On Computation of Queue Length Distribution at an Isolated ...

On Computation of Queue Length Distribution at an Isolated Signalized Traffic Intersection ? Mohammad Motie ∗ Ketan Savla ∗ ∗

Sonny Astani Department of Civil and Environmental Engineering at the University of Southern California, Los Angeles, CA {motiesha,ksavla}@usc.edu

Abstract In this paper, we compute queue length distribution, primarily under fixed-time policies, for an isolated signalized traffic intersection at the junction of two one-way streets with finite queue capacities. The queues on the streets have independent Poisson arrivals, and the vehicles accelerate safely from rest to cruise speed when departing during green phase. The saturation headway for such a departure process is derived, and, following standard procedure, is used to approximate the departure process with uniform headway over an appropriately reduced green time. Under a fixed-time policy, the two queue dynamics are decoupled. Motivated by vacation queues, we identify an imbedded Markov chain, corresponding to queue lengths at the end of cycles, whose transition probabilities are computed from analytical transient solutions of M/D/1/N queues. Convergence guarantees to steady-state distributions are provided, including for adaptive control policies. While the steady-state average queue lengths under standard M/D/1 approximations for the considered setup are convex in green times, the associated workload is proven to be upper bounded by the workload obtained from our proposed model. Illustrative simulations are included, which suggest that the queue lengths obtained from the proposed framework exhibit good agreement with the widely used Webster’s formula. Keywords: Queueing Systems, Transient Analysis, Traffic Signals 1. INTRODUCTION

underestimate, e.g., maximum queue lengths, and hence underestimate spillbacks.

Traffic signal control can be classified into fixed time and adaptive control policies. One well-known fixed time rule is to allocate green time in proportion to the maximum saturated flow ratio associated with each phase. Such a policy is straightforwardly throughput optimal. Analysis of such traffic queues has been the topic of research for decades, e.g., see Webster (1958); Miller (1963); Heidemann (1994). The main challenge from an optimization perspective is to obtain a tractable framework for computation and optimization of queue lengths in terms of traffic signal control parameters. One well-known semi-heuristic approach, which is also widely used in practice, is the Webster’s formula Webster (1958) that characterizes the average delay per vehicles in intersections with a fixed time policy. The simplest analytical approach is to approximate each link by a simple queue where the service rate is equal to the product of the saturation headway and the ratio of the allocated green time to the total cycle time. While such an approach is used in many models such as store-andforward model Aboudolas et al. (2009), and is the natural stochastic counterpart of a fluid model, it fails to model the cyclic patterns in queue length distributions induced by fixed-time policies. Such approximations can possibly

Several studies have focused on the characterization of queue length distribution for signalized traffic intersections. For example in Heidemann (1994) the steady-state distribution of queue length at fixed-time traffic signals is derived, but oscillatory behavior of queues is not analyzed. In another work, Brilon and Ning (1990) used a Markov chain theory to derive the queue length distribution at a specific point of each cycle for fixed-time policies. However, the evolution of distribution within cycle and other control policies are not investigated. Moreover, in that model it is assumed that during green periods, system works with maximum rate; however, this assumption is not valid for under-saturated regimes.

? This work was supported in part by NSF CMMI Project No. 1636377.

In this paper, we propose and analyze a queueing model for isolated signalized intersections under fixed-time and adaptive policies, and derive the transient probability distribution of number of vehicles present in the intersection. Specifically, We consider an intersection at the junction of two one-way streets with finite queue capacities. The queues on both streets are modeled as queues with independent Poisson arrivals, and the vehicles accelerate safely from rest to cruise speed when departing during green phase. The saturation headway for such a departure process is derived, and, following standard procedure, is

used to approximate the departure process with uniform headway over an appropriately reduced green time. Under a fixed-time policy, the two queue dynamics are decoupled. While each of these queues is reminiscent of vacation queues. Vacation queues are often used to model communication systems or machines with breakdowns Doshi (1986). In this queueing model, the server may become unavailable for a period of time. The performance of vacation queues highly depends on the implemented policy which is either exhaustive or non-exhaustive Tian and Zhang (2006). In exhaustive policies, the server is allowed to take a vacation only when the system becomes empty. Therefore, one can consider regeneration points at the beginning of the vacation periods to analyze such policies. However, these regeneration points in nonexhaustive policies usually do not exists which makes the analysis more complex. Application of vacation queues to transportation system is investigated by Daganzo (1990), and under an exhaustive policy for a polling system the mean waiting time and queue length is derived. However, due to the non-exhaustiveness of green periods in traffic signals, this model cannot be applied to traffic signals. Among non-exhaustive policies, Non-gated Time-limited (NT) policy is the closest policy to traffic signal control policies. Under NT policy, the server takes a vacation when either 1) its service time reaches a given maximum value or 2) the system becomes empty Tian and Zhang (2006). Leung and Eisenberg (1991) introduced vacation queues with NT policies, and characterized steady-state distributions; then, this result was applied to traffic signals de Lurdes Sim˜ oes et al. (2005). However, results showed that mean delay obtained from this model is smaller than simulation results. The reason is that, unlike NT policy, traffic signals may not switch to a red period (and start serving other queues) when there is no vehicle in the queue that is being served. This shows that we cannot use existing vacation queue models for computing queue length distributions. We identify an imbedded Markov chain, corresponding to queue lengths at the end of cycles, whose transition probabilities are computed from analytical transient solutions of M/D/1/N queues. Convergence guarantees to steady-state distributions are provided, including for adaptive control policies. While the steady-state average queue lengths under standard M/D/1 approximations for the considered setup are convex in green times, the associated workload is proven to be upper bounded by the workload obtained from our proposed model. The main contributions of the paper are as follows. First, we propose a queueing model for signalized traffic intersections which is reminiscent of queueing systems with vacation. Second, we identify an imbedded Markov chain for the proposed queueing models which is amenable to analysis. Third, we formally prove that the workload under the proposed model is greater than the workload obtained from the M/D/1 approach. On a minor note, we also derive an expression for saturation headway in terms of acceleration behavior of vehicles departing from a queue under green. Finally, through simulations, we illustrate that the queue length estimates obtained from our proposed approach are in good agreement with the well-known Webster’s formula.

2. PROBLEM FORMULATION We consider two one-way single-lane traffic streams interacting at an isolated signalized intersection. We refer to the eastbound (resp. nothbound) traffic stream as the first (resp. second) stream. A control policy, P, determines the duration of green phases for each traffic stream. We consider both fixed time and adaptive control policies that are described in Section 5. Vehicles arrive at the tail of queues in each stream according to two independent nonstationary Poisson processes with rates λ1 (t) and λ2 (t). Let the number of vehicles that are in queue in each stream be denoted by {ni (t), t ≥ 0}, i ∈ {1, 2}. We shall refer to this quantity as the queue length. Let D denote the saturation headway of traffic. The saturation flow rate is denoted by µ = 1/D. The length of each cycle is denoted by C and the effective red and green periods for the i-th stream are denoted by Ri and Gi , respectively. Moreover, under fixed time control policies and stationary arrival process, the effective saturation flow, µe , and the degree of saturation, ρ, are defined as (µe )i = Gi µ/C, ρi = λi /(µu )i . (1) Let Ni be the capacity of the i-th link i.e. the maximum number of vehicles that can be fit in link i; therefore, if ni = Ni , new vehicles cannot joint the queue and they leave the system. This capacity can be chosen arbitrary (i) large. Let πj (t) := Pr(ni (t) = j) be the probability that the queue in the i-th stream contains j, j ∈ {0, 1, · · · , Ni }, vehicles at time t ≥ 0. Correspondingly, we denote the instantaneous probability distribution of queue length in (i) (i) (i) the i-th stream by π (i) (t) = (π0 (t), π1 (t), · · · , πNi (t))T . We shall drop the index of stream of the intersection from notations whenever it is clear from the context. In this paper, we develop analytical tools in order to characterize the transient probability distribution of queue lengths, {π (1) (t), π (2) (t)}, by considering effects of implemented control policy. We model each leg of the intersection by an M/D/1/N queue in which server becomes unavailable during red phases. This model can be interpreted as a vacation queue with time limited policy. 3. EXISTING MODELS FOR FIXED-TIME POLICIES In this section, we provide details of some of the existing models for analyzing traffic queues under fixed-time control policy, and discuss the shortcomings of these models. 3.1 Uninterrupted Queues One common approach for fixed time traffic queues with constant arrival rate is to approximate the queue with another queue that is not interrupted by red phases, and consider the effective service rate for the system. This simplifying assumption does not allow characterization of queue oscillations due to green/red phases. We refer to this model as the uninterrupted model and the model that considers interruptions caused by red periods as the on/off model. Uninterrupted Model as a Lower Bound Now, we discuss that the uninterrupted model gives a lower bound on the queue length at the end of red periods. This quantity is important because it essentially characterizes the maximum queue length over a cycle. For our analysis, in this section, instead of queue length, we use the notion of

workload

workload that is the total remaining unfinished work in the system. In an M/D/1 queue, workload can be a good proxy for analyzing the queue length. Let wi (t) denote the workload of the i-th stream at time t ≥ 0. Figure 1 shows the workload evolution of an example first cycle where both uninterrupted and on/off models start with empty initial condition. In this figure, green and red periods are equal. Therefore, by (1) the service rate of the system in the uninterrupted model is half of the service rate in the on/off model. This implies that, as shown in Figure 1, the workload decreases at a rate that is half of the rate of on/off model. Figure 1 illustrates that, at the end of the red period, the workload in the on/off model is lower bounded by the workload of uninterrupted model. Although, Figure 1 provides the intuition, the following result formally states this observation for one of the streams whose green period is denoted by G.

0

Actual Model Approximate Model

b (1 )

~ b (1 )

C

time

Figure 1. Evolution of a workload in on/off and uninterrupted models over the first cycle for an example realization of arrival process. Proposition 1. For a fixed time policy, with G ∈ (0, C) and C > 0, at the end of red periods, workload in the on/off model is lower bounded by workload in the uninterrupted model. ˜ , and M denote the uninterrupted and Proof 1. Let M on/off models, respectively. Let consider the same realization of the arrival process for both systems. The proof is by induction. The base case is concerned with workloads at end of the first red period. The proof of the base case is the same as the proof of induction step, so we prove the induction step. Let w(k) and w(k) ˜ denote the workload at the ˜ , respectively. end of k-th red period in system M and M Assume that w(k) ˜ ≤ w(k). For the induction step, we show that w(k+1) ˜ ≤ w(k+1). Note that w(0) = w(0); ˜ therefore, the proof of the base case is also similar. In a worst case scenario, let w(k) ˜ = w(k). Moreover, let wa (k + 1) be the workload added to the system during during (k + 1)-th cycle (i.e. from the end of k-th red period to the end of (k + 1)-th red period). wa (k + 1) is equal to the number of arrivals during the (k + 1)-th cycle multiplied by D. Moreover, wa (k + 1) is the same for both models since the considered realization of the arrival process is the same. Let b(k + 1) (resp. ˜b(k + 1)) denote the total duration that ˜ ) is busy during (k + 1)-th cycle (see Figure 1 M (resp. M for an illustration). Therefore, w(k + 1) = w(k) + wa (k + 1) − b(k + 1) w(k ˜ + 1) = w(k) ˜ +w ˜a (k + 1) − ˜b(k + 1)G/C (2) ˜ is considNote that in (2), the slower decrease rate in M ered by the factor G/C. It is easy to see that b(k + 1) ≤ G. ˜ never gets interrupted by red periods, so Moreover, M it will work during the entire cycle, if work exists in ˜ works at a slower rate, the the system. Also, since M

work that has taken M an b(k + 1) amount of time, ˜ an b(k + 1)C/G amount of time. Therefore, would take M ˜b(k + 1) ≥ b(k + 1)C/G. Using this fact and the induction assumption combined with (2) gives w(k ˜ + 1) ≤ w(k + 1). 3.2 Webster’s Model For under-saturated queues, Webster (1958) proposed the following approximate expression for steady-state average delay of vehicles in queues under fixed-time control policy   13 ρ2 C G(1 − G/C)2 ρ2+5G/C + − 0.65 E[w] = 2(1 − G/Cρ) 2λ(1 − ρ) λ2 (3) where ρ = λ/µe and µe are given in (1). The first term in (3) is derived assuming uniform arrivals and using a geometric interpretation of cumulative curves of arrivals and departures of vehicles. The second term is the mean waiting time attributed to the randomness of vehicle arrivals. This term is the mean waiting time of a customer in an uninterrupted M/D/1 queue. The third term is an empirical correction factor added to fit the equation to the simulation results. However, this model does not provide any insight about the uncertainty of the prediction, it cannot characterize the transient and oscillatory behavior of queues, and it is not applicable to other control policies. 3.3 Time-dependent Models Although Webster’s model is not time-dependent, in another line work, researchers have provided time-dependent characterization of mean delay and queue length. In these models a coordinate transformation, proposed by Kimber and Hollis (1979), is used to transform the steady-state models so that they become asymptotic to the deterministic over-saturated models. However, there is no rigorous theoretical basis for this method Hurdle (1984). As an example, Ak¸celik (1980) used this method to characterize the expected overflow queue length no (queue length immediately after a green phase) over a finite evaluation period T. This expression for ρ > ρ0 is given as ! r CT 12(ρ − ρ ) 0 ET [no ] = ρ − 1 + (ρ − 1)2 + (4) 4 CT where ρ0 = 0.67 + µG/600. Although the notion of time exists in this model, it only characterizes queue length at the end of green periods. Similar to Webster’s model, it does not characterize the distribution. 3.4 Queues with on/off Service Lioris et al. (2017) proposed a model to consider on and off service periods. In this work, a Markov chain whose state is (q, s) is considered where s ∈ {on, off} represents the state of server. In this model, a Poisson clock model is used to determine transition rates. For example, when the k-th state is (qk > 0, off), the next state will be  (qk + 1, off) w.p. λ/(λ + γ2 ) (5) (qk , on) w.p. γ2 /(λ + γ2 ) where γ2 (resp. γ1 ) is the number of switches from off (resp. on) to on (resp. off) service per unit of time . For example, for a cycle with length 40 seconds and G = R = 20 seconds, γ2 = γ1 = 1/40 switches per seconds. Under this

model, they derive the following expression for mean queue length λγ 2 + 2λγ1 γ2 + λγ1 µ + λγ22 n ¯= 1 (6) (γ1 + γ2 )(γ2 µ − λ(γ1 + γ2 ) We compare this result with the result obtained from our proposed model in the simulation section; however, one can argue that this on/off service model will over estimate the queue length. Once the server switches from on to an off state, the number of transition it takes to return to an on state has a geometric distribution with mean γ2 /(γ2 + λ). For practical values of C, R, and G this random variable has a very large variance. This gives the intuition that, on average, the system remains in off service periods longer than actual system; therefore, (6) will over-estimate the mean queue length. 4. TRANSIENT QUEUE LENGTH DISTRIBUTION OVER ONE CYCLE In this section, we derive transient probability distribution of queue length. We focus only on one of the traffic streams over one cycle, say k-th, of length C seconds. During this cycle, duration of red and green phases are known. It is assumed that the amber time and the lost time are appropriately incorporated in R and G such that R + G = C (See Figure 2 for an illustration). Assume that cycle starts with a red period and is followed by green period. Therefore, the end of cycles are the end of green periods. Let t = rk be the beginning of the k-th cycle or red period, and t = gk be the onset of green period. rk

R

gk

G

rk

1

C

Figure 2. k-th cycle of traffic signal.

L

During the green period, we approximate the system with an M/D/1/N queueing model where D is the saturation headway. However, during a red phase, queue length can only grow; therefore, queue length process is modeled by a pure birth process. Moreover, since a finite buffer queue is considered, if an arrival finds N vehicles in the system, it will not join the queue. From a queuing theoretic perspective, one can interpret the system as an M/D/1/N queue in which server is not available (off) during red phases. Since the green (on) phases are finite and system alternates between on and off periods, transient analysis of the queue length distribution, π(t), is of great importance. In the next section, we apply the results of this section to fixed and adaptive control policies. In the following, the analysis of transient queue length distribution over red and green periods is provided. 4.1 Red Periods The dynamics of queue length over red periods is modeled by a pure birth process with rate λ(t). Therefore, by considering the Chapman-Kolmogorov equations, the dynamics of π(t) during a red phase is given as: π˙ 0 (t) = −λ(t)π0 (t), π˙ N (t) = λ(t)πN −1 (t) π˙ j (t) = −λ(t)πj (t) + λ(t)πj−1 (t), j = 1, · · · , N − 1

and in matrix form, ˙ π(t) = A(t)π(t),

(7)

where A(t) ∈ R(N +1)×(N +1) is λ(t)H and H ∈ R(N +1)×(N +1) is a constant matrix given by,  −1 0 0 ··· 0  1 .. . 0 0

H=

−1 .. . ··· ···

0 ··· 0 . . . . ..  . . . . 1 −1 0 0 1 0

Note that dynamics in (7) is a linear time variant system. Let Φ(t, rk ) be the corresponding state transition matrix from time rk to t. Therefore, given the distribution of queue length at the beginning of k-th red phase, i.e. t = rk , the trajectory of π(t) during the red phase is given as, π(t) = Φ(t, rk )π(rk ) (8) Since A(t) is the product of a constant matrix with a time dependent scalar, the state transition matrix is given as (see e.g. Chen (1999)) Rt A(s)ds Φ(t, t0 ) = e t0 (9) Although the state transition matrix can be computed by (9), the following result exploits the fact that during red interval we have a pure birth process and characterize a simpler expression for the trajectory of π(t). In our preparation to state the next result, we define the probability of m arrival during interval [t1 , t2 ] as, R m − R t2 λ(s)ds t2 λ(s)ds e t1 t1 am := t1 ,t2 m! which follows since, under a non-stationary Poisson process, the number of arrivals observed during an R t interval [t1 , t2 ] is a Poisson random variable with mean t12 λ(s)ds (see e.g. Ross (1996)). Theorem 1. For any k ∈ {0, 1, 2, · · · }, t ∈ [rk , rk + R], π(rk ), and {λ(t); t ≥ 0}, π(t) is given as   0 ar

0

k ,t

a1r

a0r

k ,t

    

k ,t

.. .

.. .

−1 aN r ,t

1−

PNk−1 j=0

k ,t

PNk−2

1−

j=0

0

0

···

0

..

.. .

0

. ···

−2 aN r ,t

ajr

···

ajr

k ,t

..  .  π(rk ) 0



a0r

k ,t

··· 1−a0r

k ,t

(10)

1

Proof 2. In order to compute π(t), we condition on the queue length at the beginning of the red period i.e. t = rk . In particular, for i ∈ {1, 2, · · · , N − 1}, i i X X Pr(n(t) = i) = Pr(n(rk ) = j)ai−j πj (rk )ari−j rk ,t = k ,t j=0

j=0

(11) where the second equality follows by the definition of πj (t). Similarly, for i = N , conditioned on n(rk ) = j, one has to compute the probability of having more than N − j − 1 arrivals during interval of length [rk , t]. Therefore, !! NX −j−1 N X m Pr(n(t) = N ) = πj (rk ) 1 − ark ,t (12) j=0

m=0

By writing (11) and (12) in matrix form, the result of this theorem is obtained. Remark 1. Note that the matrix in (10) is equal to Φ(t, rk ) in (8). It should be noted that in computing (10), the only Rt integration involved is rk λ(s)ds which is common for all

am rk ,t , m ∈ {0, 1, · · · , N }. Therefore, Theorem 1 provides an efficient alternative to (9). 4.2 Green Periods Our analysis in this section is based on the results in Garcia et al. (2002). We extend their results for the case when arrival rate is time varying. During the green period, queued vehicles are allowed to depart the system while new vehicles will be still added to the queue. Therefore, we approximate the system with a finite capacity M/D/1/N queue whose departure process is determined by the saturation headways. Let Sk (t, h) be the probability that a vehicle departs between time t and t + h, and leaves k, k ∈ {0, 1, · · · , N − 1}, customers in the queue by its dePN −1 parture. Therefore, S(t, h) := k=0 Sk (t, h) denotes the probability of a departure between time t and t + h. Furthermore, let sk (t) := limh→0 Sk (t,h) , k ∈ {0, 1, · · · , N −1}, h S(t,h) and s(t) := limh→0 h . These limits are interpreted as the departure rates of the system. In particular, at time t, s(t) is the total rate of departure, and sk (t) is the rate of departure that leaves k vehicles in the queue. Characterization of departure rates is useful for deriving transient queue length distribution. The computation of π(t) is carried out recursively, one period after another. In other words, starting from t = gk , we consider bR/Dc contiguous intervals of length D (where D is the deterministic inter-departure times) Moreover, a last interval of length R − bR/DcD is also considered to cover the entire green period. Therefore, the m-th interval during k-th green period is defined as  m=0 [rk − D, rk ) k Im := [rk + (m − 1)D, rk + mD) m ∈ {1, · · · , bR/Dc}  [R − bR/DcD, rk+1 ] m = bR/Dc + 1 Theorem 1 characterizes π(t) from the onset of red period to the beginning of the green period. The following results, k characterize the dynamics of system over interval Im by assuming that π(t) and departure rates sk (t) are known k during the pervious interval (i.e. Im−1 ). Let S(t) := (s0 (t), s1 (t), · · · , sN −1 (t))T denote the vector of departure rates at time t. The following result characterizes the dynamics of S(t). k Proposition 2. Given π0 (z) and S(z) ∀z ∈ Im−1 , the k following gives S(t) for any t ∈ Im , S(t) = C(t)S(t˜)+B(t), where t˜ := t − D, and B(t) ∈ RN is given as " #T N −2 X N −2 0 1 i B(t) = λ(t)π0 (t˜) at˜,t at˜,t · · · at˜,t 1 − at˜,t i=0 N ×N

and C(t) ∈ R  0

and is given as

0 ··· 0 PN0−2 i −2 a0t˜,t a1t˜,t a2t˜,t ··· aN 1− at˜,t ˜

     

0

0

a0t˜,t a1t˜,t ···

.. . 0

..

0

···

. ···

..

.

..

.

0

a0t˜,t

0

0

t,t −3 aN ˜,t t

.. . a1t˜,t a0t˜,t

i=0 PN −3

1−

i=0

ait˜,t

..

P1.

1−

ai˜,t i=0 t 0 1−at˜,t

T      

Proof 3. (sketch). A departure will occur in interval (t, t + ∆t) and leave k ∈ {0, · · · , N − 2} vehicles in the system if one of the following two condition holds.

(1) The system was empty at time t − D and one vehicle joined the queue in interval (t − D, t − D + ∆t), and during its service, k new vehicles have arrived. (2) A vehicle departs in (t − D, t − D + ∆t) and leaves i ∈ {1, · · · , k + 1} vehicles behind in the queue. During the service of the first of these vehicles, there were k + 1 − i new arrivals. The first condition considers the departure of a vehicle that starts a busy period, and the second condition considers the departure of any vehicle that arrives within a busy period. By considering the aforementioned two conditions, one can write an equation for Sk (t, t+∆t) in terms of π0 (t˜), at˜,t (j), and Sj (t˜, t˜ + ∆t), j ∈ {0, · · · , k}. By dividing this equation by ∆t and taking the limit as ∆t → 0, the result of this proposition is established. The following set of differential equations give the dynamics of π(t) during interval Irk . π˙ 0 (t) = −λ(t)π0 (t) + s0 (t), π˙ N (t) = λ(t)πN −1 (t) − sN −1 (t) π˙ j (t) = −λ(t)πj (t) + λ(t)πj−1 (t) − sj−1 (t) + sj (t), By adding a zero to vector S(t) and transposing it, we ˜ define S(t) := (s0 (t), s1 (t), · · · , sN −1 (t), 0)T . Therefore, the above equations can be written in matrix form as, ˜ ˙ π(t) = A(t)π(t) − H S(t) (13) Although the above can be solved using numerical integration methods, the following result provides the analytical solution of (13) for empty initial condition. Theorem 2. If the queue is empty at t = gk i.e. π(gk ) = (1, 0, · · · , 0)T , then for any t ∈ [gk , tk+1 ], π(t) = Rt ˜ Φ(t, gk )π(gk ) − Φ(t, z)H S(z)dz gk

Proof 4. The result follows by (13). Moreover, Φ(t, gk ) is the state transition matrix given in Theorem 1. Theorem 2 characterizes the trajectory of π(t) for the empty initial condition. However, when the green period starts the queue may be non-empty i.e. n(gk ) > 0. Then t = {gk + D, · · · , gk + n(gk )D} will be the only departure epochs up to time gk + n(gk )D, with probability one. In between these point, queue length process is a pure birth process. The service of j-th vehicle, j ∈ {1, · · · , n(gk )}, starts at tj := (j − 1)D and ends at tj := jD. Given π(tj ), one can determine π(tj+1 ) as π T (tj+1 ) = π T (tj )Ptj ,tj+1 (14) where Ptj ,tj+1 is given as   0 PN −2 i −2 a1t ,t a2t ,t ··· aN at ,t 0 at ,t tj ,tj+1 1− j j+1 j j+1 j j+1 j j+1 i=0 P N −2 i   a0 N −2 1 2  tj ,tj+1 atj ,tj+1 atj ,tj+1 ··· atj ,tj+1 1−Pi=0 atj ,tj+1 0    N −3 N −3 a0t ,t a1t ,t ··· at ,t 1− ait ,t 0  0 j j+1 j j+1 j j+1 i=0 j j+1   .. .. ..   .. .. .. . . 0 . . .  .   .. P1 i .. .. ..  . . . a1t ,t . 1− a 0 j j+1 i=0 tj ,tj+1   0 0 0

0

···

0

0

0

···

0

at

j ,tj+1

1

1−at

j ,tj+1

0

0

In words, the element in the i-th row and k-th column of Ptj ,tj+1 gives the probability of having k − 1 vehicles in the queue after the departure at of j-th vehicle at t = tj given that the number of vehicles at t = tj−1 was i. This probability is equivalent of the probability of having k − i arrivals during (tj , tj+1 ). Proposition 3. For any n(gk ) ∈ N and t ∈ (gk , rk+1 ], π(t) is given as

  Φ(t, tj−1 )π(tj−1 ), t ∈Z(tj−1 , tj ), j ∈ {1, · · · , n(gk )} t ˜ Φ(t, z)H S(z)dz, t > tn(gk ) Φ(t, t )π(t ) −  n(g ) n(g ) k k  tn(gk )

Proof 5. Queue length distribution at the first n0 departure points is given by (14). In between these points, the system works as a pure birth process and the result follows by Theorem 1. Once the last initial vehicle leaves the system at t = tn(gk ) , dynamics of π(t) can be determined by (13). Therefore, similar to Theorem 2, the trajectory of π(t), t > tn0 , is determined. Remark 2. While Proposition 3 gives the trajectory of π for deterministic non-empty initial condition, one can obtain similar result for general π(gk ) by conditioning on the initial queue length. 5. QUEUE LENGTH DISTRIBUTION UNDER WELL-KNOWN CONTROL POLICIES In this section, we study the transient and steady-state distribution of queue length under different control policies. The considered control policies determine the green phases for each traffic stream for a given cycle length. In particular, given the control policy, P(f ), green phases are determined as, G1 = f (n1 , n2 )C, G2 = (1 − f (n1 , n2 ))C (15) where f is a function of queue lengths at the beginning of the cycle and f : [n1 , n2 ] → [0, 1]. In the following, two different f are considered that lead to two different policies. 5.1 Fixed Time Policy Let G1 and G2 be the pre-timed green phases for stream 1 and 2, respectively. In this case, the dynamics of queue length in each stream is independent from the other stream. Therefore, we focus only on the first stream. Similar analysis can be applied for the other streams. Given n1 (0), G1 , C, and {λ1 (t), t ≥ 0}, the results in Section 4 can be readily used to characterize π (1) (t) for t ∈ [0, C]. Then, by having π (1) (C), one can evaluate π (1) (t) for t ∈ (C, 2C]. This recursive procedure can be applied to find the trajectory of π (1) (t) for all t ≥ 0. If λ1 is stationary (i.e. time invariant), it is imaginable that queue length reaches a steady-state distribution. In the following, we show that, under aforementioned conditions, the queue length at the end of cycles reaches an steady state. Consider a Markov chain whose state is the queue length at the end of each cycle (end of green periods) i.e. X(k) = n1 (kC), k ∈ {0, 1, · · · , }. Let P denote the probability transition matrix of this Markov chain. The j-th column of P gives the probability distribution of the queue length at the end of a cycle if at the beginning of the cycle the queue length was n1 = j which can be obtained by the results of Section 4. It can be also realized that since λ1 is stationary, P is time invariant. Note that since the underlying queuing model has a finite capacity, the queue length will remain bounded for any λ ≥ 0. The following result shows that the steady-state exists and characterizes it. Proposition 4. For time invariant λ1 > 0, Markov chain {X(k), k = 0, 1, · · · } is ergodic and the steady state dis-

tribution is the eigenvector associated with the eigenvalue 1. Proof 6. It is easy to show that all the entries of P are positive. Therefore, the Markov chain is both aperiodic and irreducible. This, combined with the fact that the Markov chain has a finite state space, implies that it is ergodic and the stationary distribution exists and equals to the eigenvector associated with eigenvalue 1 (see e.g. Ross (1996)). 5.2 Adaptive Policy We consider a proportionally fair policy. Under this policy, for each cycle, green phases are proportional to the queue lengths of each stream at the beginning of the cycle. In particular, f in (15) is ( n1 n1 + n2 6= 0 f (n1 , n2 ) = n1 + n2 (16) 1/2 n1 + n2 = 0 Under this policy, queue length processes in stream 1 and 2 are not independent. However, given queue lengths at the beginning of a cycle, the queue length distributions can be independently characterized up to the end of the cycle. Consider a Markov chain that its state consists both queue lengths at the end of a cycle. In particular, the state of this Markov chain is denoted as X(k) = (n1 (kC), n2 (kC)), k ∈ {0, 1, · · · } and probability distribution of X(k) is essentially the joint distribution of n1 (kC) and n2 (kC) which is denoted by T ˜ π(k) := [˜ π0,0 (k), π ˜0,1 (k) · · · , π ˜N,N (k)] (17) where π ˜i,j denotes the probability of n1 (kC) = i and n2 (kC) = j. Moreover, let P (k), k ∈ {0, 1, · · · }, be the associated probability transition matrix. Consider a state X(k) = (i, j), i, j ∈ {0, 1, · · · , N }. Given X(k) by using (15) and (16), G1 and G2 are determined and by using results of Section 4, independently for each stream, queue length distributions at the end of cycle, π (1) ((k +1)C) and π (2) ((k + 1)C), are evaluated. Therefore, the probability of a transition from X(k) = (i, j) to X(k + 1) = (m, n) is (2) (1) given as πm ((k + 1)C)πn ((k + 1)C). Therefore, P (k) is characterized and the transient joint distribution of queue ˜ + 1) = Pk π(k). ˜ length is given as π(k

If λ1 and λ2 are both stationary, Pk does not depend on k and is the same for all cycles. Let P be the probability transition matrix in this case. The following result shows that steady-state joint probability distribution of queue lengths at the end of cycles exists and characterizes it. Proposition 5. For time invariant λ1 and λ2 , Markov chain {X(k), k = 0, 1, · · · } is ergodic and the steady state distribution is the eigenvector associated with the eigenvalue 1. Proof 7. Similar to to the proof of Proposition 4. ˜ also gives Remark 3. The steady state distribution of π the steady state distribution of the duration of green phases. 6. SIMULATIONS In this section, simulation results are reported to further illustrate the proposed model and contrast it with existing models. Figure 3 depicts the transient mean queue

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Figure 3. Mean queue length under proposed model and approximate model over 10 cycles with fixed time policy. Parameters are: α = 1.3, dc = 4 m, vf = 16 m/s, am = 4 m/s2 , λ = 0.45, N = 20, C = 40, G = 20, and R = 20. As it is described in Section 5, the proposed model can be utilized to analyze the performance of control policies. Figure 4 illustrates total mean queue length at the end of cycles under both fixed time and proportionally fair (16) policies. The considered intersection includes two streams and both start with empty initial condition. In the fixed time policy, the green periods are allocated equally. Since both streams start with empty initial condition, under proportionally fair policy (16), during the first cycle green periods are equal; therefore, up to the end of the first

cycle, mean total queue length under both policies is the same. After the first cycle, however, proportionally fair policy converges to a steady state that is smaller than the one associated with the fixed time policy. One can also observe that fixed time policy converges faster than the proportionally fair policy. 12 10 8

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length at one of the streams under fixed time policy. Red and green periods are shown on the horizontal axis. As expected, mean queue length increases over red periods and decreases over green periods, and the proposed model captures these oscillations. Although, over cycles, these fluctuations remain in the system, one can observe that, as time goes by, from one cycle to other, these fluctuations become similar. This observation confirms the fact that queue length at the end of cycles reaches a steady state value, as stated in Proposition 4. In Figure 3, the uninterrupted model, explained in Section 3.1, is also illustrated. The transient behavior of the uninterrupted model is obtained using Theorem 2 by considering an uninterrupted M/D/1/N queue with scaled saturation headways. It can be observed that, as discussed in Section 3.1.1, mean queue length at the end of red cycles in the proposed model (which is an on/off model) is lower bounded by the uninterrupted model. One can see that the difference is not negligible. Moreover, in Figure 3, we compare the proposed model with Webster (3) and Akcelic (4) models. We use Little’s law to compute the mean queue length from mean delay equation in Webster’s formula (3). Figure 3 shows that mean queue length obtained from Webster’s formula lies almost in the middle of the range of the proposed model. Furthermore, the mean overflow queue length (queue length at the end of green periods) obtained from Akcelik model is close to the proposed model. We also computed the mean queue length using the on/off service model (6) proposed by Lioris et al. (2017). The mean queue length for the set of parameters in Figure 3, obtained from (6), is 99. As we expected and explained in Section 3.4, this model (6) over-estimates the mean queue length.

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Figure 4. Mean total queue length at the end of cycles under both fixed time and adaptive policy (16). Parameters are: α = 1.3, dc = 4 m, vf = 16 m/s, am = 4 m/s2 , λ1 = 0.4, λ2 = 0.3, N = 20, C = 40, and Gmin = 5. For fixed time policy: G1 = G2 = 20. In this case, ρ is 0.9. 7. CONCLUSIONS In this paper, we proposed a framework to compute queue length distributions at a signalized traffic intersection, primarily for fixed time policies. We utilized an imbedded Markov chain approach, where transition probabilities are obtained from transient analysis of an appropriate M/D/1/N queue. The expected queue lengths obtained from these computations show good agreement with Webster’s formula, and are more accurate than conventional M/D/1 approximations. The proposed techniques are an important step towards providing an analytical basis for existing heuristic formula for computing mean queue length at a signalized traffic intersection. Beyond this immediate advantage, this also lays foundations for an analytical framework for optimal selection, as well as estimation, of traffic signal parameters, which we plan to pursue in future. We also plan to extend our analytical results to formalize the relationship between queue length computations from our model and other proposed approximate queueing models in the literature. To this purpose, we plan to investigate conditions for existence of steady-state distributions of the proposed queueing model in the unbounded queue capacity case. While the simulations results for finite queue capacity are encouraging, the main technical challenge in formal proof appears to be the lack of translation invariance of the transition matrix of the imbedded Markov chain. REFERENCES Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2009). Store-and-forward based methods for the signal control problem in large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 17(2), 163–174.

Ak¸celik, R. (1980). Time-dependent expressions for delay, stop rate and queue length at traffic signals. Australian Road Research Board. Internal Report AIR 367-1. Brilon, W. and Ning, W. (1990). Delays at fixed-time traffic signals under time-dependent traffic conditions. Traffic engineering & control, 31(12), 623–631. Chen, C. (1999). Linear System Theory and Design. Oxford University Press, 3 edition. Daganzo, C.F. (1990). Some properties of polling systems. Queueing Systems, 6(1), 137–154. de Lurdes Sim˜ oes, M., de Oliveira, P.M., and Costa, A.P. (2005). Queues with server vacations in urban traffic control. In Proceedings of International Symposium on Applied Stochastic Models and Data Analysis. Doshi, B.T. (1986). Queueing systems with vacations—a survey. Queueing systems, 1(1), 29–66. Garcia, J.M., Brun, O., and Gauchard, D. (2002). Transient analytical solution of M/D/1/N queues. Journal of applied probability, 853–864. Heidemann, D. (1994). Queue length and delay distributions at traffic signals. Transportation Research Part B: Methodological, 28(5), 377–389. Hurdle, V. (1984). Signalized intersection delay models–a primer for the uninitiated. In Transportation Research Record, 971, 96–105. TRB, National Research Council, Washington, DC,. Kimber, R. and Hollis, E.M. (1979). Traffic queues and delays at road junctions. Technical report, TRR Laboratory Report 909, UK. Leung, K.K. and Eisenberg, M. (1991). A single-server queue with vacations and non-gated time-limited service. Performance Evaluation, 12(2), 115–125. Lioris, J., Pedarsani, R., Tascikaraoglu, F.Y., and Varaiya, P. (2017). Platoons of connected vehicles can double throughput in urban roads. Transportation Research Part C: Emerging Technologies, 77, 292–305. Miller, A.J. (1963). Settings for fixed-cycle traffic signals. Journal of the Operational Research Society, 14(4), 373– 386. Ross, S.M. (1996). Stochastic Processes, 2nd Edition. John Wiley & Sons. Tian, N. and Zhang, Z.G. (2006). Vacation queueing models: Theory and Applications, volume 93. Springer Science & Business Media. Webster, F. (1958). Traffic signal settings, road research technical paper no. 39. Road Research Laboratory.