Modeling Queues At Signalized Intersections

FrancescoViti, Henk J.Van Zuylen

Modeling Queues At Signalized Intersections

Francesco Viti Delft University of Technology Faculty of Civil Engineering and Geosciences Transportation Planning and Traffic Engineering Section P.O. Box 5048 2600 GA Delft The Netherlands Telephone: 31.15.2784912 Fax: 31.15.2783179 E-mail: [email protected]

Henk J. van Zuylen Delft University of Technology Faculty of Civil Engineering and Geosciences Transportation Planning and Traffic Engineering Section P.O. Box 5048 2600 GA Delft The Netherlands Telephone: Fax: 31.15.2783179 E-mail: [email protected]

This paper contains 5400 words and 12 figures

Submitted for presentation at the 83nd Annual Meeting of the Transportation Research Board January 11-15, 2004, Washington D.C. and for publication in Transportation Research Record

Submission date:

FrancescoViti, Henk J.Van Zuylen

ABSTRACT Traffic control is one of the most effective techniques for a road manager to adapt the network capacity to the flows. It also gives possibilities to modify users’ choices of modes and routes. An important area of research is formed by the dynamic assignment problem, taking into account the effect of delays at controlled intersections. This requires realistic queuing models. If the degree of saturation of an intersection is close to or larger than 1.0 the queue size has a dynamic character. That is also the case when the initial queue is non-zero. The existing queue models have important limitations. A Markov model has been developed to calculate the dynamics of the queue. It appears that the variance of the queue length is an important determinant of the queue dynamics. This paper first analyzes the case of decreasing queues showing that available models in this case underestimate the mean values being hardly applicable in modeling time-of-day dynamics. An approximate expression for the case of decreasing queues is provided and analyzed with its parameters. Finally the same approach suggests an approximate formulation also for the evolution of the standard deviation along time, providing a better knowledge of the uncertainties at signalized intersections. The analytical model introduced in this paper is suited for application in dynamic assignment problems.

FrancescoViti, Henk J.Van Zuylen

INTRODUCTION Delays at controlled intersection are an important component of the travel time in urban networks. Therefore, route choice and departure time choice are influenced by traffic control. Modeling this interaction requires knowledge about the dynamics of traffic queues. Since travelers are not only choosing modes and routes that minimize their costs, but also want to reduce uncertainty, also knowledge about the variance of the travel times is necessary if one models their behavior. The existing models for queues at an intersection can be distinguished in microsimulation models, macroscopic functions and mesoscopic models, e.g. Markov models (1), (2). This paper uses the approach of a Markov model because such a model makes it possible to analyze in much more detail the dynamics of the queue and its variance, without the need of repeated simulations. The objective is to derive macroscopic functions for the queue and variance of the queue and to calibrate the coefficients. Newell (3) provided in 1971 a rigorous method for calculating the time-dependent queues in a general context using the diffusion equation. He was the first to characterize the queue state distribution as normally distributed and attests the importance to give an estimate also of the variance. His assumption of a normal probability distribution limits the validity of his model, since the queue length distribution has a standard deviation that is often higher than the expectation value. A normal distribution would mean that the probability of negative queue lengths is non zero. On the other hand his method doesn’t lead to a simple expression and the solution can be found only through numerical approximations. For the situation that initially the queue is zero, Kimber and Hollis (4) developed a suitable heuristic analytic model for queue length and delay using the coordinate transformation technique. Their model has been applied in e.g. the TRANSYT program (5). Troutbeck and Blogg (6) compare the two above mentioned methods with simulated data using also non-zero initial queues. They start from a known value of the initial queue and analyze the evolution of the queue state along time. They finally conclude that the Kimber and Hollis model cannot be used for decreasing queues since it leads to a discontinuity in the discharge rate and the queue prediction can give an error of more than 4 vehicles per cycle, while the Newell method is consistent with results from simulations but difficult to use for practitioners. Akcelik (7) has developed a macroscopic function, which is probably used most for practical applications. For the situation that the initial queue is non-zero, Catling (8) gives an approximation, which is basically just a deterministic decrease of the queue if the degree of saturation is less than 1.0. This assumption does not take into account that arrivals of vehicles at the intersection have a stochastic character that gives deviations from the linear decrease as soon as the standard deviation of the number of arrivals becomes equal to the length of the linear decreasing queue (1). This paper develops a function for both the decreasing queue and its standard deviation starting with a Markov model for fixed time traffic control (1), (2). The approach is applied to isolated intersections with fixed time control and Poisson arrivals. The results of that model are used to calibrate analytical functions for the queue length and the standard deviation. In section 2 a short overview is given of time dependent queue models. Section 3 introduces the Markov model, section 4 discusses some results from that model. In section 5 the mathematical form of the macroscopic formula is derived and the next section gives the result of the calibration of the parameters. In section 7 the same is done for the standard deviation. Section 8 discusses possible extensions of the method to intersections in a network, with other statistical properties of the arrival process and to traffic dependent controlled intersections. Section 9 gives some conclusions and directions for further research. TIME-DEPENDENT QUEUE MODELS A most important component in the model for the delay at controlled intersection is the random or oversaturation delay component. If the green phase is not long enough to handle all the cars, the queue at the end of the green phase is not empty. If the number of arrivals is structurally higher than the number of cars that can depart during the green phase, this term will grow every cycle and it will depend on the duration of the period over which the delay is calculated. If the number of arrivals is, on the average, less than the number that can depart in green, the green phase is under saturated on the average, but due to stochastic variations in the number of arrivals, there is a finite probability that still one or more cars will have to stop twice because the green phase could not handle the whole queue. The queue that evolves in this case is defined as the overflow queue. Modeling overflow queue sizes is a research topic that has been studied since the work of Webster (9). Akcelik (7) formulated the expression that is most frequently used at present. He provided a continuous formulation of the queue evolution using the coordinate transformation technique:

FrancescoViti, Henk J.Van Zuylen

cT 4

Q(T ) = • • • •

•

 12 ⋅ ( x − x0 )  2  x − 1 + ( x − 1) +  cT  

(1)

where T is the time step (expressed in number of cycles); Q is the number of vehicles in queue at the starting of a green phase; c = S.g / C is the capacity of the arm, equal to the saturation flow S multiplied by the green ratio g / C; x is the degree of saturation, ratio between the flow and the capacity; S ⋅ g is the limit value of the degree of saturation above which the stochastic effects are x0 = 0, 67 + 600 relevant.

This formula is the most used model in case of flows approaching capacity and initial queue Q0=0. On the other side it fails in all the other cases were the queue starts from a non-zero initial value limiting its applicability. A recent extension of the former model for including the effect of an initial queue can be found in the aaSIDRA manual (10). The authors add to the flow rate q a term that takes the residual queue generated in the previous intervals into account. The model uniformly distributes the extra flow over the calculation period adding Q0/T where Q0 is the expected residual queue from the previous time step and T is the time step length. A more detailed description of existing models can be found in Rouphail et al. (11). The case of decreasing queues hasn’t more accurate analytical formulation than the one provided by Catling (9) who models the evolution as a linear function that follows at the starting the deterministic decrease represented by the linear function:

Q = Q0 + ( x − 1)cT

(2)

until it reaches the equilibrium value that depends on the degree of saturation and not on the initial value. The Highway Capacity Manual (HCM 2000) (12) derives an applicable formula for the delay also at cases of non-zero initial queues and platoon arrivals. Basically, it uses formula (2) to calculate the delay. As will be shown in the next sections, this approximation is not valid if the standard deviation of the queue length becomes equal to the expectation value. At that moment stochastic effects will influence the development of the queue, just as it did for the case that the initial queue is zero. A better model is needed to describe properly time dependent queues, not only for calculating the delay but also for several other aspects (spillback effects, queue duration, lane design etc.). Olszewski (2) studied queues using a Markov model. He showed that the queue behavior is affected by the variance of its distribution and obtained good results of modeling the mean of the queue with the same technique. However, up to now no analytical macroscopic models can be considered valid for all the relevant cases: e.g. for optimization of traffic signals, calculation of queue sizes in dynamic situations, estimating travel times and uncertainty in travel times. This paper contributes to the development of a more comprehensive set of mathematical formulas for this purpose.

THE MARKOV MODEL FOR CONTROLLED INTERSECTIONS The average length of the queue E[Q(0)] at the end of a (fixed time) green phase on a controlled intersection is one of the terms in the formula that determine the expected value of the delay, E(W) (13):

E[W ] =

q⋅r 2 1 1  {r + E[Q(0)] +  1 + } 2(1 − y ) q s  1− y 

with • • • •

r = duration of the red phase (s) q = flow of an intersection arm (veh/h) s = saturation flow (veh/h) y = load ratio q/s.

(3)

FrancescoViti, Henk J.Van Zuylen

For the value of E[Q(0)] several expressions have been developed, e.g. (7), (8), (9), (10), and (14). In the case of an oversaturated intersection this quantity will grow from cycle to cycle. That happens also for intersections that are still undersaturated, but where the queue length is lower than the equilibrium value. If the queue length is longer than the equilibrium value, the queue will decrease from cycle to cycle. Olzewski (2) studied the queue length dynamics with a model in which the probability distribution of the queue length is calculated from cycle to cycle:

+∞

P(n, j ) = ∫ ds' −∞

2 2 1 e − ( s '− s ) / 2σ σ 2π

j + s 't g

(

∑ p lP n − 1, j − l + s ' t g

l =0

)

(4)

with • • •

P(n.j) σ pl

= probability of a queue of j vehicles at the end of the nth green phase = the variance of the saturation flow and = probability of l arrivals in the cycle.

Figure 1 gives an example of the probability distribution, starting with an empty queue. It is clear from visual inspection that the probability distribution is not normal, as Newel assumes (3). [Figure 1] An alternative to a Markov model for the probability distribution, microscopic simulation can be done. It is rather unlikely that anything will come out of micro-simulation models that significantly differs from the Markov model since most microscopic simulation models use the same assumptions about the arrivals and departures of cars as the Markov model does. [Figure 2] Figure 2 shows the results of a calculation with the Markov model. It is remarkable that expectation value of the queue is of the same order of magnitude as the standard deviation of the probability distribution. If we start the Markov model with a zero queue, the standard deviation of the initial distribution is zero too. As can be seen in figure 1, the zero-queue remains the most probable also after several cycles, but the probability that the queue at the end of the green phase is bigger than zero begins to rise. If we start the simulation with a queue length of 4 cars, the dynamics of the queue differs from the graph of figure 2. The behavior of the queue in fig. 3 after the moment that the queue size is 4 is different from the behavior of the queue starting with a queue 4. This is due to the fact that the state of the queue with length 4 and variance 0 is different from the state in which the queue length is 4 and the variance is unequal to zero. In figure 3 the graph of the initial queue starts at cycle 0 and the other graph with the initial queue 4 starts at the cycle where the first queue arrives at the expectation value 4. [Figure 3] If we would try to verify the Markov model, we could use a microscopic simulation program or observations of queues in reality. Since the standard deviation of the queues is about the same as the expectation value, the number of observations or simulations has to be high in order to determine the expectation value with sufficient accuracy. For a 10% accurate estimation, the number of observations has to be at least 100. Troutbeck and Blogg (6) used microscopic simulation to verify the models of Kimber and Hollis (5) and Newel (3) and found that even with 500 simulation runs the resulting curves are not smooth. Apparently this is due to the large standard deviation. This makes it infeasible for real-world observations, because to do 100 observations under the same conditions is practically impossible. The simulation with the Markov model gives a reasonable good fit to Akcelik’s (7) model for the situation that the initial queue is zero. However, if the initial queue is larger than zero, strange effects occur. [Figure 4] Olzewski (2) found that a queue that started slightly higher than the equilibrium value, first decreased until under the equilibrium value and then asymptotically approaches the equilibrium value. This behavior is more general; it applies to all queues that have an initial value larger than zero and a zero initial standard deviation. Figures 2, 3

FrancescoViti, Henk J.Van Zuylen

and 4 give examples of this peculiar behavior and figure 5 shows the dynamics of the queue for different initial values. [Figure 5] The explanation of this peculiar phenomenon is that in the initial state the queue has a point distribution without standard deviation. The probability distribution gets a larger standard deviation after a few cycles and that property gives the expectation value of the queue length its asymptotic behavior that is also characteristic for Akcelik’s formula. This means that the dynamics of the queue length depends on the standard deviation. If the standard deviation is zero, the queue has a deterministic, linear dynamics. In the paper we give several examples that show the importance of the standard deviation as a determinant of the queue dynamics and how the initial value determines the queue to be, as average, deterministic for a certain period of time. If the initial state is a distribution of queue lengths instead of a known queue length, the initial linear behavior of the expectation value of the queue does not occur, at least not if the standard deviation of the initial queue distribution is of the same order of magnitude as the queue length itself (15).

PROBLEM STATEMENT In the present paper we reintroduce the problem raised in (1) and (15), i.e. how to find an analytical function that describes the behavior of the queue. The Markov model is very well suited to study the dynamics of the queue and its statistical properties, but it is not suited for the optimization of traffic signals or the calculation of travel times in realistic networks. In the present paper we will validate the results published in (1), (2) and (15) trying to derive from these results an analytical expression that could cover the cases that are still missing in the present analytical queue models. [Figure 6] Figure 6 shows the evolution of the queue in four cases where the only characteristic that changes is the initial queue (Q0=0, 8, 14, 20) using for the first case the Akcelik’s formula and using the results of a Markov chain applied in Van Zuylen and Taale (14) for the others. In the same figure are reported also the linear approximation functions as proposed by Catling (8). As it can be noted the three queue functions with a non-zero initial queue follow the linear approximation for a number of cycles increasing with the difference between the starting queue and the equilibrium one. When the cycles pass the queue behavior follows the Akcelik’s function more and more and it can be shown that they all reach the same asymptote. Van Zuylen and Taale (15) emphasize also the strange behavior noted when the initial queue is slightly higher or lower than the equilibrium value where the queue initially decreasing and changes his evolution after few cycles (as also visible in figure 3). Van Zuylen and Viti (1) explain this behavior as the result of the initial state where the standard deviation of the queue is zero. As long as the standard deviation remains small, the linear model of Catling (8) is applicable. When the standard deviation gets the same order of magnitude as the expectation value of the queue, the queue follows and exponential approach to the equilibrium value.

AN ANALYTIC MODEL FOR DECREASING QUEUES The change in the decrease of the queue length can be shown in a convincing way by the derivative of the queue with respect to time. Figure 7 shows the trend of the logarithm of the decrease of the queue for Q0 = 50 and x = 0,95. After a period in which the derivative remains constant, it assumes quickly a linear trend, which means that the queue has an exponential behavior. [Figure 7] Since the queue is constrained to follow the deterministic function at the starting and gradually becomes an exponential function a weight is applied to the two trends in order to approximate a transitional phase between the two functions. The queue size as a function of time has then three regimes: 1. Initial linear decrease

Q = Q 0 - (x-1) ⋅ c T 2.

Transition between linear and exponential decrease, which can be approximated as:

(5)

FrancescoViti, Henk J.Van Zuylen

Q = α (T) ( Q 0 - (x-1) ⋅ c T) + (1-α (T)) [Q e +γ exp(-β T)] 3.

(6)

Exponential approach to the equilibrium value

Q = Qe +γ exp(-β T)

(7)

The weight function α(T) is a continuous function that balances the linear deterministic behavior and the exponential function only in the transitional phase, assuming the value respectively 1 and 0 for the first phase and the third phase. The continuity of the function is assured by the continuity of the three functions and from its initial and final values. The function is then, even if divided into three regimes, continuous with continuous derivatives in all its points. [Figure 8]

CALIBRATION OF THE PARAMETERS The function for the queue contains two β and γ characterizing the exponential part in eq. 7. Furthermore a closed form has to be sought to describe and to specify α(T). The parameters β and γ can be calibrated considering only the data concerning the third phase. Since the function α(T) becomes 0 after several cycles keeping this value from then on, it is possible to calibrate the parameters β and γ with a simple regression model using the logarithm of the derivative of the function. Once β and γ are modeled as functions of the external variables we can analyze α(T) with an inverse problem. Knowing the first and the third regimes we search for every time step in the transition phase the value of a in the interval [0,1] so that:

a ( Q 0 - (x-1) ⋅ c T) + (1-a ) [Q e +γ exp(-β T)]=Q mc

(8)

where Qmc is the value of the queue calculated with the Markov model. After determining α, we can associate an approximate expression and determine the parameters in this expression. This is a fully heuristic approach, because no reason exists why a certain form of a function for α(T) should come out. The parameters β and γ can indirectly be calculated using a linear regression on the latter function knowing that in the third phase:

 ∂Q  log   ≅ β 't + γ '  ∂t 

(9)

Thus, the parameters β and γ will be:

β = β ', γ =

eγ ' β'

(10)

Further analysis shows the dependence of β and γ on the degree of saturation and the initial queue. The degree of saturation influences mainly the β’ while the initial value will modify it only offsetting the function into a parallel one. It is then possible to associate a closed form mathematical expression that reproduces the parameters β’ and γ’, and consequently β and γ. Experimental analysis suggest the following simple relationships:

β = −6.67 ⋅ (1 − x)2 γ ' = (1 − x) ⋅ Q0 + 1.5

(11)

The variable x, the degree of saturation, includes also the other variables acting in the problem, green time and saturation flow.

FrancescoViti, Henk J.Van Zuylen

The third phase of the queuing behavior is then modeled with a closed form function. In figure 9 an example is shown of a queue behavior calculated with the Markov model and the exponential function representing the third part. The exponential function perfectly follows the queue behavior for a large range of time and leaves it only approaching the initial state. [Figure 9] For the transitional phase we have introduced the weight function α(T) that gradually transfers the system from a linear – deterministic - behavior to the exponential –stochastic - one. If the exponential function is exactly tangent to the deterministic function, the function α(T) reduces to a step function that associates the value 1 to the linear part at the starting and after the intersection point gives 1 to the exponential distribution. A reasonable approximation of the transition phase can be obtained using a logistic function:

α=

1 1+ e

− µ (T −T0 )

(12)

where T0 and µ represent the two parameters of the function, one representing the position and one the extension of the transition phase. Analyzing the trends shown from the pictures we can approximate the expressions of T0 and µ1 as:

0.15 ⋅ Q0 ⋅ x 0.01 + + 5.4 (1 − x) (1 − x)3 0.11 ⋅ x µ= (1 − x) 2

T0 =

(13)

In the following figure 10 a queue evolution calculated with the Markov model is compared with the approximate expression calculated from the expressions (5)-(7) using the parameters derived from the expressions (8)-(12). [Figure 10] The initial linear decrease of the queue length is the result of the fact that the initial state has a standard deviation that is smaller than the expectation value of the queue, as stated in the last paragraph of section 3. To proof this statement another simulation was made with a queue starting from zero. This queue increases till the standard deviation is comparable with the mean. After that point the degree of saturation is reduced from 0.975 to 0.95. The expectation value of the queue decreases directly in an exponential way, without the initial linear decrease. Figure 11 shows the evolution of the expected value of the queue and its standard deviation. [Figure 11] The simulation gives an evolution of the expectation value of the queue in which the initial linear is missing, which is consistent with the assumption that the linear part only is an initial stage when the standard deviation is still less than the expectation value. The formula that applies to the whole time after the maximum queue value in figure 11 is:

Q(T ) = Qe + (Q0 − Qe ) ⋅ e− β T

(14)

where the parameter β assumes exactly the same expression (10) . These results show that for the prediction of decreasing queues under decreasing traffic demand, the simple exponential formula (13) can be applied.

The analysis of the data shows a slight variation of µ w.r.t. the initial value but this dependence is negligible compared to the value itself and the variability w.r.t. the degree of saturation. 1

FrancescoViti, Henk J.Van Zuylen

A MODEL FOR THE STANDARD DEVIATION Using the Markov model it is also possible to calculate the evolution over time of the standard deviation, according also to (1), (2), and (15). As we can notice from the diagrams the standard deviations all start from zero, according to the starting hypothesis and follow initially a trend well approximated by √T. This is rather obvious, since we assume a Poisson arrival pattern where the standard deviation of the total number of arrivals is the square of the number of arrivals. Following the same analysis we have done for the mean value we can similarly model the behavior of the standard deviation of the queue over time with an approximate expression similar to the one adapted for the mean value. 1.

Initial square root behavior:

Q= 2.

c ⋅ x ⋅ T + Q0 + 0.01 ⋅ Q0

2

(15)

Transition phase:

Q = (1-αSD (T)) c ⋅ x ⋅ T + Q0 + 0.01⋅ Q0 +α SD (T) [QSD,e +γ exp(-βT )] 2

3.

(16)

Exponential approach to the equilibrium value:

Q = QSD,e +γ exp(-β T)

(17)

The third phase differs also in the asymptotic value, which is slightly higher than the Qe. A multiplicative factor can be associated to Qe for the asymptotic value of the standard deviation. This factor depends only on the degree of saturation and not on the starting value. Data analysis from the Markov model output a sort of direct relationship between the mean value and its deviation. Since the mean value (in the hypothesis of deterministic capacity, Poisson arrival process and zero initial queue) is well represented by the former relationship, the standard deviation can then be represented by the Akcelik’s function multiplied by an appropriate scale factor. The factor between the standard deviation and the expectation value of the queue depends on the degree of saturation as shown in figure 12. The multiplier can be represented as QSD,e =6.67 – 5.66 X (shown already in (1)). [Figure 12] The analysis of the logarithm of the derivative of the queue with respect to time shows a linear regime after a certain number of cycles, just as in the case of the expectation value of the queue. The parameter β appears to be exactly equal to the previous case (eq. 10), while the parameter γ has a slightly different form:

β = −6.67 ⋅ (1 − x)2 γ ' = (1 − x) ⋅ Q0 − 2.2

(18)

EXTENSION OF THE METHOD TO AN INTERSECTION IN A NETWORK AND DYNAMIC CONTROL The application of the Markov model has been made for a fixed time signal control. This is a limitation if it is applied to a real situation, since often there is some kind of traffic dependent length of the green phase. In such cases the length of the expectation value of the queue will be strongly influenced by the details of the control program. The Markov can be extended to deal with this to a certain extent. In vehicle dependent control the green phase is extended up to the moment that the gap in the flow exceeds a certain maximum or until a maximum green time has elapsed. This seems too complicated to deal with in our Markov model but it is possible to analyze green phases that are determined by the queue length at the end of the red phase and the average flow. In such a Markov model, the process of the conflicting flows on the intersection has to be

FrancescoViti, Henk J.Van Zuylen

included. It is obvious that for such kind of control the expectation value of the queues and the standard deviation are considerably less than for a fixed time control. For networks we have to include the effect of the finite storage space, the spillback to upstream links and the metering effect of up-stream controlled intersections has to be included. This complicates the transition probability, but it creates no fundamental problem for the application of the Markov model. The present study is limited in its scope and applicability, but goes still further than the existing models, while its possibilities for generalization seem to be good.

CONCLUSIONS The mesoscopic Markov model for queues at a controlled intersection appears to be a good basis for deriving, calibrating and validating macroscopic formulas for the queue length and its standard deviation. The analysis described in this paper provides a new, easy-to-use mathematical formulation for the expectation value and standard deviation of time-dependent queues at signalized intersections. The extension of the mathematical formulas to the standard deviation opens the possibility to investigate the role of the variability of travel time. The new formulas are an extension to existing mathematical models for the queue length like Akcelik’s (7) and Catling’s (8) formulas. Akcelik’s formula has the limitation that it is only applicable if the initial queue is zero. Catling’s formula ignores the stochastic aspect in the dynamics of the decreasing queue. The role of the standard deviation appears to be very important for the dynamics of the expectation value of the queue. It explains the strange effects that the expectation value may initially decrease and afterwards increases. It can be explained as the transition from a linear decrease in the situation that the standard deviation is smaller than the expectation value of the queue, to an exponential approach towards the equilibrium value of the queue size (15,16). The standard deviation of the queue length is very large, about the same as the expectation value. This makes it difficult to verify the model with measurements in practice and even with microscopic simulations. Probably the standard deviation will be much lower for intersections in networks and for traffic dependent control. Further analysis to extend the models for these conditions will be done in the future. For dynamic traffic assignment the formulas that are derived can be applied, although a generalization is needed to include network effects like metering and spill back. The model for the standard deviation will give a way to represent uncertainty in travel time in route and departure time choice.

ACKNOWLEDGMENTS The authors thank Dr.Yusen Chen and Drs.Karel Lindveld for their useful comments. The work of both authors is financially supported by the Transport Research Center of Rijkswaterstaat. It is a part of the research program Dynamic Traffic Management of the Research School TRAIL.

REFERENCES 1.

Van Zuylen H.J. and Viti F., 2003. Uncertainty and the Dynamics of Queues at Signalized Intersections. Proceedings CTS-IFAC conference 2003, 6-8 August, Tokyo. Elsevier, Amsterdam 2. Olszewski P., 1990. Modeling of Queue Probability Distribution at Traffic Signals. Transportation and Traffic Theory, ed M.Koshi, Elsevier, Amsterdam, pp. 569-588. 3. Newell G.F., 1971. Applications of Queuing Theory, Monographs of Applied Probability and Statistics, Chapman and Hall, London. 4. Kimber and Hollis, Traffic queues and delay at road junctions, TRL LR 909, 1979 5. Robertson, D,I.. (1969). TRANSYT: A Traffic Network Study Tool. Road Research Laboratory Report LR 253, Crowthorne. 6. Troutbeck R. and Blogg M. (1998). Queuing at Congested Intersections. Transportation Research Record No.1646, TRB, National Research Council, Washington D.C., pp.124-131. 7. Akcelik R., 1980. Time-Dependent Expressions for Delay, Stop Rate and Queue Length at Traffic Signals. Australian Road Research Board, Internal Report, AIR 367-1. 8. Catling, I. 1977. A Time Dependent Approach to Junction Delays. Traffic Engineering and Control, 18, pp. 520-526. 9. Webster F.V, 1958. Traffic Signal Settings. Road Research Laboratory Technical Paper No. 39, HMSO, London. 10. Akcelik, R., 2002. aaSIDRA Traffic Model Reference Guide. Akcelik & Associates Pty Ltd. 11. Rouphail, N. A. Tarko, J. Li, 2000. Traffic flows at signalized intersections, Chap. 9 of the update of Transportation Research Board Special Report 165, “Traffic Flow Theory”, 1998.http://wwwcta.ornl.gov/cta/research/trb/tft.html. 12. Transportation Research Board (2000), Highway Capacity Manual, special report 209, National Research Council, Washington D.C., TRB.

FrancescoViti, Henk J.Van Zuylen

13. Miller, A.S. (1968), The capacity of signalised intersections in Australia, Australian Research Board bulletin 3 14. Mayne, A.J. (1979). An outline of new comprehensive formulas for queues and delays. Working paper Transport Studies Group, University College, London. 15. Van Zuylen H.J. and Taale H., 2002. Dynamic and Stochastic aspects of Queues at Signal Controlled Intersections. 13th Mini Euro Conference – Handling Uncertainty in the Analysis of Traffic and Transportation systems. Bari, 2002. 16. Taale H. And Van Zuylen H.J., 2001. Testing the HCM 1997 Delay Function for Dutch Signal Controlled Intersections. Proceedings of 80th Annual Meeting of Transportation Research Board, Washington D.C., January 2001.

FrancescoViti, Henk J.Van Zuylen

List of figures Figure 1 Probability distribution for the queue length Figure 2 Evolution of the queue length and its standard deviation Figure 3 Growth of a queue starting from queue length 0 or 4 Figure 4 Queue dynamics for initial queue of 8 and equilibrium value 7.8 for 60 cycles of 60s each Figure 5 The dynamics of the queue for different values of the initial queue Figure 6 Evolution of the queue considering different starting values Figure 7 Logarithm of the derivative of the queue along time Figure 8 Example of α(T) for the decreasing of the queue Figure 9 Queue calculated through the Markov Chain process and exponential distribution Figure 10 The queue calculated through the Markov model and proposed function Figure 11 The mean queue (solid line) and its standard deviation (dashed) calculated through the Markov model for x=0.975 in the first 500 cycles and then with x=0.95 Figure 12 Ratio between the standard deviation and the expectation value of the queue

FrancescoViti, Henk J.Van Zuylen

queue length distribution probability 1 0.8 0.6 0.4

0 8

0.2 16

Figure 1 Probability distribution for the queue length

13

10

7

4

1

number of queued vehicles

0 24

cycle

FrancescoViti, Henk J.Van Zuylen

queue and stadard deviation x = 0.967 9 8 7 6 5

QUEUE SIZE

4

ST.DEV.

3 2 1 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 cycle

Figure 2 Evolution of the queue length and its standard deviation

FrancescoViti, Henk J.Van Zuylen

Grow th of the queue starting at length 0 and 4 9

expected queue length

8 7 6 5

Initial queue 0

4

Initial queue 4

3 2 1 0

cycle Figure 3 Growth of a queue starting from queue length 0 or 4

FrancescoViti, Henk J.Van Zuylen

10 9 8 7 6

QU EU E SIZE

5

ST,D EV,

4

D ETER M IN ISTIC

3 2 1 0

1

8

15 22 29 36

43 50 57

64 71 78 85

Figure 4 Queue dynamics for initial queue of 8 and equilibrium value 7.8 for 60 cycles of 60s each

FrancescoViti, Henk J.Van Zuylen

Figure 5 The dynamics of the queue for different values of the initial queue

FrancescoViti, Henk J.Van Zuylen

25,00 20,00

Queue Qo=8

queue

linear approximation

15,00

Akcelik's function Queue Qo=14 linear approximation

10,00

Queue Qo=19 linear approximation

5,00 0,00 cycles

Figure 6 Evolution of the queue considering different starting values

FrancescoViti, Henk J.Van Zuylen

Figure 7 Logarithm of the derivative of the queue along time

FrancescoViti, Henk J.Van Zuylen

Figure 8 Example of α(T) for the decreasing of the queue

FrancescoViti, Henk J.Van Zuylen

Figure 9 Queue calculated through the Markov Chain process and exponential distribution

FrancescoViti, Henk J.Van Zuylen

Figure 10 The queue calculated through the Markov model and proposed function

FrancescoViti, Henk J.Van Zuylen

Figure 11 The mean queue (solid line) and its standard deviation (dashed) calculated through the Markov model for x=0.975 in the first 500 cycles and then with x=0.95

FrancescoViti, Henk J.Van Zuylen

Figure 12 Ratio between the standard deviation and the expectation value of the queue