Properties of Simulated Path Travel Times and Implications for Dynamic Network Assignment and Control Algorithms

Hani Mahmassani
Northwestern University

Traffic simulation models, at all scales of representation (macro, micro and meso) are in widespread use in the study of traffic flow in transportation networks, for both research and engineering applications, and are increasingly embedded in optimization procedures for the design and control of transportation networks. However, the mathematical properties of quantities produced by such simulators are not sufficiently understood due to the rule-based and algorithmic elements integral to most simulators.
Of primary interest in this presentation are the path travel times produced by simulation models of traffic in a network. Although such path travel times can be produced by simulation analysis, other essential properties of this variable may not be readily established. The properties of interest in this study include continuity, monotonicity, convexity and sensitivity of the path travel time vector as a function of the path flow vector. These properties are essential in seeking to establish the existence, uniqueness and convergence of optimization models and algorithms in which these simulation procedures are embedded. Furthermore, the lack of analytical derivatives of experienced (in the simulation) path travel times with respect to path departure rates cause these models to fail in certain applications. Evaluating each derivative numerically is equivalent to repeating the simulation several times, which is computationally very demanding considering the number of input and output variables of the system, especially in real-time applications.
In the existing literature, desirable properties for context-specific functional forms of path travel time have been established. This presentation however is concerned with the extent to which these properties may hold for a general traffic flow network. The approach followed to accomplish this objective consists of breaking the traffic flow model into its smallest elements, sub-flow trajectories, and analyzing the anatomy of flow movement. Each property is investigated for a general discrete-event traffic simulation model that only imposes a weak first-in first-out (FIFO) assumption at the segment level. The analysis is illustrated through simulation results on several actual networks. Finally, an algorithmic framework for capturing the sensitivity of path travel time to path flow is proposed and its performance is tested through implementation for several networks. The algorithm is also implemented for solving the system optimal (SO) dynamic network assignment problem. In this regard, perturbation analysis results in a new class of “ordered selection” SO assignment algorithms.

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