Queue Length Simulation for Signalized Arterial Networks and Steady State Computation under Fixed Time Control

Queue Length Simulation for Signalized Arterial Networks and Steady State Computation under Fixed Time Control

Abstract

We consider traffic flow dynamics for a network of signalized intersections, where the outflow from every link is constrained to be equal to a given capacity function if the queue length is positive, and equal to the minimum of cumulative inflow and capacity function otherwise. In spite of the resulting dynamics being discontinuous, recent work has proved existence and uniqueness of the resulting queue length trajectory if the inter-link travel times are strictly bounded away from zero. The proof, which also suggests a constructive procedure, relies on showing desired properties on contiguous time intervals of length equal to the minimum among all link travel times. We provide an alternate framework to obtain queue length trajectories by direct simulation of delay differential equations, where link outflows are obtained from the provably unique solution to a linear program. Existence and uniqueness of the solution to the proposed model for traffic flow dynamics is established for piecewise constant external inflow and capacity functions, and the proposed method does not require travel times to be bounded away from zero. Additionally, if the external inflow and capacity functions are periodic and satisfy a stability condition, then there exists a globally attractive periodic orbit. We provide an iterative procedure to compute this periodic orbit. A periodic trajectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. The update for a given link is based on the periodic trajectories computed in the previous iteration for its upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit.

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1 Introduction

Modeling of traffic flow dynamics for signalized arterial networks has to strike a tradeoff between the ability to capture variations induced by alternating red/green phases and computational complexity of the resulting framework for the purpose of performance evaluation and control synthesis. Store-and-forward models, e.g., see [1], approximate the dynamics by replacing a time-varying outflow due to alternating green and red phase on a link with an equivalent average outflow. Such models have been used for optimal green time split control, e.g., see [2, 3]. Continuous-time versions of these models have also been used for green time control, e.g., in [4]. However, the approximation does not model the effect of offsets and cycle lengths. These limitations are overcome by discrete-event models, which have been utilized for optimal control synthesis for isolated signalized intersections in some cases, e.g., see [5, 6].

[7] proposed and analyzed a model, which captures offset and cycle times in the same spirit as discrete-event models. In particular, in [7], a fixed-time control setting is considered, where every link is endowed with a given capacity function, that specifies the maximum possible outflow from a link as a function of time. In order to maintain non-negativity of queue lengths, the outflow from every link is constrained to be equal to the capacity function if the queue length is positive, and equal to the minimum of cumulative inflow and capacity function otherwise. In spite of the resulting dynamics being discontinuous, it was shown in [7] that the traffic dynamics admits a unique queue length trajectory if the inter-link travel times are strictly bounded away from zero. The proof, which also suggests a constructive procedure, relies on showing desired properties on contiguous time intervals of length equal to the minimum among all inter-link travel times.

We provide an alternate framework to obtain queue length trajectories by direct simulation of delay differential equations, where link outflows are obtained from the provably unique solution to a linear program. For given queue lengths, this linear program solves for maximum cumulative outflow from all links subject to constraints imposed by the link capacity functions, and subject to maintaining non-negativity of queue lengths. Existence and uniqueness of the solution to delay differential equations is established for piecewise constant external inflow and capacity functions, and the method does not require travel times to be bounded away from zero. The existence and uniqueness result also extends to adaptive control policies, as long as the resulting capacity functions remain piecewise constant. This would happen, e.g., if traffic signal control parameters (green time, cycle length, and offsets) at every intersection are updated once per cycle. The piecewise constant assumption is practically justified because a common model for a capacity function is that it is equal to the saturated capacity during the green phase and zero otherwise, and external inflows can be modeled as a sequence of rectangular pulses representing arriving vehicle platoons. The key idea in the proof is that, under constant inflow and capacity, the set of links with zero queue lengths is monotonically non-decreasing, which implies overall finite discontinuities over any given time interval under the piecewise constant assumption. The ability to model zero inter-link travel time is particularly desirable for possible extensions to model finite queue capacity, under which inter-link travel time approaches zero as the downstream queue approaches capacity.

If, additionally, the external inflow and capacity functions are periodic and satisfy a stability condition, then there exists a globally attractive periodic orbit. This result and its proof follows the same structure as in [7], but is adapted to the proposed modeling framework. One consequence of this adaptation is that we work with the norm, instead of the sup norm in [7], for continuity arguments in our proofs.

Our most novel contribution is a procedure to explicitly calculate the globally attractive periodic orbit. Indeed, this was noted as an important “outstanding open problem” in [7], due to its usefulness in directly quantifying relevant performance metrics for a given fixed-time control. We provide an iterative procedure to compute this periodic orbit. A periodic trajectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. This update for a given link is based on the periodic trajectories computed in the previous iteration for upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit.

The representation of periodic orbit in terms of the time instants when queue length transitions between being positive and zero, as is implicit in our computational procedure, is to be contrasted with sinusoidal approximation consisting of a single harmonic proposed, e.g., in [8]. While it is compelling to improve this approximation by including higher harmonics [9], such an approach can potentially face several challenges: computing Fourier coefficients is not easy due to discontinuous dynamics; no bounds exist on approximation error for a given number of harmonics; and most importantly, because of discontinuity, including arbitrarily high number of harmonics may not give a zero approximation error due to the well-known Gibbs phenomenon. On the other hand, our proposed procedure computes the periodic orbit with arbitrary accuracy.

In summary, the key contributions of the paper are as follows. First, we provide a delay differential equation framework to directly simulate queue length dynamics under fixed-time or adaptive control, by establishing that it has a unique solution as long as the external inflow and capacity functions are piecewise constant. Second, under additional periodicity and stability condition, we adapt a recently proposed technique to establish existence of a globally attractive periodic orbit in our setting. Third, we provide a procedure to compute this periodic orbit with arbitrary accuracy. Illustrative simulations, including comparison with steady-state queue lengths from a microscopic traffic simulator, are also included.

The outline of the paper is as follows. Section 2 contains the proposed delay differential equation framework to simulate queue length dynamics. Section 3 provides the (non-iterative) framework to compute the periodic orbit for an isolated link. This forms the basis for an iterative procedure to compute periodic orbits for a network in Section 4 where we also establish uniform monotonic convergence of the iterates to the desired periodic orbit. Section 5 presents illustrative simulation results and concluding remarks are presented in Section 6. The proofs for most of the technical results are collected in the Appendix.

We conclude this section by introducing key concepts and notations to be used throughout the paper. , , , and will stand for real, non-negative real, strictly positive real, non-positive real, and strictly negative real, respectively, and denotes the set of natural numbers. For , we let denote the non-negative part of . A function is called piece-wise constant if it has only finitely many pieces, i.e., can be partitioned into a finite number of contiguous right-open sets over each of which is constant. The road network topology is described by a directed multi-graph with no self-loops, where is the set of intersections and is the set of directed links.

2 Problem Formulation

2.1 Traffic Flow Dynamics

The network state at time is described by the vector of queue lengths, corresponding to the number of stationary vehicles, and the history of relevant past departures from the links, , which quantifies the number of vehicles traveling in between links. The quantity shall be described formally soon. Let and be saturated flow capacity and external inflow functions, respectively, for link . Let the matrix denote the routing of flow, e.g., denotes the fraction of flow departing link that gets routed to link . Naturally if link is not immediately downstream to link . We shall assume that is sub-stochastic, i.e., all of its entries are non-negative, all the row sums are upper bounded by 1, and there is at least one row whose row sum is strictly less than one. We further assume the following on the connectivity of .

Assumption 1
  1. is weakly connected, i.e., for every , there exists a directed path in from to , or from to .

  2. For every , either the sum of entries of the -th row in is strictly less than one, or there exists a directed path from to at least one link such that the entries of the -th row in is strictly less than one.

Remark 1

The weak connectivity aspect of Assumption 1 is without loss of generality: if is not weakly connected, then our analysis applies to each connected component of , as long as each of these connected components satisfies (ii) in Assumption 1. Indeed, part (ii) of Assumption 1 implies that, for every vehicle arriving into the network, either it is possible for the vehicle to depart directly from the arrival link, or there exists a directed path to an another link from which the vehicle can depart the network. Formally, part (ii) of Assumption 1 implies that the spectral radius of , and hence also of , is strictly less than one. In particular, this guarantees that is invertible.

We now describe a model for traffic flow dynamics. The queue length dynamics is described by a standard mass balance equation: for ,

where denotes the outflow from link at time . In (1a), is the travel time from link to , and is a concise notation for . It would be convenient to rewrite the queue length dynamics as: for ,

(1a)

where

(2)

is the net inflow to link due to external arrivals and arrivals due to vehicles from upstream which were traveling until , and

(3)

is the set of links upstream of with zero inter-link travel time. Let be the maximum among all travel times from link to its downstream links. We let

(1b)

be the history of relevant past departures1, and be the number of vehicles traveling in between links at time .2 Finally, let , , , and denote the collection of corresponding quantities over all links. (1a)-((1b)) collectively describe the evolution of starting from initial condition . We propose link outflows for be obtained as solution to the following linear program, for any :

(1c)

where

is the set of links with no stationary vehicles. ((1c)) computes the maximum cumulative outflow, weighted by , in the network, subject to two constraints. The first one imposes link-wise capacity constraint, and the second one imposes the constraint that, for a link with zero queue length, its outflow is no greater than its inflow. The second constraint is to ensure non-negativity of queue lengths. The well-posedness of our proposed method for computing link outflows, i.e., uniqueness of solution to ((1c)) for a given , and independence w.r.t. is established in the next section. Thereafter, we establish existence and uniqueness of the solution to our traffic flow model in (1a)-((1b))-((1c)), which we shall collectively refer to as (1).

In order to present our results on existence and uniqueness concisely, we introduce a couple of more notations. Let and be the, respectively, maximum and minimum among all inter-link travel times.

2.2 Existence of Solution to (1a)

The proof of the next result is provided in the Appendix.

Proposition 1

Given , , and , ((1c)) has a unique solution, which is independent of . Moreover, the optimal solution satisfies

(3)

Proposition 1 implies that in (1a) is well-defined. With regards to (3), indeed for , , except possibly at time instants when there is a change in . It is rather straightforward to see that (1) admits a unique solution in between such changes. The frequency of such changes in general depends on , , and the initial condition . We bound the frequency of changes, and thereby establish existence and uniqueness of the solution to (1) for all , under the following practical assumption.

Assumption 2

, , and are all piece-wise constant.

The proof of the next result is provided in Appendix.

Proposition 2

Let , and the initial condition satisfy Assumption 2. Then, there exists a unique solution for all , to (1).

Remark 2
  1. (1) allows direct simulation of queue length dynamics under fixed-time control. Moreover, unlike [7], existence of a unique solution to (1) does not require to be strictly greater than zero. However, this comes at the expense of piecewise constant assumption.

  2. Assumption 2 is practically justified because a common model for a capacity function is such that it is equal to the saturated capacity during the green phase and zero otherwise, and external inflows, as well as past departures before can be modeled as a sequence of rectangular pulses modeling vehicle platoons.

  3. Proposition 2 holds true also when the capacity function is state-dependent (referred to as adaptive traffic signal control), but piecewise constant. For example, let the capacity function be equal to if , and equal to zero otherwise, where is the offset, and is a sequence of green times. Such green times can be determined as a function of queue lengths. One such simple proportional rule, when the capacity functions for all the incoming links at every intersection are mutually exclusive, is:

    The summation in the denominator is over all links incoming to the intersection to which is incident, and is the maximum queue length during the -th cycle on link .

2.3 Periodic Solution

It is straightforward to see that the solution to (1) can be equivalently described in terms of . Therefore, we shall use and interchangeably to refer to the solution to (1). We now develop a result analogous to the one in [7] on the existence of a globally attractive periodic orbit , under the following periodicity assumption.

Assumption 3

The external inflow functions and capacity functions are all periodic with the same period .3

Let

(4)

be the external inflow and capacity functions averaged over one period. Let and denote the collection of external inflow and capacity functions, respectively, for all links. The following stability condition will be one of the sufficient conditions for establishing periodicity of at steady state.

Definition 1 (Stability Condition)

There exists such that .

The proof of the following theorem is provided in the Appendix.

Theorem 1

Let , and the initial condition satisfy Assumptions 2 and 3, and the stability condition in Definition 1. Then, there exists a unique periodic state trajectory with period for (1), to which every trajectory converges.

2.4 Problem Statement

While one can use (1) to obtain the steady state by direct simulations, in this paper, our objective is to develop an alternate framework to obtain .

3 Steady State Computation for an Isolated Link

Let be the cumulative inflow into link . Referring to (1a), this quantity is given by . For an isolated link , . It is easy to see that if for all . In order to avoid such trivialities, we assume that the set has non-zero measure. The key in our approach is a procedure to easily compute for some . Thereafter, for all can be easily obtained by simulating (1) over a time interval of length . The natural candidates for such a are the time instants when the queue length transitions between zero and positive values. We now provide a detailed procedure to compute such a transition point. We implicitly assume throughout this and the next section that Assumption 2 and the stability condition in Definition 1 holds true.

Definition 2 (Transition Points)

Let be the time instants in when transitions from being zero to being positive.

Figure 1 illustrates the transition points for a sample scenario.

Figure 1: Illustration of transition points, and negative/positive sets. In this case, and . Subscript is not shown for brevity.
Remark 3
  1. Under the stability condition in Definition 1, , as also noted in [7, Theorem 2].

  2. For a given and , Theorem 1 implies uniqueness of the resulting , and hence of .

  3. As noted earlier, the knowledge of at any single time instant is sufficient to determine over the entire period . Indeed, if . By construction such a corresponds to a periodic orbit for (1). Once is computed, inspired by Proposition 1 and remarks immediately following it, let be given by:

    (5)

    Periodicity of , and imply that in (5) is periodic, i.e., is a periodic orbit. The uniqueness result in Theorem 1 implies that this is indeed the desired object to be computed.

The time instant referenced in Remark 3 (iii), for whose computation we now provide a procedure, is . We need the notion of negative and positive sets, defined next.

Definition 3 (Negative and Positive Sets)

Let be contiguous subsets of of non-zero size in which , and let be contiguous subsets of of non-zero size in which .

Remark 4
  1. Since the set is assumed to have non-zero measure, under the stability condition in Definition 1, we have and .

  2. The sets and do not necessarily form a partition of . Specifically, they exclude sets where .

Illustration of negative and positive sets are included in Figure 1. In preparation for the next result, let , and , be closures of and , respectively.

Proposition 3

Consider a link with inflow function and capacity function , both periodic with period . Let the transition points and positive/negative sets be given by Definitions 2 and 3 respectively. Then, there exists a strictly increasing such that for all .

{proof}

We drop the subscript for brevity in notation. The strictly increasing property of , if it exists, is straightforward; we provide a proof for existence. For a given , we let denote the time instant in between and when the queue length transitions from being positive to being zero. Similarly, we let be the time instant in between and when the queue length transitions from being positive to zero if it exists, or else we let . We also let be the time instant in between and when the queue length transitions from being positive to zero if it exists, or else we let .

Assume, by contradiction, that there exists such that . Let

if it exists, and is equal to zero otherwise. Similarly, let , if it exists, and is equal to zero otherwise. Since and can not both be equal to zero, we have . Therefore, consider the following cases, where we use the convention that :

  1. : From the definition of , we have (i) if , or (ii) otherwise. In case (i), such that , implying for all . Similar argument holds true for case (ii). Therefore, which is in contradiction to , since .

  2. : The definitions of and imply that . Therefore, such that , which implies that for all . Therefore, , which contradicts .

This establishes the proposition.

Proposition 3 narrows down our search for . We now sharpen this result to the point where it readily yields . In prepartion for this result, we need a few more definitions. For , let

Let be such that , and, for ,

(6)

where is implicitly defined by the value of where the set over which is taken in (6) is empty. In words, (6) implies that, for , is the index of the next positive set before which there exists a negative set over which the solution to (1), assuming , hits zero. The “or” in the second line of (6) is to ensure that the time instant when the trajectory hits zero does not coincide with , which is a candidate for for some (cf. Proposition 4). See Figure 2 for an illustration.

Figure 2: Illustration of the procedure in (6) to compute . Specifically, the figure illustrates how (6) determines to be equal to , given . Subscript is not shown for brevity.

Clearly, , which from Proposition 3 is known to contain . The next result shows that in fact the last entries of correspond to .

Proposition 4

Consider a link with inflow function and capacity function , both periodic with period , and the corresponding set defined via (6). Then , and, in particular,

(7)
{proof}

We drop subscript for brevity in notation. Assume that there exists a such that . let be the largest element in such that . Since (by definition), taking into account Proposition 3, is well-defined. Recall the definition of from the proof of Proposition 3, and in particular that is the immediately after . If , then , giving a contradiction. Therefore, . It is easy to see that for some . Therefore, . This in turn would give .4 However, referring to (6), this would imply , giving a contradiction. This proves the first claim in the proposition.

In order to prove (7), observe that if for some , then (6) implies . If we assume that with , then (6) implies that is a point where the queue length transitions from being zero to being positive, giving a contradiction. Therefore, .

Since is not known, (7) can not be used to compute all , . However, (7) readily gives

(8)

from which , , can then be computed as explained in Remark 3 (iii). In order to execute this last step, it is more convenient to use:

(9)

which is obtained by integrating (1) from to , and recalling (6) for the definition of . The entire procedure is summarized in Algorithm 4.

{algorithm}

[htb!] \Input- periodic inflow function and periodic capacity function initialization: ,   compute from (8) and from (9)  compute , , by simulation of (1) with initial condition ; compute , , from (5)  Computation of for isolated link

Let the relationship between and , as determined by Algorithm 4, be denoted by and respectively. These notations will be used in extending the procedure to compute steady state for the network.

4 Steady State Computation For a Network

Algorithm 4 formally describes the steps to compute steady-state for a general network. The number of iterations in the while loop in Algorithm 4 is determined by a termination criterion. While one could explicitly specify the number of iterations for termination criterion, a better criterion can be formulated as follows. For , let be the average outflow from link at steady-state. Integrating (1) over at steady state, we get that , where we use notation from (4). This then gives (cf. Remark 1 for invertibility of ). Therefore, considering monotonicity of the iterates of Algorithm 4 as established in Proposition 5, and letting , a termination criterion could be , for a specified .

{algorithm}

[htb!] \Inputperiodic inflow functions and periodic capacity functions , initialization: ; , , for all while termination criterion is not met do for all :
  compute and from Algorithm 4     compute , Computation of ,

Proposition 5

Consider a network with -periodic external inflows and -periodic capacity functions . The link outflows computed by Algorithm 4 satisfy the following for all : and for all .

{proof}

We prove by induction. Algorithm 4 implies that, for all , . Therefore, Corollary 1 (in Appendix .4) implies that and , and hence for all , , .

Assume that for all . Since , Corollary 1 implies that and for all , , .

for all . An upper bound on can be shown along similar lines as Lemma 3 (in Appendix .3). Combining this with monotonicity from Proposition 5 implies that converges to . Periodicity of for every implies periodicity of . It is easy to see from the construction of Algorithm 4 that, for every iteration : for all . Therefore, for any :

where the first equality follows from periodicity of by construction. Therefore, taking the limit as , we get that, for all :

This implies that