Modeling Traffic Signal Control Using Petri Nets - IEEE Xplore

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 5, NO. 3, SEPTEMBER 2004

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Modeling Traffic Signal Control Using Petri Nets George F. List, Member, IEEE, and Mecit Cetin

Abstract—This paper focuses on the use of Petri nets (PN) to model the control of signalized intersections. The application of PN to an eight-phase traffic signal controller is illustrated. Structural analysis of the control PN model is performed to demonstrate how the model enforces the traffic operation safety rules. This is followed by a discussion of why this modeling tool has future value as the use of more advanced control strategies continue to expand. Index Terms—Petri nets (PN), traffic signal control.

I. INTRODUCTION

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HIS PAPER discusses the use of Petri nets (PN) in modeling traffic signal control. At a signalized intersection, movements (e.g., left turns, through movements, etc.) are combined to form phases, time intervals during which specific movements are given greens simultaneously. To ensure the safe and proper operation of the intersection, the signal controller must not allow conflicting movements to have the right of way simultaneously. However, it should be able to serve all signal phases and transition from one phase to another when specific conditions are met. Before any field implementation takes place, the control logic governing a signal controller should be tested to verify that these requirements are met. In this paper, it is shown that the PN formalism is an effective and advantageous way to develop such signal-control logic. It is also shown that the safety requirements listed above can be easily ensured by analyzing the structural properties of the PN model. The topic of traffic signal control can be separated into two categories: 1) determining what signal-indication sequence to follow in order to optimize the system performance and 2) ascertaining how to implement the signal-control logic. This paper is focused on the second category. Before turning to our main discussion, a brief review on the optimal signal control is useful. There is a vast amount of literature focused on finding optimal control strategies. A variety of mathematical programming methodologies and artificial intelligence (AI) techniques have been used to model the traffic flow and control logic. For example, SCOOT [1], one of the first major real-time traffic control systems, makes continual incremental adjustments in real-time of cycle lengths, splits, and offsets throughout a sig-

Manuscript received August 16, 2001; revised May 3, 2004. This work was supported in part by the U.S. National Science Foundation under Grant 0085694 and by the New York State Energy Research and Development Authority under Grant 1936-EEED-POP-93. The Associate Editor for this paper was P. Ioannou. G. F. List is with the Civil and Environmental Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail: [email protected]). M. Cetin is with the Civil and Environmental Engineering Department, University of South Carolina, Columbia, SC 29208 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TITS.2004.833763

nalized network by using the degree of saturation. SCATS [2], another major system, runs voting algorithms to find the best signal plan from a predetermined set of cycle lengths, splits, and offsets, which can be used in particular traffic-flow conditions. More recent control strategies, such as OPAC [3], PRODYN [4], and UTOPIA [5], operate in a different manner than SCOOT and SCATS and do not explicitly consider cycle lengths, splits and offsets. These systems decide whether or not to switch the traffic lights at each time step [6]. OPAC and PRODYN use dynamic programming to optimize signal timings. Another recently developed and implemented control strategy, traffic-responsive urban control (TUC), utilizes efficient optimization techniques for coordinated control of large-scale networks [7]. In addition to the systems mentioned above, a second category of control strategies uses AI techniques. Signal-control strategies that use neural networks [8]–[10], genetic algorithms (GA) [11]–[13], fuzzy logic [14]–[18], and neurofuzzy control [19] are still under research and are not usually implemented (except [15]). This second group is gaining attention due to the difficulties in mathematical modeling of traffic flow. However, GA-based controllers [11] and fuzzy controllers [17] are claimed to be more efficient in terms of computation as compared to conventional methods for adaptive real-time applications. In order to implement or test any control strategy in the field or simulation environment, the control logic has to be translated into computer code. The public literature does not address the way in which the signal-control logic is implemented. Even though it is possible to find instances where the logic is explained by means of pseudocode [15] or by flowcharts [3], in general, the method (or paradigm) used to translate the logic or the mathematical model, which might be expressed in terms of an objective function and a set of constraints, to executable computer code is not discussed. Even for the systems that have been implemented in the field, it is hard to find discussions about the manner of implementation. It must be clearly defined how/when the indication of signal displays changes. Moreover, the control logic must prevent any conflicts in traffic to occur. As discussed in [20] and [21], PN formalism, among other benefits, provides a clear means for presenting simulation and control logic and can also expedite the generation of control code from the PN graph. As the trend for more sophisticated advanced real-time control strategies grows, the need for an efficient development and evaluation tool will become even more apparent. Making a change in the control logic of a model will require a substantial amount of change to the computer code, whereas PN-based models, with the help of a graphical tool such as NETMAN [22], enable the direct generation of the control code after desired changes are made in the graphical environment.

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Another benefit of using PN, as will be demonstrated in this paper, is the invariant analysis that makes it possible to verify that the control model enforces all the safety rules. This feature becomes more essential, especially for AI-based control strategies in which signal-switching sequences can be relatively “arbitrary.” For example, optimal signal-control strategies produced by a fuzzy-logic controller may not be admissible in terms of traffic safety requirements [23]. In these circumstances, it is the role of the PN to assure that all signal plans are acceptable. Even though the PN framework seems ideally suited to modeling signal control, not much has been done to explore its potential. In 1986, [24] described how a colored PN could be used to model traffic-signal control. Other researchers [25] developed a PN model that described the mutual exclusivity of phases that must be maintained by a multiphase signal and augmented this with a “scheduler” for servicing the phases. It is demonstrated that this “Petri net control with a scheduler” could be implemented on a microcontroller [26]. In [27], a PN model that encompasses both traffic signal-control logic (including phase selection) and traffic flow is presented. And it is demonstrated that the performance evaluation of such models could be accomplished using PN-based simulators such as NETMAN [22] and UltraSAN [28]. The work in [27] has been extended by exploring the application of PN to a widening range of traffic-flow control and modeling situations [21], [29]. As illustrated in [27], PN formalism can be used to model both signal-control logic and flow of vehicles in a network of signalized intersections. Using PN formalism for simulating large-scale signalized transportation networks to evaluate system performance might be computationally inefficient due to the state explosion problem [30]. To address this issue, a model of a network of intersections is developed, where traffic flow at the intersection level is modeled by stochastic timed PN (STPNs) and the movement of vehicles between intersections by random distributions [30]. Another PN-based simulation model of a transportation network that can run on a multiprocessor system is presented in [31]. Most of the PN models reported above are not flexible enough to implement various control strategies. For example, the model in [27] follows a predefined sequence of phase transitions. In contrast, models in [23] and [32] are capable of switching the signal indications in any order instructed by an “optimization layer” (optimization layer is not PN-based). Those latter models have signal-indication subnets (consisting of places and transitions) for each traffic movement (e.g., south bound through, left turns, etc.). In an adaptive traffic-control environment, the phase-transition logic has to facilitate almost any kind of phase-transition sequence. If the signal-control model is interfaced with an “optimization layer” that determines which phase should be activated next, then the control model must be capable of transitioning from any current phase to any other phase. The PN model for traffic-signal control presented here enables such phase transitioning and has also the following advantageous features. — Modularity: Overall PN model is made up from individual submodels. This feature enables easy model development and improves readability.

Fig. 1. Movements and phases. The first column of the table in this figure indicates the phase number, whereas the second shows the movements that are allowed for each signal phase. All eight possible movements are shown in the diagram.

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Functionality: PN model includes the elements needed to model a real-world actuated or semi-actuated signal controller. For example, it explicitly models minimum green, max green, and forceoffs, etc. This model also includes aspects that make it suitable to be used in a simulation environment and it can easily be interfaced to other system layers, such as optimization and traffic-flow layers. The PN model emphasizes modularity and the functionalities mentioned above more than the other PN models for trafficsignal control that we surveyed. Another important contribution of this paper is the behavioral analysis of the PN model. We use -invariants to ascertain that the control logic is reliable, which has not been done before in the traffic-signal-control literature. We also prove that the PN model is live, i.e., no possibility of a deadlock, using the reachability tree method. After describing the traffic-signal-control issues in the next section, a PN model that has the characteristics mentioned above is presented in Section III. Section IV presents the results of the behavioral analysis of this PN model. A discussion on the use of a PN paradigm in modeling traffic-signal control is provided in Section V and, finally, the conclusion is given in Section VI. II. TRAFFIC-SIGNAL CONTROL Before proceeding further with the logic description, it is useful to define two terms—movement and phase—with the help of Fig. 1. A movement is a specific traffic flow that occurs at the intersection. In our case, each intersection has eight movements, 0–7. Movements 0, 2, 4, and 6 are left turns, while movements 1, 3, 5, and 7 are through-right combinations. (Technically, the throughs and rights are separate movements, but in our model they are combined.)

LIST AND CETIN: MODELING TRAFFIC SIGNAL CONTROL USING PETRI NETS

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Fig. 3. Indication display module. There are eight of those modules, one for each movement. Solid bars are immediate transitions and small rectangles are timed transitions. We refer to the places of this net by using either the numbers inside the circles or the labels next to them.

Fig. 2. Movements and phases. The first column of the table shown in the diagram above indicates the phase number, whereas the second one shows the movements that are allowed for each signal phase. All eight possible movements are shown in the diagram.

Movements are paired to form phases. When a phase is selected, greens are displayed for both of the movements involved, while all other movements receive a red. For a typical intersection, like this one, eight phases are usually specified. Listed in the middle of the diagram, they run from phase 0, which combines movements 0 and 4, to phase 7, which combines movements 3 and 7. Without a loss of generality, we can assume that phase 0 represents east- and westbound lefts, phase 1 represents green for the westbound movements, and phase 7 represents green for the north- and southbound throughs and rights. A typical pretimed phase rotation would be 0, 3, 4, and 7. This provides greens for all lefts, throughs, and rights in a simple predictable pattern. For a fully actuated signal, all seven phases can be displayed with skips allowed (e.g., 0, 2, 3, 7, 1, 3, 4, 6, and 7). Ideally, an optimum signal-switching strategy aims to provide a solution in which no vehicle ever stops at an intersection. In principle, phase transitions at all intersections organize the traffic such that a green band exists for every vehicle from every origin to every destination. So far, no such control strategy exists. Most common control strategies are: pretimed (fixed phase sequence and durations), fully actuated (all phase lengths depend on the traffic demand, detected by loops in the pavement), semi-actuated (minor street phase is demand responsive), and queue management (signal switching based on queue lengths). In actuated operations, minimum and maximum greens for each movement as well as extension interval must be specified. A minimum green is the duration that a phase has to remain green before any switching can be made. Maximum green is the maximum duration that a signal can display green. Extension interval is the time that the green is extended for each ve-

hicle arrived at the detector from the instant of the arrival at the detector. The PN model described in the next section is designed so that any one of those control strategies can be accommodated easily. III. PN REALIZATION A similar version of the signal-control PN model described here was used in [33] as a part of a larger PN model for simulating traffic in a network of signalized intersections. Beside the signal-control model (layer), there are two other system layers needed to simulate traffic under various signal-control strategies: an optimization layer and a simulation layer. The interfacing of these layers to the signal-control model (layer) is achieved through some places and transitions of the control net, which are explained at the end of this section. Control over the signal timings is split between the PN model code (optimization layer). The PN and user-generated C covers the logic concerning signal indications (green, yellow, and red) and the transitions between indications (one light goes red before another goes green). The choice of which phase to service next and signal timing are contained in the user-generated code. The signal-control net for an eight-phase signal controller is presented in Fig. 2. The net actually contains 64 separate subnets. The eight subnets in the upper part, each of which looks like a backward letter “N,” control the red and green signal indications for each movement. The 56 subnets that look like paired spiders control the transitions from one signal indication (phase) to another (phase). Redundant in structure but different enough that the repetition is useful, they cluster into eight sets of seven. Each set of seven captures the logic involved in transitioning from one specific signal indication (phase) to the other seven possible signal indications. Each of these subnets is described in more detail below.

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Fig. 4. Phase-transition module for phase 0. The index next to the capital letters inside all circles (except C01, C02,. . .,C07) identifies to which movement the place belongs. For example, RR4 is the RR place of movement 4 (see Fig. 3). The upper transitions (t12, t22,. . .,t72) are timed transitions; this time corresponds to the all-red duration.

Initial marking of the indication display logic subnets determines the initial signal indications. A generic indication display subnet is shown in Fig. 3. There are eight indication display subnets, one for each movement. For each subnet there is a token either in place Go Green (GG) or in Rest Red (RR) initially. A. Greens and Reds Fig. 3 shows the logic for displaying greens, yellows, and reds for a given movement, as well as for enabling vehicle flow. Any transition in this net, with the exception of transitions actuations (Act) and force off (FO), becomes enabled once a token is deposited in all input places and fires after the specified time expires. The enabling of Act and FO is predicated on other conditions, as will be described later. It is easiest to start the description by assuming a single token has been deposited in the GG place. This enables the transition immediately above (t1) and results in tokens being deposited in the initial green (place 1), Display Green (DG), and Movement (M) places. The token in M enables vehicles to move, and the token in DG will enable a green termination sequence to start once a token is deposited in Go Red (GR). We can leave that for now. Proceeding up the left-hand portion of the subnet, once the transition for initial green (Min) has finished firing, a token is deposited in Extend Green (EG). This token remains until one of three conditions is met. Either a timeout occurs between Act if the intersection is operating in a demand responsive mode (fully or semi-actuated); an FO occurs (in coordinated semiactuated mode, the phase is terminated by supervisory control to maintain synchronization) or maximum green (Max) is reached. As long as there are vehicles sensed in the approach, the “Act” transition will reinitialize and not fire. When any one of these three transitions fires, the token in EG is removed and a token is deposited in Rest Green (RG). This does not necessarily mean that the green will terminate, but just that it can.

As will be explained in the next paragraphs, phase-transition modules remove the token in RG and deposit a token in GR of a movement if the green for this movement is to be terminated. Once a token is deposited in GR, the transition at the bottom of the right-hand leg (t2) becomes enabled. When t2 fires, the green for this movement terminates and a token will be deposited in Yellow (Y) and transition “Yel” will begin timing. When transition “Yel” fires, t3 becomes enabled and fires immediately. Tokens in place 8 and M will be removed and a token will be deposited in RR. Removing the token from M stops vehicles from being able to enter the intersection and depositing a token in RR enables the signal to indicate a red. B. Phase Transitions Transitions from one phase (movement combination) to another are handled by the eight seven-cluster subnets across the bottom sections of Fig. 2. Displayed close up in Fig. 4, each of these makes a provision for the signal to transition from one phase to any one of the other seven phases that are possible. Note that all places involved in phase-transition subnets are the same places in indication display nets, except the places in the middle of each subnet (C01, C02, etc.). The control network is drawn in this way for clear interpretation. The timed transitions in the upper part of Fig. 4 (t12, t22, etc.) account for all-red durations. In other words, once one of these transitions is enabled, all signals display red for a short duration (usually 2 s) before the next phase can start. Each vertical subnet in Fig. 4 represents a transition (phase change) from a common initial phase condition (phase 0 in this figure) to another (phases 1–7). The three or four places across the bottom of each subnet represent the signal indication conditions that must be met for that transition sequence to commence. For example, for the cases where four places are involved, the two movements currently in green must be in RG and the other two movements must be in RR.


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For illustrative purposes, let us assume that there is a phase transitioning from phase 0 (movements 0 and 4) to phase 7 (movements 3 and 7). For the transition t71 (at the bottom of the subnet on the right) to be enabled, movements 3 and 7 must be in RR (RR3 and RR7) and movements 0 and 4 must both be in RG (RG0 and RG4). When t71 fires, tokens are removed from all the bottom places and tokens are deposited in Go Red 0 (GR0) and Go Red 4 (GR4). A token is also deposited in C07 as an indicator that this phase change is underway. When the signal indication module (see Fig. 3) reaches the point where movements 0 and 4 have both arrived in red (RR0 and RR4), then the upper transition times down and fires and tokens will be deposited in the four topmost places: RR0, RR4, GG3, and GG7. This same logic pertains to all the other phase transitions with one exception. Where a phase transition does not involve having one or the other of the two movements go red, only three conditions must be satisfied before the transition can occur and only three final actions need to be taken. Examples are phase changes from phase 0 to either 1 or 2; from 1 or 2 to 3; from 4 to either 5 or 6; and from either 5 or 6 to 7. Not shown in Fig. 4 is a common resource place (RS) that is connected to all transitions in Fig. 4, as well as to the transitions of all other remaining seven phase-transition modules. This resource place contains one token as an initial marking. The immediate transitions (t11, t21, 31, etc.) remove this token and the upper timed transitions (t12, t22, t32, etc.) put it back when a phase transition is completed. This resource place prevents two phase transitions from occurring simultaneously. Therefore, when a phase transition is in progress, no other phase transition can occur.

phase to service determines which of the seven possible immediate transitions is to be enabled. There also is a connection between the optimization layer and the FO transition of indication display modules. The transition FO (see Fig. 3) can fire (provided that there is a token in EG) if instructed to do so by the optimization layer. Firing FO forces a phase-transitioning sequence to start. (In actuated controllers operating in coordinated mode, force offs are used in exactly this fashion to terminate greens.)

C. Interfacing to Other Layers

This section presents the results of place invariant ( -invariant) analysis. This analysis confirms that the control logic implemented in PN enforces the safety rules of traffic operation. As explained in Section III, there are eight signal-indication display modules (Fig. 3) and eight phase-transition modules (Fig. 4) in an intersection-control model (Fig. 2). The signal display for each movement changes to green, yellow, and red when t1, t2, and t3 of Fig. 3, respectively, fires. For example, when t3 fires, the signal changes to red and stays in red until t1 fires. Hence, if there is a token in place 9, then the signal is red. However, a lack of tokens in place 9 (RR) does not mean that the signal cannot be in red. In the invariant analysis, when we refer to a signal being in RR, we only refer to the case when there is a token in place 9 (or RR). During the phase transitions, that will change the signal of a particular movement from red to green; the token in place 9 of this movement will be removed and deposited in place 0. The duration of this phase transition is equal to the durations of yellow plus all-red. For that time period, there is no token in place 9, but the signal still displays red. For brevity in the analysis, places 0–3 (the left-hand branch of Fig. 3) are combined and denoted by G, 6–8 by Y, 5 by M, and 9 by RR. For example, G6 represents all four places (0–3) of movement 6. Similarly, RR6 refers to the RR place of the same movement. The reason for this notation is that those places appear together in many -invariants. It is important to realize that at any given instance there cannot be more than one

The signal-control model (layer) described previously can be interfaced with an optimization layer and a layer for simulating traffic flow. The places and transitions of indication display and phase-transition modules that are involved in the interfacing are briefly described later. For traffic simulation, we need to determine when the vehicles are allowed to move. As mentioned before, depositing a token in M (see Fig. 3) allows vehicles to move and removing the token stops vehicles from being able to enter the intersection. The fact that the token in M is not removed until the yellow has finished timing means that vehicles are allowed to still enter the intersection on yellow, which is conventional practice in most states. Another element of the signal display modules that is used for interfacing between signal control and simulation is the transition labeled “Act.” This transition cannot fire unless a timeout occurs between actuations, if the intersection is operating under demand-responsive mode. The signal-control model also interfaces with the optimization layer (the C code), which determines the best phase sequence. Before a phase transition occurs, the optimization layer determines which phase to serve next. Therefore, transitioning from one phase to another depends on the input from the optimization layer. The interfacing of the phase-transition modules to the optimization layer is achieved via the immediate transicode that selects the next tions (e.g., t11 and t21). The C

IV. ANALYSIS OF THE CONTROL NET A traffic-signal-control model must have certain features for proper and safe operation. Before a control strategy or model is implemented, it is necessary to make sure that the model is error proof. For example, the controller should not lock up (deadlock) due to some unexpected combination of actions (inputs), should not allow conflicting movements to have right of way simultaneously, should be able to serve all signal phases and return to some initial state, and should not allow phase changes unless all the required conditions are met. This list can be expanded. A major strength of PN is the availability of methods for analyzing the properties of the model. Those properties of the PN model reveal whether the model is reliable or not. In this section, those properties of the control net are investigated and related to the operation of the signalized intersection described before. For brevity, not all properties of the control net are presented. A. P-Invariants

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marked place among places 0–3. The same is true for places 6, 7, and 8. Places in the middle of phase-transition modules are where represents the phase that is going to denoted by terminate and is the phase to be activated. For example, C05 of Fig. 4, when marked, represents a phase change from phase 0 to phase 5. Before we start discussing the -invariants and their implications, it is useful to observe some facts about the structure of the network. The signal-display logic depicted in Fig. 3 ensures that the signal display follows the same sequence of color display all the time; green, yellow, and red. As mentioned in Section III, the control logic is being initialized by depositing a token in places GG or RR of all eight modules. Careful examination of the modules in Figs. 3 and 4 reveals that there always is one token (not two) in any given place if this place is marked (it is also proven in the next section that the net is 1-safe). Fig. 4 shows the phase transitions from phase 0 to all other phases. Notice that, during a phase transition, there are not any tokens in the display module of a movement that is going to change from red to green. For example, during the transition from phase 0 to phase 3, the tokens in places RR1 and RR5 of movements 1 and 5 will be removed and, after the phase transition is completed, a token will be deposited in places GG1 and GG5 of those two movements. The -invariants are found by solving the homogeneous , where is the incidence matrix. The equation solutions to this equation give the -invariants. Solving this equation for our PN model yields 28 minimal -invariants. (All places in the net are covered by these invariants, so the net is bounded.) To interpret the meaning of these -invariants, they . For any marking , are multiplied by the initial marking . Some of the -invariants of the control network are explained below. Referring to Figs. 1, 3, and 4 while reading the analysis below would help in understanding the interpretation of the -invariants. We assume that initial marking , GG GG , of the control net is as follows: RS and RRj for j 1, 2, 3, 5, 6, and 7; all other places have no tokens. This marking corresponds to phase 0 being green initially. Overall, there are two kinds of -invariants in the control PN model. One group deals with individual movements in particular (invariants 1 to 16) and the other with the interactions of different movements. It is the second group that deserves more attention, since those invariants will provide evidence that the conflicting movements are not being allowed simultaneously. 1) Invariants 1–16: There are two minimal invariants for each one of eight signal-indication modules (Fig. 3) that add to 16 invariants, which are as follows: ( stands for marking and is omitted later on). : This invariant states that if there is one token in place 5, then there must be a token in either place 4, 7, or 8. In other words, if there is traffic movement on one approach, then the signal indication for that approach has to be either green or yellow. : If the signal indication is green , then either the minimum duor there is actuation ration is not completed or the green is about to teron that approach or ). minate (

Y RR for 0, 1, 2, 2) Invariant 17: G 6, 7. 3, 4, 5 and This invariant states that there can be only one token in any ’s repreone of the places involved in the equation above. sent, when marked, the state of the system when there is a phase transition from other phases to phases 6 and 7 (these are the only phases that contain movement 3). For example, C means that there is a phase transition from phases 0 to 7 (see Fig. 4). This invariant can be interpreted as follows. If there is a transition (from a phase that does not contain movement 3) to movement 3 (signal 3 will be green soon), then all the places of the indication display module of 3 must be empty. In other words, this signal cannot be in green, yellow, or RR. As mentioned before, RR only refers to the case where a token , exists in place RR. If, for example, we assume that C then the token in place RR of movement 3 is removed. However, the signal display for movement 3 is still red, since t1 of ’s movement 3 has not fired yet. In summary, if one of the is equal to one, then the signal display is cannot be in yellow or green. There exist similar invariants for the other remaining seven movements. RR Y RR Y 3) Invariant 18: Y RR Y RR . This invariant asserts that, at any given time, the sum of tokens in the right-hand side branch of display indication modules (Fig. 3) of movements 4, 5, 6, and 7 must be 3. This constraint can be interpreted as follows. At any given time, movements 4, 5, 6, and 7 cannot be all in RR or yellow. This invariant does not declare that at least one of those movements is green at any given time (this assertion is true if we ignore phase transitions). During phase transitions, all of the movements can actually display red. Notice that movements 4, 5, 6, and 7 are all conflicting movements. So, no two of those movements can be yellow (because of the single resource token in RS) or green (because of this invariant) simultaneously. If we assume that 4 is green and that all others (5, 6, and 7) are red, then this equation is satisfied. If 4 is yellow (movement 4 is about to terminate), there is a phase transition (from phase 0 or 2 to any other phases); then, exactly one of movements 5, 6, and 7 is neither in yellow nor in RR. This last sentence says that either one of 5, 6, or 7 will be green in the next phase. If we assume Y ; then, by Invariant 17, there cannot be that C RR . any tokens in Y or RR places of movement 7, Y Similar conclusions can be drawn for the other movements. There exists a similar invariant for movements 0, 1, 2, and 3 as well. RR Y RR Y 4) Invariant 19: Y RR Y RR . Ignoring any phase transitions, this equation claims that if movements 2 and 3 are both in RR then, 6 and 7 must be in RR as well. If either 2 or 3 is green, then either 6 or 7 has to be green. During phase transitions, say from phase 7 to phase 6, movement 3 will be green (2 must be red), 7 will be yellow (or RR , because red), and, finally, 6 is red, but this transition will remove the token in place RR of movement 6. G G G C C C 5) Invariant 20: G C C C C C .


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We can start the analysis of this invariant by assuming that . In order to satisfy the above movement 0 is green G equality, one of the positive terms has to be true (i.e., 1). Among four positive phase transitions, only two can be true: C01, C10, C23, or C32 can not take place because those will require movement 1 to be green (0 and 1 can not be green by Invariant 18). To be brief, if movement 0 is green, then 4 or 5 must be green or there must be a transition from either • •

phase 0–phase 1, C phase 1–phase 0, C

: G : G

Similarly, if movement 1 is green G be green or there is a transition from either • •

phase 2–phase 3, C phase 3–phase 2, C

G G

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, then 4 or 5 must Fig. 5.

Reduced indication display logic.

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6) Invariant 21: RS C C C C C C C C G G G G . This invariant verifies two properties about the phase-transition logic.

control logic modeled by PN is accurate and it does not allow any mishaps to occur.

, If there is no phase transition in progress RS then one of the movements 4, 5, 6, and 7 has to be green. If one of the phase transitions (in the equation above) is in progress, then one of the movements 4, 5, 6, and 7 has to be green.

As stated in [34], there are three basic ways to analyze a PN: 1) reachability tree method; 2) matrix equation approach; and 3) reduction or decomposition method. Each technique has its advantages and disadvantages. The first approach that involves the enumeration of all possible states is employed here to analyze the liveness of the control PN. Since the size and number of states are limited in our PN model, this approach seems to be feasible. Also, it is intuitive in this approach to see that there is no possibility of a deadlock in the network. In order to minimize the size of the reachability tree, the indication display logic module is reduced to the form shown in Fig. 5. This transformation preserves the liveness property of the net [34]. In essence, the phase transition can also be reduced (i.e., all the self loops can be eliminated). As explained in Section III, there are eight of those signal indication modules, one for each movement. Three transitions labeled as Ti1, Ti2, and Ti3 change the signal display to green, yellow, and red, respectively, when fired. The subscript stands for the number of the movement. For example, T33 is the transition, when fired, that changes the signal indication of movement 3 to red. To see how states of the systems evolve, we can asRS sume that the initial marking is M GG GG RRj for and all other places have no tokens . There are only two transitions that are enabled under this marking, T01 and T41, which can fire in any order and proand , as shown in Fig. 6. In order to save duce marking space, only those places that contain a token are included in the and nodes of the tree. Other places have zero tokens. From , we get by firing T41 and T01, respectively. corresponds to phase 0 where movements 0 and 4 have green. As shown in Fig. 6, any one of the transitions t11, t21, t31, t41, t51, t61, and t71 (lower case is used to distinguish the transitions of the phase-transition modules; see Fig. 4) can fire and produce their respective markings. Note that when one of these transitions fires, then the others cannot fire afterward, because the common resource token will be consumed (without this re, T42 fires striction there is a possibility of a deadlock). From

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As depicted in Fig. 1, all phases involve exactly two movements. All phase transitions that appear in this invariant keep the signal indication for one of the movements in green and terminate green of the other movement. For instance, C02 reflects a phase transition that terminates only movement 0 and initiates green for movement 1. During this phase change, movement 4 stays green. Y RR Y RR 7) Invariant 22: Y RR G for 2, 3, 4, 5, 6, 7 and 0, 1. Among other conclusions, this invariant states that if movement 0 is green, then 1, 6, and 7 are RR. In other words, neither 1, 6, nor 7 can be yellow if 0 is green. If any one of those three movements is yellow, then the above equality would not hold. For instance, if 1 is yellow, then there must be a transition from either phase 2 or 3 to one of the other phases 4, 5, 6, or 7. From Invariant 17, we know that if there is a transition to a phase, then the signal of that phase cannot be in green, yellow, or RR. Therefore, this phase transition cannot take place, because it will remove the token from place RR of either movement 6 and 7. Of course, this will violate the equality above. , then 1, 6, and 7 are all in RR or one Similarly, if any of them is yellow and the other two are RR. In summary, if 0 is green or is about to become green, then 1, 6, and 7 have to be RR. Invariant 19 proved that if 6 and 7 are red, then 2 and 3 must be red as well. Invariants 19 and 22 prove that when vehicles on approach 0 are moving, all possible conflicting movements (1, 2, 3, 6, and 7) are stopped. Since the signal display modules for each movement are identical, it can be concluded, by symmetry, that there is no possibility for any two conflicting movements to occur simultaneously. More information on the rules supported by the -invariants can be inferred. The partial analysis reported above proves that the

B. Deadlock Analysis

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Fig. 6. Reachability tree.



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and changes the signal of movement 4 to yellow. Then, T43 and t12 fire to transition the signal system to phase 1. enables both T02 and T42. After firing these two will be transitions and T03 and T43 in some order reached. From this marking, t32 becomes enabled and fires. This will produce marking , which corresponds to the initiation of phase 3. enables T02. The structure of this branch is similar to the first (the one that results in phase 1). In essence, if we follow any , we reach one of the other phases, i.e., phases branch after 1–7. Incomplete branches are left out intentionally to save space. In summary, if we initialize the control net with a marking that corresponds to one of the phases, then there is no possibility of a deadlock. Fig. 6 demonstrates this proof for phase 0. Because of the symmetry in the control PN model, the reachability trees of any other phases will look exactly the same as Fig. 6, except that the markings and transitions will be different. From this analysis, it can be concluded that: 1) the network is safe because all markings in the tree contain only 0’s and 1’s; 2) there are no dead-end states in the tree, so therefore the net is live; and 3) the net is reversible because we can always find a firing sequence that brings the system back to the initial marking.

Further, with PNs it is possible to test whether the logic can deadlock—that is, reach a state from which it is impossible to transition to any other. This means that before implementing the control logic in the field, one can be sure that the controller will not “hang up,” that is, receive some combination of inputs that cause it to reach an impasse from which it cannot recover. As we hope is apparent from Figs. 2–4 and the discussion that pertains to them, PNs are fairly easy to understand. Unlike most computer programs, once one is familiar with the elements involved and how they interact, it is possible for one person to look at the PN developed by another and quickly gain a fairly clear understanding of the control logic being followed. PNs are, in effect, a graphical language by which one analyst can communicate his or her intentions with another with a fairly low probability that those intentions will be misinterpreted. Moreover, the syntax of the “language” is very compact. As a matter of fact, Fig. 2 presents the logic that is sufficient for a network of a limitless number of intersections; the differentiation among intersections can be affected through the use of colored tokens [32], [33]. The PN model presented in this paper has also been applied to a six-intersection network [33], where a number of signal timing strategies (e.g., fixed, queue management, fully actuated, etc.) were tested and evaluated. In that paper, the interface between the signal-control layer and the optimization and traffic simulation layers was described. Optimal signal timing and traffic performance optimization are outside the scope of this paper. However, the reader can refer to the references for details on these topics. Finally, in addition to modeling signal-control logic, the PN formalism can be used in other application areas. The authors have explored its use in the context of toll booth facilities [35], unsignalized intersections, roundabouts, and nonhighway domains such as airport runways [36].

V. DISCUSSION Without a doubt, one of the main questions we must address is: why should researchers in the traffic engineering field be asked to consider yet another tool for modeling traffic network situations? The answer lies in the features and advantages that PNs afford. Most importantly, once the control logic presented in a diagram, such as that shown in Fig. 2, is deemed acceptable, there are programs available that can translate that logic directly into executable code—typically using C as an intermediate step—ready for downloading into a controller’s microprocessor. This means error- and bug-free code can be created immediately, without an intervening step in which a computer programmer translates the logic articulated in the diagram into field-ready programs. This feature of a PN also means that the lead time from development to implementation can be shrunk dramatically. It is conceivable that a control strategy can be developed in the laboratory on one day and implemented in the field the next. Particularly in light of the ITS environment, where the list of desirable features and capabilities is likely to outstrip our ability to create the code for carrying them out, this automatic code generation capability of PNs is likely to have significant value. Moreover, the need to debug this code is eliminated. It is also important to note that the code that is being tested in the laboratory is the exact same code that will be implemented in the field, with all of its features and attributes. There is no difference between that code and the code that will be implemented in the field. If the traffic-flow simulator employed is faithful to the way traffic actually behaves, there should be no surprises or unexpected side effects from implementing the control logic.

VI. CONCLUSION This paper has described how PN can be used to model traffic control situations in urban networks. The model presented achieves modularity through the use of subnets; changes can be made to one component without affecting the subnets for other parts. The models allow easy changes to the traffic control logic, timing, and coordination and the assumptions on network input flows. Structural analysis of the traffic-signal-control model is performed. -invariant analysis proves that the control net does not allow any conflicting movements to occur simultaneously. It also verifies that the rules of the control logic (such as which movements can be green at the same time) related to signal indications are preserved in the PN model. The reachability tree analysis shows that the model is deadlock free. The authors believe that this area of research is one that has significant promise for the future, especially in light of the increasing demands for more features and capabilities, especially real-time control, that are being placed on advanced traffic-management systems.

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Our own future research will concentrate in two areas: developing innovative intelligent distributed traffic-control strategies that take full advantage of advanced technologies and the development of tools that allow PN to be used to their fullest potential in intelligent transportation system research and implementation. These include analysis, synthesis, performance evaluation, and control.

ACKNOWLEDGMENT The authors would like to thank A. A. Desrochers for his assistance with the analysis of the PN model.

REFERENCES [1] R. Robertson and R. D. Bretherton, “Optimizing networks of traffic signals in real time—the SCOOT method,” IEEE Trans. Veh. Technol., vol. 40, pp. 11–15, Feb. 1991. [2] A. G. Sims and K. W. Dobinson, “The Sydney coordinated adaptive traffic (SCAT) system philosophy and benefits,” IEEE Trans. Veh. Technol., vol. VT-29, pp. 130–137, May 1980. [3] N. H. Gartner, P. J. Tarnoff, and C. M. Andrews, Evaluation of optimized policies for adaptive control strategy, in Transportation Research Record 1324, Trans. Res. Board, Washington, DC, 1991. [4] J. J. Henry, J. L. Farges, and J. Tuffal, “The PRODYN real time traffic algorithm,” in Proc. 4th IFAC/IFORS Conf. Control Transportation Systems, Baden-Baden, Germany, 1983, pp. 305–309. [5] C. DiTarnato and V. Mauro, UTOPIA, in Reprints of Control, Computers, Communications and Transportation (CCCT 89), IFAC, IFIP, IFORS, Paris, France, 1989. [6] W. M. Wey, R. Jayakrishnan, and M. G. McNally, “A local feedback controller for oversaturated intersection control based on dynamic road traffic models,” in Proc. IEEE. Vehicle Navigation Information Systems Conf., 1995, pp. 172–178. [7] V. Dinopoulou, C. Diakaki, and M. Papageorgiou, “Simulation investigation of the traffic-responsive urban control strategy TUC,” in IEEE Proc. Intelligent Transportation Systems Conf., Dearborn, MI, 2000, pp. 458–463. [8] J. Hua and A. Faghri, Development of neural signal control system—toward intelligent traffic signal control, in Transportation Research Record 1497, Trans. Res. Board, Washington, DC, pp. 53–61, Jul. 1995. [9] D. C. Chin, J. C. Spall, and R. H. Smith, “Evaluation of system-wide traffic signal control using stochastic optimization and neural networks,” in Proc. 1999 American Control Conf., vol. 3, 1999, pp. 2188–2194. [10] R. K. Saraf, “Adaptive traffic control using neural networks,” Ph.D. Dissertation, Dept. Civil Environ. Eng., Vanderbilt Univ., Nashville, TN, 1994. [11] G. Q. Memon and A. G. R. Bullen, Multivariate optimization strategies for real-time traffic control signals, in Transportation Research Record 1554, Trans. Res. Board, Washington, DC, pp. 36–42, 1996. [12] G. Abu-Lebdeh and R. F. Benekohal, Genetic algorithms for traffic signal control and queue management of oversaturated two-way arterials, in Transportation Research Record 1727, Trans. Res. Board, Washington, DC, pp. 61–67, 2000. [13] B. B. Park, C. J. Messer, and T. Urbanik, Enhanced genetic algorithm for signal-timing optimization of oversaturated intersections, in Transportation Research Record 1727, Trans. Res. Board, Washington, DC, pp. 32–41, 2000. [14] A. Tzes, W. R. McShane, and S. Kim, “Expert fuzzy logic for traffic signal control for transportation networks,” in Proc. Institute of Transportation Engineers 65th Annual Meeting, Denver, CO, 1995, pp. 154–158.

[15] J. Nittymaki, “Installation and experience of field testing a fuzzy signal controller,” Eur. J. Operat. Res., vol. 131, pp. 273–281, 2001. [16] Q. Lin, B. W. Kwan, and L. J. Tung, “Traffic signal control using fuzzy logic,” in Proc. IEEE Conf. Syst., Man, Cybern., vol. 2, 1997, pp. 1644–1649. [17] J. H. Lee and H. Lee-Kwang, “Distributed and cooperative fuzzy controllers for traffic intersections group,” IEEE Trans. Syst., Man, Cybern., pt. C, vol. 29, no. 2, pp. 263–271, 1999. [18] T. M. Sayers, M. G. H. Bell, T. Mieden, and F. Busch, “Improving the traffic responsiveness of signal controllers using fuzzy logic,” in Proc. IEE Colloq. Urban Congestion Management, 1995, pp. 6/1–6/4. [19] E. Bingham, “Reinforcement learning in neurofuzzy traffic signal control,” Eur. J. Operat. Res., vol. 131, pp. 232–241, 2001. [20] G. F. List and R. Troutbeck, “Advancing the frontier of simulation as a capacity and quality of service analysis tool,” Transport. Res. Circ., pp. 485–502, 2000. [21] F. DiCesare, P. Kulp, M. Gile, and G. F. List, “The application of Petri nets to the modeling, analysis and control of intelligent urban traffic networks,” in Proc. 15th Int. Conf. Application Theory Petri Nets, R. Valette, Ed., Zaragoza, Spain, June 1994, pp. 2–15. [22] F. DiCesare, M. Gile, and P. Kulp, “A code generation tool for simulation and control, based on colored Petri nets,” in Proc. IEEE Conf. Syst., Man, Cybern Intelligent Systems 21st Century, vol. 4, 1995, pp. 3063–3068. [23] J. L. Gallego, J. L. Farges, and J. J. Henry, “Design by Petri nets of an intersection signal controller,” Transport. Res., pt. C, vol. 4, no. 4, pp. 231–248, 1996. [24] K. Jensen, “Colored Petri nets,” in Petri Nets: Central Models and Their Properties, ser. Lecture Notes in Computer Science . New York: Springer-Verlag, 1986, vol. 254, pp. 248–299. [25] A. Cois, A. Fanni, and A. Giua, “An expert system for designing and supervising a discrete event model,” in Proc. IEEE Int. Workshop Intelligent Motion Control, Istanbul, Turkey, 1990, pp. 103–107. [26] A. Giua, “A traffic light controller based on Petri nets, CML final project report,” Dept. Elect., Comput., Syst. Eng., Rensselaer Polytechnic Inst., Troy, NY, 1991. [27] H. Wang, “Modeling and analysis of traffic signal systems using Petri nets,” Ph.D. dissertation, Dept. Civil Eng., Rensselaer Polytechnic Inst., Troy, NY, 1992. [28] W. H. Sanders, W. D. Obal II, M. A. Qureshi, and F. K. Widjanarko, “The UltraSAN modeling environment,” Perform. Eval., vol. 24, no. 1, pp. 89–115, 1995. [29] H. Wang, G. F. List, and F. DiCesare, “Modeling and evaluation of traffic signal control using timed Petri nets,” in Proc. IEEE Conf. Syst., Man, Cybern. Syst. Engineering Service of Humans, vol. 2, 1993, pp. 180–185. [30] J. Wang, C. Jin, and Y. Deng, “Performance analysis of traffic networks based on stochastic timed Petri net models,” in Proc. 5th IEEE Int. Conf. Engineering Complex Computer Systems (ICECCS’99), 1999, pp. 77–85. [31] A. Tzes, K. Seongho, and W. R. McShane, “Application of Petri networks to transportation network modeling,” IEEE Trans. Veh. Technol., vol. 45, pp. 391–400, May 1996. [32] F. DiCesare, G. F. List, T. Myers, M. Gile, and P. Kulp, “Petri net modeling and simulation for intelligent control of traffic networks,” presented at the IEEE–SMC Conf. Computational Engineering Systems Applications, Lille, France, July 1996. [33] G. F. List and F. DiCesare, “A prototype real-time control system for signalized networks: Final report,” Center Infrastructure Transport. Studies, Rensselaer Polytechnic Inst., Troy, NY, July 1998. [34] T. Murata, “Petri nets: Properties, analysis and applications,” Proc. IEEE, vol. 77, pp. 541–580, Apr. 1989. [35] L. Elefteriadou, J. D. Leonard II, G. F. List, H. Lieu, M. Thomas, R. Giguere, G. Johnson, and R. Brewish, Beyond the highway capacity manual: Framework for selecting simulation models in traffic operational analyzes, in Transportation Research Record 1678, Trans. Res. Board, Washington, DC, pp. 96–106, Nov. 1999. [36] M. Cetin, “Integrating information and physical flow layers in transportation systems models,” Ph.D. dissertation, Dept. Civil Eng., Rensselaer Polytechnic Inst., Troy, NY, 2002.


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George F. List (S’74–M’79) received the B.S.E.E. degree from Carnegie Mellon University, Pittsburgh, PA, in 1971, the M.E.E. degree from the University of Delaware, Newark, in 1976, and the Ph.D. and C.E. degrees from the University of Pennsylvania, Philadelphia, in 1984. He is Chair of the Department of Civil and Environmental Engineering and Director of the Center for Infrastructure and Transportation Studies, Rensselaer Polytechnic University, Troy, NY. He also holds an appointment in the Department of Decision Sciences and Engineering Systems. He is best known for his work in the modeling, simulation, and optimization of transport systems and networks. His current research focuses on the creation of observability in transportation systems and the use of real-time data to enhance the operation of traffic networks. Dr. List was a 1999 Finalist in the Edelmann Prize Competition (INFORMS) and is the 2003 Recipient of Rensselaer’s Darrin Counseling Award. He is a Member of the Highway Capacity and Quality of Service Committee, the Transportation Research Board, and the Executive Board for the Intelligent Transportation Society, New York (State) (ITS-NY). He is a Fellow of the ASCE and a Member of TRB, ITE, and INFORMS.

Mecit Cetin received the B.S. degree in civil engineering from Bogazici University, Istanbul, Turkey, in 1995 and the M.S. degree in civil engineering and the Ph.D. degree in transportation engineering from Rensselaer Polytechnic Institute (RPI), Troy, NY, in 1999 and 2002, respectively. He was a Postdoctoral Research Associate with RPI and currently is with the Department of Civil and Environmental Engineering, University of South Carolina, Columbia, as an Assistant Professor. His main areas of expertise are intelligent transportation systems (ITS), modeling and simulation of transportation systems, and freight transport. Dr. Cetin is a Member of ASCE and INFORMS.