZfZPP Zf

(7)

Because the normal operation condition and disabled condition are in

opposition to each other, the reliability R and the probability of congestion satisfy 1 fl PR .

Researches show that the traffic demand appears to be a normal distribution within a long time unit and with the different demand volume, the traffic capacity appears to be a normal distribution or a skewed normal distribution.[3] Under conditions of normal use of roads, which means only the road lasting disturbance is taken into account, but not the abnormal happenings as accidents, disasters, if it is known that the probability density function of traffic capacity kC and demand flow D is )(kCfkC

and )(Df D and kC and D are mutual independent, then

)()(),()( DfkCfDkCfZf DkCZZ (8)

And the congestion probability at entry lane is

1/

)()()1()1(DkC

DkCf dCdDDfkCfD

kCPZPP (9)

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As the cross section, the urban road intersection consists of several entrance lanes from different directions. With the different structure of intersections and level of road combination, there are two models to calculate the overall traffic reliability at the intersection.

The first model: The overall benefit of intersection depends on the traffic flow from a certain number of flow directions. Guarantee of the traffic capacity at the intersections on arterial roads priors to that on sub-trunk roads or branch roads. In the eyes of reliability, the intersection can be regarded as a series system combined with several entrance lanes of the main roads, which means the reliability of these several important entrance lanes should be guaranteed. Given that the overall traffic reliability at the intersection is R, which includes n critical entrance lanes, whose traffic reliability is iR , then the overall traffic reliability at the intersection is

n

ifi

n

ii PRR

11

)1( (10)

The second model: The road combination levels of the intersection are

consistent and the change of the flow with the time tends to harmonization. Besides that, these exists mutual disturbance between each flow direction, for example at the intersection of sub-trunk roads or branch roads, and then we can see the intersection as a road unit and determine the congestion and calculate the reliability according to the function Z . The congestion probability at the intersection is

1

0)()1( ZdZfZPP Zf

(11)

And the overall traffic reliability at the intersection is

fPR 1 (12)

(2) Reliability model of connection road sections

The traffic reliability model of connecting road sections is established on the basis of Bhargab model’s determination standard of road congestion. Suppose a given linked intersection ALB made up of the upstream intersection A, the connecting roads section L and the downstream intersection B. Firstly we establish the congestion probability model of the connecting road section L.

According to Bhargab model of congestion level, we suppose that the road section can be considered as congested, when the value VCG is above the threshold

value 0CG . Then the congestion probability of the connecting road section L is fLP .

0

)()()( 0 CGfL CGdCGfCGCGPP (13)

Within the given time T, which is divided into n smaller time intervals, we can

determine in each small interval whether the road section is congested or not. There are only two conditions of a road section, so we can determine the congestion

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probability through the ratio of congested time to the total time as follows.

T

t

CGCGPP CGCGii

fLV

0:

0 )( (14)

In this fuction it means the time interval i，i＝1,2,…,n；

0: CGCGi

i

V

t is the summation of congested time intervals

T is the given time interval Then traffic reliability of road section between intersections is

fLL PR 1 (15)

(3)Reliability model of linked intersections

The function of the three basic units in the linked intersections system should be guaranteed at the same time. That is to say, the traffic function of linked intersections can be realized only when the basic units functions are put into full effect at the same time. Therefore, the upstream intersection A, the connecting road section L, and the downstream intersection B can be seen as the three basic units of series system. According to the reliability calculation model of series system, the reliability R of the linked intersection ALB is

BLA RRRR (16)

Analysis of Model Applicability The reliability model based on D/C ratio requires the known traffic demand

and capacity distribution, both of which can be obtained through some detecting methods and statistic analysis. The change of the traffic demand within a short time internal (such as one hour) is not obvious and even to be neglected, while the characteristic of distribution within a long time internal is obvious. Therefore, in this perspective models are applicable to the analysis on traffic reliability in a long time internal.

In addition, considering the model’s features, the applicable conditions and requirements of traffic reliability model of linked intersections based on D/C are as follows:

(1) Relative high requirements to the condition of vehicle traveling on road sections, for example the linked intersections of main roads in the city;

(2) Relative large intersection spacing and queuing overflow is not the major problem;

(3) Applicable to evaluate the reliability of any time period. The wholeness and sustainable development of the traffic system play a very

important role in applying the model, which has remarkable significance for the construction and reconstruction of traffic facilities.

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CONCLUSION The paper analyses and defines the traffic reliability of linked intersections,

and established the reliability model based on Demand/Capacity, which can be used to evaluate linked intersections under the signal-controlled condition. The follow-up studies will further focus on the quantitative analysis of influence on the traffic reliability exerted by the specific structure of linked intersections, including road section flow wave, coordinated signal controlling condition and so on. The study on this area will play an important role in optimizing the control system.

REFERENCES

Guo, B., and Wu, X. Y. (2002). “System reliability analysis.” National University of

Defense Technology Press, Beijing. Hang, M. S. (2002). “Theories and methods of real time adaptive control on

intersections group.” Ph.D. thesis, Tongji University, Shanghai. Liang, Y. (2002). “Operation evaluation of road network in metropolis.” M.S. thesis,

Beijing University of Technology, Beijing. Lomax, T. J. (1988). “Methodology for estimating urban roadway system

congestion.” Transportation Record Research, No.1181, Transportation Research Board, Washington, D.C.

Maitra, B. (1999). “Modeling congestion on urban roads and assessing level of service.” Journal of Transportation Engineering, Vol.125, No.6.

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