Equilibrium in capacitated network models with queueing delays, queue-storage, blocking back and control

This paper considers a steady-state, link-based, fixed (or inelastic) demand equilibrium model with explicit link-exit capacities, explicit bottleneck or queuing delays and explicit bounds on queue storage capacities. The (spatial queueing) model at the heart of this equilibrium model takes account of the space taken up by queues both when there is no blocking back and also when there is blocking back. The paper shows in theorem 1 that a feasible traffic assignment model has an equilibrium solution provided prices are used to impose capacity restrictions and utilises this result to show that there is an equilibrium with the spatial queueing model, provided queue-storage capacities are sufficiently large. Other results are obtained by changing the variables and sets in theorem 1 suitably. These results include: (1) existence of equilibrium results (in both a steady state and a dynamic context) which allow signal green-times to respond to prices and (2) an existence of equilibrium result which allow signal green-times to respond to spatial queues; provided this response follows the P0 control policy in Smith (1979, 1987). These results show that under certain conditions the P0 control policy maximises network capacity. The spatial queueing model is illustrated on a simple network. Finally the paper includes elastic demand; this is necessary for long-run evaluations. Each of the steady state models here may be thought of as a stationary solution to the dynamic assignment problem either with or without blocking back.