Computing full link state distributions in the dynamic network loading problem

This paper derives a new dynamic network loading model that yields full link queue length distributions, properly accounts for spillback, and maintains a differentiable mapping from the dynamic demand on the dynamic link states. The approach builds upon an existing stationary queuing network model that is based on finite capacity queuing theory. The original model is specified in terms of a set of differential equations, which in the new model are carried over to a set of equally smooth difference equations. The representation of full dynamic link state distributions has so far been reserved to microsimulations. The approach used in this paper differs from previous work in that it (i) exploits closed-form results from queuing theory, (ii) provides the additional benefit of a closed-form expression of the system's stationary state, and (iii) consists of one integrated set of smooth equations whereas previous research deployed a switching logic between multiple linear models. Essentially, the original stationary model the authors use starts from the link state distributions from the standard queuing theory global balance equations. Coupling equations are used to capture the network-wide interactions between these single-link models. The new dynamic version of this model consists of a dynamic link model and a static node model. The global balance equations are replaced by a discrete-time closed-form expression for the transient link state distributions. This expression guides the link model's transition from the full queue distribution of one time step to the next. It is available in closed form under the reasonable assumption of constant link boundary conditions during a simulation step. No dynamics are introduced into the node model, which maintains the structure of the original stationary model. Disposing of both the dynamic model and the according stationary model is useful because it allows for the evaluation of the stationary limit of the dynamic model at a low computational cost. In the analysis of the new model, this consistency is checked by running the dynamic model until it is stationary and comparing the resulting link state distributions with those of the original model. The realism of the new model's dynamics is investigated by comparison with empirical distributions obtained from a calibrated microscopic simulation model of the city of Lausanne during the evening peak hour. There are various applications of the new model. Full dynamic link state distributions can be used as inputs for route or departure time choice models that capture risk-averse behavior. The analytically tractable form of the stationary model has enabled engineers in the past to use it to solve traffic control problems using gradient-based optimization algorithms. Since the dynamic formulation preserves the smoothness of the original model, the authors expect it to be of equal interest for problems that involve derivative-based algorithms, including solution procedures for the dynamic traffic assignment problem.

Topic: Transportation Planning and Optimization
Subfield: Transportation
Field: Social Sciences
Domain: Social Sciences