Computing Full Link State Distributions in the Dynamic Network Loading Problem
C Osorio, G Flötteröd, M Bierlaire, EPFL, CH
We present a differentiable dynamic network loading model that describes stochastic space-queues of vehicular traffic.
We derive 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 queueing network model that is based on finite capacity queueing theory (Osorio and Bierlaire; 2008, 2009). The original model is specified in terms of a set of differentiable 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 work of Sumalee et al. (2008) on a stochastic version of the cell-transmission model being a notable exception. Our approach differs from this work in that it (i) exploits closed-from results from queueing 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 Sumalee et al. (2008) deploy a switching logic between multiple linear models).
Essentially, the original stationary model we start from derives the link state distributions from the standard queueing 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 (Morse, 1958). 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 to evaluate 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 stationarity 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 (Dumont and Bert, 2006).
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 us in the past to use it to solve traffic control problems using gradient-based optimization algorithms (Osorio and Bierlaire, 2008). Since the dynamic formulation preserves the smoothness of the original model, we expect it to be of equal interest for problems that involve derivative-based algorithms, including solution procedures for the dynamic traffic assignment problem.
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