Dynamic Network Traffic Assignment and Simulation ... - Springer Link

Networks and Spatial Economics, 1: 2001 267±292 # 2001 Kluwer Academic Publishers, Manufactured in the Netherlands.

Dynamic Network Traf®c Assignment and Simulation Methodology for Advanced System Management Applications HANI S. MAHMASSANI The University of Texas at Austin, Austin, Texas 78712, U.S.A. e-mail: [email protected]

Abstract Evaluation and operation of intelligent transportation system technologies in transportation networks give rise to methodological capabilities that require description of the dynamics of network traf®c ¯ows over time and space. Both descriptive and normative dynamic traf®c assignment capabilities are required in this environment. Several dynamic network ¯ow modeling problem formulations that arise in this context are discussed, and simulationassignment procedures are described for these problems. A dynamic traf®c assignment (DTA) system for advanced traf®c network management is described. It is built around a traf®c simulation-assignment modeling framework, which describes the evolution of traf®c patterns in the network for given traf®c loading under particular control measures and route guidance information supply strategies to individual motorists. The simulator is also embedded in an interactive search algorithm to determine optimal route guidance instructions to motorists. Numerical experiments with the model illustrate the relative effectiveness of different information supply strategies under different user behavior response rules. Keywords: Network modelling, dynamic traf®c assignment, intelligent transportation systems

1. Introduction The representation of the dynamics of traf®c ¯ow and user decisions in response to information and control actions in networks is a problem of considerable scienti®c interest and practical signi®cance. Traf®c networks are complex systems where decisions made by individual tripmakers interact in nonlinear ways over space and time. As such they bring together elements of social and economic systems, strongly interacting with both automotive and information technologies. Modeling such systems is of particular importance to the analysis and operation of intelligent transportation systems (ITS), which are predicated on extensive use of sensing and detection technologies to provide real-time information to traf®c controllers for the operation of the transportation infrastructure. The opportunities of ITS technologies have motivated considerable research over the past decade towards the development of analysis methodologies and algorithms for realtime operation of transportation networks. Much of this work has been characterized by a strong emphasis on the dynamic aspects of transportation phenomena in networks. This focus on dynamics stands in contrast to several decades of transportation network research

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during which the principal paradigm addressed steady-state ¯ow conditions in networks. The latter perspective may be adequate for long-term strategic planning horizons, but is not appropriate for the analysis of the kind of tactical measures enabled by ITS technologies, nor as a basis for real-time traf®c prediction and control in an operational setting. While static models for planning applications could rely on simpli®ed representations of traf®c processes, dynamic network models for operational purposes require proper description of ¯ow propagation in networks, including queueing at junctions, under time-varying loading patterns. Achieving such representation has been a goal of much research in dynamic traf®c assignment. However, this work has not yet discovered an entirely satisfactory analytic representation that satis®es the laws of physics and traf®c science, while at the same time yielding a mathematically tractable and well-behaved mathematical formulation. The complexity of the spatial and temporal interactions have to date precluded operationally realistic analytic models. For this reason, computer simulation of tripmaker decisions and associated traf®c processes in networks plays a central role in solution methodologies proposed for various problem formulations that arise in conjunction with the evaluation and operation of ITS-enabled networks. Simulation allows representation of a complex array of entities and description of their interaction in a large-scale traf®c network, and has been the method of choice for traf®c modeling in an integrated simulation-assignment methodology and software system implementation developed by the author and associates primarily at the University of Texas at Austin (and the current institutions of former associates). The system is known under the DYNASMART acronym (DYnamic Network Assignment-Simulation Model for Advanced Road Telematics). Recognizing different types of application requirements, two different model systems have been developed around the same core simulation-assignment methodology, as described in the next section. The core methodology represents a con¯uence of two major categories of procedures that have heretofore developed along separate tracks for differing purposes: (1) network assignment models, used primarily in conjunction with demand forecasting procedures for strategic (long-term) planning applications; and (2) traf®c simulation models, used primarily for traf®c operational studies. The adopted simulation logic is noteworthy in that it combines a microscopic level of representation of individual tripmakers and drivers, with a macroscopic description of some of the interactions taking place in the traf®c stream. This allows the computation of robust solutions with acceptable accuracy at a fraction of the computational cost that would have been required with a completely microscopic representation of traf®c maneuvers. Computational considerations are particularly important because simulation is used for many applications as a component in iterative solution schemes, which may need to be executed repeatedly for quasi-real-time operational decision-making. This paper presents an overview of the DYNASMART simulation-assignment logic, the principal problem formulations that it is intended to support in the context of ITS network applications, and the speci®c dynamic traf®c assignment procedures developed for these formulations. The principal objective is to illustrate the inter-relation among these problem formulations and associated methodologies, and assess the state-of-the-art and many remaining challenges in this rich and growing area of transportation network research and

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application. Most of the details of the procedures discussed in this paper have appeared elsewhere, and the reader is referred to the sources for in-depth exposition. Similarly, the software implementation aspects, and various features of available tools are not within the scope of this paper, which focuses on the conceptual and scienti®c underpinnings of the work. The paper is structured as follows. In the next section, we explain the principal functions of various dynamic traf®c assignment models that arise in the context of ITS network applications. This is followed, in section 3, by a description of the conceptual framework and structure of the DYNASMART simulation-assignment model, which is the core of the DTA system. In section 4, we review a simulation-based algorithm for the provision of real-time route guidance instructions, which is an essential DTA function, in which the DYNASMART simulator is used iteratively as part of the search algorithm for performance evaluation and to provide a search direction for the next iteration. In section 5, we discuss the real-time application of the above algorithm in a rolling horizon implementation of the DTA capabilities for on-line control, and the principal issues affecting the real-time execution of the DTA system functions. This centralized approach to real-time route assignment and guidance is contrasted to a decentralized approach which relies on heuristic local rules that react to observed measurements. In section 6, we present an application to an actual network, and discuss numerical experiments to evaluate the effectiveness of different information supply strategies under incident conditions. Concluding comments are presented in section 7. 2. Descriptive and Normative Dynamic Traf®c Assignment Capabilities Deployment of ITS technologies for network traf®c management typically includes a traf®c management center (TMC), which integrates system surveillance and monitoring functions with management and control functions. The latter involve development and implementation of control actions over the traf®c network in a metropolitan area. The TMC receives information in real-time about prevailing conditions from both ®xed and mobile sources. This serves as a basis for traf®c control actions, including incident response, traf®c signal setting, variable message signs to inform users and in¯uence their route choices, as well as the provision of route guidance instructions to vehicles equipped with two-way communication capabilities and in-vehicle display units (Catling, 1994; Whelan, 1995; Branscomb and Keller, 1996). Several methodological capabilities are required to support TMC functions to process the volumes of incoming information, analyze network operations, and determine control actions that optimize network performance. Central among these methodologies are dynamic traf®c assignment techniques, along with several associated support functions, which are conveniently integrated into a dynamic traf®c assignment (DTA) system. Two essential capabilities are required of the DTA system. The ®rst is descriptive, and consists of describing how ¯ow patterns develop spatially and temporally in a traf®c network, typically given a set of desired trips between origins and destinations. This descriptive capability allows estimation of current state of the network especially when the latter is only partially observable, as well as prediction of future network states over time.

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At its core is the ability to model the outcomes of tripmaker decisions, primarily the decision of which path to take between origin and destination, but also possibly the decision of when to depart, and what mode to use. The premise of ITS is the ability to sense prevailing conditions and rapidly devise actions to optimize system performance in real-time. Because the dynamics of traf®c systems are complex, as they depend on the interaction of many independent agents (drivers) acting non-cooperatively in a physically connected network, it is important to devise strategies that anticipate unfolding conditions instead of adopting a purely reactive approach. Real-time simulation of the traf®c network forms the basis of a state prediction capability that fuses historical data with sensor information, and uses a description of how traf®c behaves in networks to predict future conditions, and accordingly develop control measures. Because these actions are predicated on network conditions, which in turn depend on the users' decisions, network states have to be determined simultaneously with the tripmaker choices, generally in an iterative scheme. The estimated state of the network and predicted future states, in terms of ¯ows, travel times and other time-varying performance characteristics on the various components of the network, are used in the on-line generation and real-time evaluation of a wide range of measures, including information supply to users. The core of the descriptive DTA capability is a traf®c simulation model, which seeks to capture the dynamics of traf®c ¯ow movement in the network. Our approach relies on the DYNASMART simulation model, described in the next section (Mahmassani and Jayakrishnan, 1990; Jayakrishnan et al., 1994). To the extent that the actual traf®c network is also monitored continuously via a variety of sensing devices and probe vehicles capable of two-way communication with the control center, an important external support function for on-line DTA consists of ensuring consistency of the simulation-assignment model results with actual observations, and updating the estimated state of the system accordingly. Another external support function is intended to perform the estimation and prediction of the origin-destination (O-D) trip desires that form the load onto the traf®c network, and are as such an essential input to the simulation-assignment core. The second major DTA capability required for ITS network operation is normative; it aims to provide route guidance information to tripmakers, generally to attain some systemwide objectives, taking into account the individual welfare of tripmakers and the longer-term credibility and acceptance of the information system. In this sense, the provision of route guidance information is viewed as an integral element of traf®c network operations management, working in tandem with the traf®c control systems and incident response management systems to optimize overall network performance and productivity. There are different ways of seeking to achieve this capability. The most natural is to search for path assignments that are in some way optimal from the system's perspective, subject to certain reasonableness and acceptability requirements from the standpoint of individual users. This would again require joint determination of paths and associated network conditions. This approach is in fact required for any information supply strategy for which the supplied information is to be accurate and/or achieves the intended objectives when users' reactions to the information are taken into account. Another approach would be to guide individual users in a more ``local'' manner, link to link, with the actual path followed


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resulting from the succession of links along which the user would have been guided. The latter, appropriate for a decentralized control architecture, though also implementable in centralized architecture, could be entirely based on measured network conditions, and as such be highly responsive to sudden changes in traf®c conditions due to supply shocks such as incidents. It may however signi®cantly underperform a system optimal routing strategy when traf®c conditions are relatively stable and predictable. Both descriptive and normative DTA capabilities are necessary at the TMC in two distinct operational settings: real-time on-line operation (tactical), and off-line planning (strategic). Of course, the descriptive capability (e.g. the DYNASMART simulator) is also an essential component of most algorithms and procedures designed to provide the normative capability, in order for the latter to achieve consistency between the information supplied and the users' responses to the information. On the other hand, the route guidance instructions and/or other traveler information produced as output of the normative capability will be an input to the network state prediction performed by the descriptive DTA, to the same extent that the output of the adaptive signal controllers and other control elements are continuously fed to the descriptive real-time DTA. The above two capabilities (descriptive and normative), along with their support functions, are integrated in the DYNASMART-X DTA System, intended to provide, in real-time: (1) estimates of network traf®c conditions, (2) predictions of network ¯ow patterns over the near and medium terms in response to various contemplated traf®c control measures and information dissemination strategies, and (3) routing information to guide tripmakers in their travel. The functionality of the system relies on its ability to describe how ¯ow patterns develop spatially and temporally in a traf®c network, typically given a set of desired trips between origins and destinations. The structure of the system, depicted in ®gure 1, includes the following modules: O-D estimation, O-D prediction, realtime network state simulation, consistency checking, updating and resetting functions, and network state prediction. These modules are integrated through a ¯exible distributed architecture, using CORBA (Common Object Request Broker Architecture) standards, for real-time operation in a rolling horizon framework with multiple asynchronous horizons for the various modules. The system interacts continuously with multiple sources of realtime information, such as loop detectors, roadside sensors, and vehicle probes, which it integrates with its own model-based representation of the network traf®c state. The supporting consistency checking, updating and resetting functions compare measured values of selected state variables in the actual system to the corresponding values in the simulator, and update the internal representation within the simulator to ensure consistency with actual conditions. For off-line operational planning applications, such as ITS deployment planning and impact evaluation, the above simulation-assignment logic has been implemented in a ``planning version'' of the system, referred to as DYNASMART-P. As such, it is not concerned with real-time execution considerations, nor with interfacing with real-time detectors. On the other hand, it allows solution of different problem formulations, more commonly encountered in strategic planning applications, and can therefore be viewed as a time-varying version of static assignment models used in practice. Its main function is to describe the evolution of traf®c ¯ows in a traf®c network which result from the travel

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Figure 1. Structure of the DYNASMART-X real-time dynamic traf®c simulation and assignment system.

decisions of individual tripmakers seeking to ful®ll a chain of activities at different locations over a network over a given period of time. The most commonly encountered special instance of this problem is to assign time-varying origin-destination trip desires to the network and model the resulting evolution of traf®c over the network components. Unlike static assignment models, this simulation-assignment methodology represents the dynamics of congestion formation and dissipation associated with traf®c peak periods. As such, it allows consideration of an expanded set of supply-side and demand-oriented peakperiod congestion relief measures, compared to both conventional static assignment models and traf®c simulation tools. This is primarily due to: (1) richer representation of traveler behavior decisions than static assignment models, (2) explicit description of traf®c processes and their time-varying properties, (3) more complete representation of the network elements, including signalization and other operational controls. The next section describes the basic components of the simulator that lies at the core of the simulation-assignment framework, and provides the primary descriptive capability. 3. The Network Simulation-Assignment Modeling Framework 3.1. Problem de®nition and user assignment rules Consider a network G(N,A) consisting of a set of nodes N connected by the set of directed arcs A. The time horizon of interest (e.g. the peak traf®c period) is discretized into T small


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time slices, referred to as assignment intervals. Let rtij denote the number of users who intend to go from origin node i to destination node j during time interval t, i, j 2 N and t 1, . . ., T. Suppose the TMC can provide information to users according to a set of information supply strategies S. Denote by R the set of possible response rules followed by network users (note that an element of R may itself be a collection of responses rules followed by different classes of users). The time-dependent assignment problem is to distribute trip desires frtij ; 8i;j;tg to the network according to an assignment rule, I, where I 2 S R in this context, in a manner that is consistent with the temporal and spatial traf®c processes that take place in the network (Mahmassani, Hu and Peeta, 1994a). The conceptual framework of the simulation-based approach is elaborated in ®gure 2. Vehicles are generated according to a time-dependent O-D matrix, and assigned to routes speci®ed by the assignment rules; depending on the particular rules adopted, the paths may be obtained either through direct application of individual path choice models, or through algorithmic steps intended to satisfy certain conditions in the network. The time-dependent ¯ow pattern can be simulated by loading vehicles and representing their movements in the network. The results from the simulation may then be used in the next iteration, if called for by the particular assignment rule. In this work, we have considered four assignment rules, corresponding to different behavioral assumptions and interpretations of the time-dependent ¯ow patterns in the network. The ®rst rule determines users' paths through the network so as to minimize overall system cost (in this case travel time). System optimal (SO) assignment corresponds

Figure 2. Conceptual framework of the simulation-assignment procedure.

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Figure 3. Structure of DYNASMART simulation-assignment model.

to an information supply strategy that guides users to individual paths that are optimal for the system as a whole (i.e., normative route guidance). A SO assignment pattern does not usually correspond to an equilibrium situation as some users may be able to improve their individual trip times at the expense of greater cost to the system. However, a SO pattern provides a benchmark against which other assignments can be gauged. The second assignment rule corresponds to a time-dependent User Equilibrium (UE), under which no user can improve his/her individual cost by unilateral route switching. Such a state could result from the long-term evolution of the system, as users somehow learn and adjust under the supplied information. However, there is no theoretical nor empirical justi®cation to expect convergence to a UE pattern under inherently dynamic conditions. The third assignment rule corresponds to a family of response rules to an information supply strategy under which users receive descriptive information on prevailing link trip times. The family of response rules consists of boundedly-rational path switching and selection rules, which include a myopic switching rule (always select the shortest path based on current conditions) as a special case. The fourth assignment rule can be viewed as a special case of the preceding one, in which a vehicle is assigned to its current best path from the trip origin. Such an assignment would arise if a departing tripmaker could consult an origin-based information system


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(e.g. television or telephone) and select the shortest path to the destination under current traf®c conditions without considering possible future congestion. The solution algorithms for these assignment rules are discussed in section 4. The next section describes the various components of the DYNASMART simulation assignment model, which is the principal methodology used here to represent the complex interactions taking place in the traf®c network. 3.2. The simulation-assignment model The structure of DYNASMART is shown in ®gure 3. Given the network representation, which includes link characteristics as well as control parameters, the simulation component will take a time-dependent loading pattern and process the movement of vehicles on links and the transfers between links according to speci®ed control parameters. These transfers, which are determined by path processing and path selection rules, require instructions that direct vehicles approaching the downstream node of a link to the desired outgoing link. The user behavior component is the source of these instructions. 3.2.1. Traf®c simulation component DYNASMART uses established macroscopic traf®c ¯ow models and relationships to model the ¯ow of vehicles through a network. Whereas macroscopic simulation models do not keep track of individual vehicles, DYNASMART moves vehicles individually or in packets, thereby keeping a record of the locations and itineraries of the individual particles. This level of representation has also been referred to as ``mesoscopic''. Multiple user classes of different vehicle performance characteristics are modeled as packets, consisting of one or more passenger car units; for instance, a bus is represented by a packet with two (or other user-speci®ed values) passenger car units. The traf®c simulation consists of two principal modules: link movement and node transfer. Link movement The link movement is a process for moving vehicles on links during each scanning time interval in the simulation (time step). Note that the network links may be subdivided into smaller sections or segments for traf®c simulation purposes. The vehicle concentration prevailing on a section over a simulation time step is determined from the solution of the ®nite difference form of the usual continuity equation, given the concentration as well as in¯ows and out¯ows over the previous time step (Jayakrishnan et al., 1994). Using the current concentration, the corresponding section's speeds are calculated according to a modi®ed Greenshield speed-density relationship, namely: Vti Vf

V0 1

Kti =K0 a V0

where, Vti ; Kti mean speed and concentration in section i during the t-th time step, Vf, V0 mean free speed and the minimum speed, respectively, K0 jam concentration, and

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a a parameter used to capture the sensitivity of speed to the concentration. Other traf®c stream models may also be incorporated based on ®eld investigation. Node transfer The node transfer module performs the link to link or section to section transfer of vehicles at nodes. For interrupted link ¯ow, it appropriately allocates the right of way according to the prevailing control strategy. The output of the node transfer includes the number of vehicles that remain in queue and the number added to and subtracted from each link section for each simulation time step. A wide range of traf®c control measures for both intersections and freeways are re¯ected in the out¯ow and in¯ow capacity constraints that govern the node transfer (Mahmassani et al., 1994b). 3.2.2. User behavior component One of the principal features that allows DYNASMART to interface with activity-based behavioral models is its explicit representation of individual tripmaking decisions, particularly for path selection decisions, both at the trip origin and en-route. Behavioral rules governing route-choice decisions are incorporated, including the special case in which drivers are assumed to follow speci®c route guidance instructions. Experimental evidence presented by Mahmassani and Stephan (1988), and more recently by Mahmassani and Liu (1999), suggests that commuter route choice behavior exhibits a boundedly-rational character. This means that drivers look for gains only outside a threshold, within which the results are satisfying and suf®cing for them. This can be translated to the following route switching model (Mahmassani and Jayakrishnan, 1991): 1 if TTCj k TTBj k > maxZj TTCj k; tj dj k 0 otherwise where dj(k) is a binary indicator variable equal to 1 when user j switches from the current path to the best alternate, and 0 if the current path is maintained; TTCj(k) and TTBj(k) are the trip times along the current path and along the best path from node k to the destination on current path, respectively; Zj is a relative indifference threshold, and tj is an absolute minimum travel time improvement needed for a switch. The threshold level may re¯ect perceptual factors, preferential indifference, or persistence and aversion to switching. The quantity Zj governs users' responses to the supplied information and their propensity to switch. The minimum improvement tj is currently taken to be identical across users according to user de®ned values. Results of laboratory experiments indicate that tj is on average equal to one minute, while Zj is about 0.2 for typical urban commutes (Mahmassani and Liu, 1997, 1999). 3.2.3. Path processing The path processing component determines the route-level attributes (e.g. travel time), for use in the user behavior component, given the link-level attributes obtained from the simulator. For this purpose, a multiple user class K-shortest path algorithm with movement penalties is interfaced with the simulation model to calculate K different paths for every O-D pair. However, in order to improve the model's computational performance, the K paths are not re-calculated every simulation time step, but at pre-speci®ed intervals. In the interim, the travel times on the set of K current paths are updated using the prevailing link travel times at each simulation time step (Mahmassani et al., 1994a).


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4. Time-Dependent Multiple User Classes Algorithm The model described in the previous section serves as the core of an algorithmic procedure for the assignment of multiple user classes (MUC). The central controller now seeks to optimize overall network performance through the provision of real-time routing information to equipped motorists, taking into account different user classes in terms of information availability, information supply strategy, and driver response behavior. The problem statement is discussed in section 4.1, followed by the solution algorithm. 4.1. Problem statement The problem assumes that the TMC has complete a priori information in the form of timedependent O-D desires for users in each class, and seeks a time-dependent traf®c assignment that provides the number of vehicles of each class on the network links and paths satisfying system-wide objectives as well as the conditions corresponding to the behavioral characteristics of each user class. This problem therefore addresses the normative DTA capability discussed in section 2, coupled with a descriptive capability to represent actual user behavior and system dynamics in conjunction with route guidance generation. As before, consider the traf®c network represented by a directed graph G(N,A). The analysis period of interest, taken here as the peak period, is discretized into small equal intervals t 1; . . . ; T: Given a set of time-dependent O-D vehicle trip desires for the entire duration of the peak period, expressed as the number of vehicle trips rtu ij of user class u leaving node i for node j in time slice t, 8 i, j 2 N, t 1; . . . ; T; and u 1; . . . ; U; determine a time-dependent assignment of vehicles to network paths and corresponding arcs. In other words, ®nd the number of vehicles rtu ijku of user class u that follow path ku 1; . . . ; Kuij between i and j beginning at time t, 8 i, j 2 N, t 1; . . . ; T; and u 1; . . . ; U; as well as the associated numbers of vehicles on each arc a 2 A over time. Here, u 1; . . . ; 4 corresponds to the following four user classes: Class 1: equipped drivers who follow prescribed system-optimal (SO) paths. The solution will assign these users to paths that impose the least marginal cost (time) on the system from the origin to the destination. Class 2: equipped drivers who follow user optimum routes. The solution will assign these users to paths that minimize their own average cost (time) from the origin to the destination, so that no member of this class could improve his/her travel time by unilaterally changing paths. Class 3: equipped drivers who follow a boundedly-rational switching rule in response to descriptive information on prevailing conditions. This emulates the behavior of users who receive real-time information in the form of best paths based on link travel times that may not recognize future conditions (that would prevail at the time of actual traversal). The behavioral rules applicable to this class are described in section 3.2.2. Class 4: non-equipped drivers who follow externally speci®ed paths, such that rt4 ijk4 are known for all i, j, k and t.

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A mathematical statement of the problem is presented in Mahmassani et al. (1994b) and Peeta and Mahmassani (1995b). 4.2. Solution algorithm Figure 4 illustrates the simulation-based solution algorithm for the above problem, obtained by extending the corresponding single class assignment algorithm (Mahmassani and Peeta, 1993, 1995; Peeta and Mahmassani, 1995a). It consists of an inner loop that incorporates a direction ®nding mechanism for the SO and UE classes based on the

Figure 4. Multiple user classes solution algorithm.


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simulation results of the current iteration, namely the experienced trip times and associated marginal trip times (Ziliaskopoulos and Mahmassani, 1993). Convergence is sought by obtaining search directions for the SO (user class 1) and UE (user class 2) classes. User class 3 (BR), which follows behavioral rules in response to descriptive current traf®c information, is not directly involved in the search process. The paths of these users are obtained based on the traf®c pattern that evolves in the network for the current path assignment (and the behavioral rules assumed), unlike classes 1 and 2 which obtain their paths based on search directions from the experience of previous iterations. Hence, from an algorithmic standpoint there is no direct guiding mechanism involved in obtaining the paths for user class 3, other than their being predicated on the assignment strategy for the SO and UE class vehicles. As illustrated in the ®gure, they form the outer loop of the iterative procedure. The unequipped users (class 4 or PS) are exogenous to the search and represent constant background information (for each iteration) as their paths remain unchanged. The DYNASMART simulator captures the interactions taking place among the four user classes in the traf®c network. It allows evaluation of the resulting network ¯ow patterns and associated system performance for a given solution to the multi-class assignment problem (i.e. for a given set of path assignments); it provides the basis for extracting the information that guides the search process of the algorithm, and actually determines the assignment solution for user class 3 internally to the simulation through the embedded behavioral rules for these users. Further detail can be found in Mahmassani et al. (1994b) and Peeta and Mahmassani (1995a, 1995b). 5. Real-Time DTA Strategies and Implementation The primary strategy considered for the implementation of dynamic assignment capabilities in real-time is a rolling horizon framework, described in section 5.1. It exempli®es a centralized predictive approach to DTA. In contrast, we also present, in section 5.2, a decentralized reactive approach, which relies on heuristic rules used by a spatially distributed system of local controllers, with varying degrees of inter-communication, operating on limited local information obtained from sensors to provide route guidance to vehicles in the network. 5.1. Rolling horizon (RH) approach for real-time implementation The principal mechanism proposed for implementing the above DTA capabilities in realtime is the rolling horizon approach, used previously for production-inventory control (Wagner, 1977), and in transportation systems for on-line demand-responsive traf®c signal control (Gartner, 1982, 1983). The underlying philosophy behind the RH approach is that current events will not be in¯uenced by events ``far'' into the future, i.e. that assignment of current vehicles may be performed with only limited consideration of vehicles to be assigned ``far'' into the future, as currently assigned vehicles would probably be out of the

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Figure 5. Rolling horizon implementation framework.

system by that time. The stage length h in ®gure 5 depicts the time horizon considered when making current assignments (its value in actual problems is network speci®c). For a given stage, the problem encountered is analogous to the complete information availability scenario, albeit, only for the duration h of the stage. The system is solved for optimality only for this duration, and O-D desires for the roll period are assigned to the paths determined. The path assignments in each stage are determined for the entire stage, but implemented only for the roll period. The time frame is now ``rolled'' forward by the roll period, and the above process is repeated till the end of the duration of system operation, possibly on a continuing basis. Hence, a series of optimizations are performed in quasi real-time. The estimation and prediction functions of the DTA system shown in ®gure 1 need to be exercised, to produce O-D desires for the entire stage. The O-D desires beyond the stage length h are assumed to be zero. From a simulation standpoint, it is necessary to ensure proper initial conditions as one advances from one stage to the next. The selection of the values of l and h represents a compromise that re¯ects quality of O-D prediction, computational requirements, rate of change in the traf®c system, and solution quality. Simulation experiments have suggested roll periods of the order of 10 to


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15 minutes, with stages of about 20 to 30 minutes in duration for DTA implementation (Mahmassani et al., 1994c). In addition to the simulation-assignment model and normative route guidance modules to generate a solution for the next roll period, the support routines generating the O-D predictions and performing consistency checking and resetting must be executing, possibly several times per roll period, in a real-time implementation of the overall DYNASMART-X DTA system shown in ®gure 1. The time frame for executing the suite of modules involved in generating a solution (route guidance/state prediction) is referred to as an execution cycle. 5.2. Decentralized reactive route control using local rules In contrast to the above centralized control architecture, with its heavy requirements in terms of input information which may be available only with varying degrees of accuracy and con®dence, algorithmic complexity and computational intensiveness, hierarchical distributed architectures provide for ``locally-oriented'' real-time strategies for vehicle routing that rely on limited available information. The decentralized approach envisions a set of local controllers scattered or distributed in the network, where every controller can only extract limited ``raw'' information (speed, travel time, concentration, etc.) from network detectors, and utilizes this information using local control rules to guide the within-territory vehicles to their individual destinations. The local control rules are intended to be operational and robust under different scenarios of spatial and temporal information availability in a real-time context. The local control unit provides communication to drivers in a territory the size of which is mainly governed by the processing capabilities of the control units. Local control rules use available partial information and heuristics to evaluate alternative sub paths emanating from the decision node towards the destination, and assign vehicles at that node among the links immediately downstream. Figure 6 illustrates the spatial extent of the area governed by one local controller. Temporally, only current travel times are known for all links in the local area. The local area is de®ned by the set of links (and nodes) with depth less than or equal to a prespeci®ed number K. Consider a vehicle v going from origin node O(v) to destination node D(v), v 1; . . . ; V. A subpath (i, j, m, K) denotes the ®rst K links of path m from the decision node i to destination j. The problem is to assign vehicle v to an outgoing link a 2 B(i), where B(i) is the set of all links incident from I; such decisions are made repeatedly upon reaching the next decision node, until v reaches D(v). Assignment decisions are reached by control units after considering the relative merit or disutility of alternative subpaths, as captured by local and non-local state variables. The latter describe the expected state beyond the knowledge level K. The logic of the proposed local control is analogous to that of the A* graph search algorithm that uses heuristic information to select the next node to be scanned. The A* algorithm relies on an evaluation function, F, which has two components: the cost of reaching the node from the start node, G, and the cost of reaching the goal from the node, H. The node chosen for expansion is the one for which F G H is minimum (Pearl, 1984).

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Figure 6. Local and non-local areas in a sample network.

Let P(O(v),D(v)) denote the path actually followed by vehicle v over its trajectory, and TT(O(v),D(v)) the corresponding trip time. In making the nodal assignments of individual vehicles to an emanating link, the desired underlying objective is to minimize SV v1 TTOv;Dv; where the summation is taken over all vehicles, v 1; . . . ; V; seeking to depart from anywhere in the network over the duration of a given planning horizon of interest. Of course, the kind of local heuristic assignment rules considered here will not in general achieve the underlying global objective. The interest is to explore the performance of such rules, for different functional speci®cations and parameter values, relative to what might be achievable by more global algorithms, recognizing the imperfect knowledge (e.g. of future O-D demands) under which the latter might operate. A penalty function G`; speci®ed as a function of a various local state variables, is de®ned to capture the performance over the subpath ` (consisting of K links) under currently prevailing conditions. The desirability of the remaining portion of the path, from the end of the K-th link to D(v), is estimated according to a heuristic function H`: Thus,


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G` H` provides the basis for evaluating alternative subpaths from the current node to the destination. The speci®cation of the heuristic function may re¯ect varying degrees of ``knowledge'' or `ìntelligence'', with varying corresponding effort in terms of computation, data acquisition, data processing and/or prediction. A family of rules have been developed on the basis of the criteria by which subpaths are evaluated and the assignment process is performed (Hawas, 1996). We highlight here the speci®cation of one of these rules which distributes vehicles among several subpaths using a splitting model operationalized using the logit form. A generalized subpath disutility or penalty function is developed. It comprises local state variables (travel-time and concentration), and non-local variables (expected travel time). The proposed penalty function can be speci®ed as the approximate current marginal travel time along the subpath. The marginal time the system incurs by adding one vehicle to a subpath may be approximated by the vehicle's own travel time, from i to j, plus the marginal effect on all other subpath vehicles (assuming that the vehicle affects the subpath vehicles only). This can be expressed as: Lt Lt Gtij ` TLt ij ` MKij `Sij `

The state variable, TLt ij `; refers to the travel time of subpath ` at time t, where (L) superscript is used to indicate that this variable is local (can be actually measured). Lt The term KLt ij `Sij ` expresses the total number of vehicles along subpath `. The coef®cient M is the average marginal effect of the added vehicle at i on any of Lt the KLt ij `Sij ` vehicles; M is expected to decrease with higher knowledge levels to account for the diminishing marginal effect on the subpath vehicles as they move further away from the decision point. The heuristic function can be speci®ed as: Htij ` ATNLt ij ` The state variable ATNLt ij ` is an approximation of the anticipated non-local travel time from the end of subpath to destination j. It can be calculated by extrapolating the local travel time, historical information, or it may be replaced by corresponding information exchanged from adjacent controllers. The superscript (NL) indicates that this variable is a non-local variable (and cannot be measured directly according to the problem assumptions). This variable is calculated using local speed estimates and heuristics (Hawas, 1996). Denote by Ftij the disutility of the most promising subpath ` ; which is the minimum value of Ftij ` Gtij ` Htij ` for all feasible subpaths. The term ``feasible subpaths'' refers to the set of subpaths of apparent acceptable performance (Hawas, 1996). The share of any feasible subpath ` is inversely proportional to the penalty value, Ftij `; and is given by: ptij ` expyFtij `

Ftij =

X f

expyFtij f

Ftij

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The above equation allocates vehicles to feasible subpath ` based on the difference between Ftij ` and Ftij : The y parameter, or dispersion factor, affects the share of each subpath by allocating higher ¯ows to subpaths of lesser disutility. In numerical experiments designed to compare network performance under the predictive centralized RH and the decentralized reactive strategies, the decentralized strategy, based on rather simple rules that react to current measurements, was found to be quite competitive in terms of overall performance with the predictive centralized RH approach, under a variety of operational scenarios (Hawas, 1996; Mahmassani and Hawas, 1997). Simulation experiments also indicated that the distributed scheme is more robust under incident conditions due to its greater ability to rapidly respond to changes in network conditions (Hawas and Mahmassani, 1997).

6. Application to Evaluation of ITS Effectiveness under Incident Conditions Numerical experiments are presented to illustrate the application of the simulationassignment modeling framework to evaluate and compare the impacts of various ITS information supply strategies in a corridor network under incident conditions. The test corridor, shown in ®gure 7, is extracted from the network of Fort Worth, Texas, and corresponds to one sector of Interstate Highway I-35W between I-20 and I-30, with a surrounding network of parallel and crossing signalized arterial streets and local streets on both sides of the freeway. The network consists of 168 nodes and 441 links. It is mapped onto an area subdivided into 13 zones. Zone centroids de®ne the destinations of the trips, while the origins are distributed over the network, using as generation location the links in each zone. Freeway nodes are connected to the street network through entrance and exit ramps. The links corresponding to the freeway, frontage road and ramps are represented as directed arcs in one direction, while the rest of the links in the network are actually two directed arcs, one per direction. The time-dependent O-D demand was synthesized using a combination of static O-D information from planning studies and temporal patterns derived from link volume counts in the area. A base case scenario was constructed and calibrated to approximately replicate the peak hourly volumes observed on the major arterials and freeway. The incident scenario simulated involves freeway link 48±41, shown in ®gure 7. The starting time of the incident is speci®ed at minute 10 of the simulation, and the ending time is at minute 20, i.e. for a traf®c interruption duration of 10 minutes. The severity of this incident is set to 0.8, meaning that the blocking effect removes 80% of the capacity normally available on that link. The following information supply strategies for route diversion are tested: Descriptive Information (DES-DTA). In this case information on prevailing trip times and associated current best paths is available and is provided to the users. The market penetration can vary and consequently different levels of information are modeled. In addition to the base incident case of 0% information penetration, two levels of descriptive information penetration are considered, 25% and 75%.


Figure 7. The Fort-Worth test network.

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Normative with Multiple User Classes (NMUC-DTA). In this case, the four classes of users described in the problem formulation of section 4 are considered, with an equal fraction of network users in each class (25% for each class). Normative with 100% Optimal Route Guidance or System Optimum (NSO-DTA). This case is used as a benchmark against which to compare the others. The SO solution assumes that all vehicles are guided along paths so as to minimize the total travel time in the network. By de®nition, it is the best that could be achieved in terms of overall performance (under the given traf®c signal control policies), thereby yielding an upper bound on the bene®ts attainable with real-time traf®c information. The experimental design includes both situations without incident and with incident. The situations considered are summarized in Table 1. The principal performance measure considered in the comparative analysis is the average travel time in the network. Figure 8 presents the results for the situation with descriptive dynamic traf®c assignment only. The ®rst group corresponds to the base case of no-incident, no-information. The second group of values corresponds to the case with incident where 0% of the vehicles receive en-route information but pre-trip information is available to all users (e.g. through radio reports or cable TV channels). This means that vehicles departing after the incident start are aware of conditions around the incident location. The third group corresponds to the incident case with 25% of the vehicles receiving en-route information. The three columns indicate the performance of vehicles with information, of those without information and ®nally, the average over all the vehicles in each particular scenario. Finally, the fourth group corresponds to the incident case with 75% of vehicles receiving en-route information. Also presented is the performance of each group, namely those users that receive en-route information (``With Info.'' in the ®gure legend) and those that do not (``W/O Info.'' in the legend). Figure 9 presents the results for the incident case with different information scenarios and supply strategies. The ®rst group corresponds to the descriptive information case, and shows cases with 0%, and 25% and 75% en-route information, respectively. The second group corresponds to the MUC case with 25% of users in each class. The third group Table 1. Experimental scenarios Route-diversion strategy

Characteristics

Scenarios without incident DES-DTA 0% NMUC-DTA NSO-DTA

Base case: 0% information Four different user classes (25% of each class in the network) 100% route guidance

Scenarios with incident DES-DTA 0% DES-DTA 25% DES-DTA 75% NMUC-DTA NSO-DTA

0% information 25% information 75% information Four different user classes (25% of each class in the network) 100% route guidance


Figure 8. Travel time for the descriptive information scenarios.

Figure 9. Travel time for different incident scenarios.

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corresponds to the lowest possible travel time under the incident scenario, with 100% of vehicles guided to follow SO paths. Figure 10 presents the results for the no-incident case under three different information scenarios: descriptive, multiple user classes (MUC) and 100% SO route guidance. These values are used for comparison to the incident scenarios. Comparing the values presented for the descriptive information scenarios, including the base case, the incident cases are de®nitely worse than the base case, as expected. On the other hand, under the incident scenario with 75% of users receiving descriptive information, the performance of the system is even slightly better than the no-incident, noinformation base case. When only 25% of the users receive descriptive information, the performance of the overall system (with the incident) is not better than under the base case, but there is improvement with respect to the no-information incident scenario, and the vehicles with information do slightly better than the vehicles in the base case (no-incident). Of course, site speci®c analysis is needed to determine the percentage of vehicles receiving descriptive information that works best under particular conditions. In any case, when more vehicles have information, the users without information also experience reduced average travel time. In ®gure 10, travel time under the normative SO (100%) case represents a lower bound for the travel time in the system. One interesting feature is that the normative MUC scenario performs very well, close to the 100% SO benchmark, even though only 25% of users are following SO paths.

Figure 10. Travel time for different no-incident scenarios.

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Comparing the results achieved under Normative 100% SO (NSO) route guidance information to those attained under the DES 0% information case, reveals that the improvement in travel time possible through optimal route guidance is about 40%, for both incident and non-incident cases. These results are suggestive of the bene®ts that might be achieved through optimal route guidance in the network. Comparing performance under the MUC (multiple user classes with 25% of each class) case to the DES 0% case, the improvement in travel time is about 30%, for both incident and non-incident cases. The results also suggest that most of the bene®ts of optimal route guidance could be achieved even though only a fraction of the vehicles may follow the routes provided through route guidance. Table 2 summarizes the relative performance of the strategies considered with the incident case.

7. Concluding Comments Dynamic traf®c assignment (DTA) is an essential capability for the operation of intelligent transportation systems, and the successful deployment of telecommunications and information technologies for traf®c network management and control. This paper has described several DTA problem formulations that correspond to different functional capabilities that arise in conjunction with on-line operation of advanced traf®c management systems, as well as off-line assessment of operational measures and information supply strategies. The adopted simulation-assignment logic is noteworthy because it combines a microscopic level of representation of individual tripmakers and drivers, with a macroscopic description of some of the interactions taking place in the traf®c stream. This allows the attainment of robust solutions with acceptable accuracy with a fraction of the computational cost that would have been required with a completely microscopic representation of traf®c maneuvers. From a substantive standpoint, simulation experiments performed to evaluate system performance have provided important insights into the effectiveness of intelligent transportation system technologies and operational concepts. For example, the results presented in section 6 suggest that bene®ts attainable through coordinated real-time route guidance strategies that seek a system optimum are quite robust vis-aÁ-vis driver

Table 2. Relative performance of information strategies under incident conditions Route diversion strategy

Characteristics

Average travel time (min)

Improvement in percentage

DES-DTA 0% DES-DTA 25% DES-DTA 75% NMUC-DTA NSO-DTA

0% information 25% information 75% information Four different user classes (25% each) 100% route guidance

25.2 23.2 21.8 17.9 15.1

Base 8.0% 13.5% 28.8% 40.0%

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compliance with the supplied instructions, as a substantial portion of the potential bene®ts may be attained with only a moderate fraction of complying users. Results obtained to date with the real-time decentralized reactive strategies, compared with the considerably more elaborate centralized rolling horizon solution approach with its heavy dependence on O-D trip information, suggest further fruitful opportunities in development of such decentralized strategies. In addition, hybrid approaches that combine the centralized predictive approaches' ef®cient utilization of predicted information in generating control plans, with the ¯exibility and reactivity of decentralized reactive local rules, hold considerable promise for effective and robust system management. The deployment of intelligent transportation systems will continue to give rise to increasingly challenging problems that require careful methodological development. It would be a misconception to assume that these problems and their solution approaches represent a minor incremental modi®cation of existing methodologies, developed primarily for static equilibrium conditions. Signi®cant challenges and opportunities exist for developments ranging from the representation of the dynamic traf®c and behavioral processes, to the formulation of the assignment and route control problems jointly with other control dimensions, such as signalization and road pricing, in addition to the development of ef®cient solution of these problems both off-line, for evaluation purposes, as well as on-line for real-time control and operational purposes. The latter will undoubtedly provide ample opportunities for exciting and signi®cant developments in the basic and fundamental aspects of these problems. Finally, it is important to recognize that many of the issues encountered in the real-time execution of simulation-based frameworks for large scale complex systems evaluation and control are not limited to the intelligent transportation systems arena. These are also encountered in a growing class of telecommunications and information-driven systems across a wide spectrum of service sectors that call for real-time decision-making and resource allocation under real-time information (such as ®nance, retailing, logistics and distribution). As such, it is desirable to seek unifying frameworks and strategic constructs that cross disciplinary and domain boundaries to address common fundamental issues. Acknowledgments The author wishes to acknowledge the contribution of current and former members of his research team at the University of Texas at Austin. All work described in the paper is joint work, and draws liberally from papers and reports co-authored with several colleagues and former students, including Drs. G.-L. Chang, Y. Hawas, T.-Y. Hu, R. Jayakrishnan, S. Peeta, and A. Ziliaskopoulos. The author is particularly indebted to Dr. Yaser Hawas for his help in preparing section 5 of this paper. The numerical experiments in section 6 rely on work conducted with recent and current graduate students at the University of Texas, including Drs. A. Abdelfatah, Y. Kang, and D. Valdes, as well as A. Abdelghany, K. Abdelghany, Y.-C. Chiu, and N. Huyhn. Dr. Russ Taylor's contribution to the software engineering aspects of DYNASMART-X is gratefully acknowledged. This paper is based in large part on work supported by the U.S. Federal Highway Administration and


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administered through the Oak Ridge National Laboratory. The contribution and encouragement of Dr. Rekha Pillai of Oak Ridge, and Dr. Henry Lieu of FHWA have been instrumental to the continuing development of this research. Additional support was provided by the U.S. Department of Transportation through the Southwest Region University Transportation Center. Of course, the author is solely responsible for the contents of this paper.

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