Congestion Costs in Bottleneck Equilibrium with Stochastic Capacity and Demand

Congestion Costs in Bottleneck Equilibrium with Stochastic Capacity and Demand


M Fosgerau, Technical University of Denmark, Department of Transport, DK


I provide explicit expressions for total and marginal internal and external congestion costs. Uncertainty increases costs up to 50 % relative to the deterministic case.


Congestion is an extremely complicated phenomenon. Consider for example the morning peak for commuters going to work. Some essential features of this situation are the following:

Commuters want to arrive at more or less the same time, but capacity is scarce so some have to accept arriving earlier or later than preferred. They are not able to coordinate their departure times and hence queueing occurs as commuters balance the larger travel time in the peak against being early or late. Moreover, travel time is essentially random from the perspective of commuters, since there are unpredictable day-to-day variations in the number of commuters going to work and in the capacity.

This situation is naturally analysed in the Vickrey bottleneck model of a congestible facility with a peak load in demand. The Vickrey bottleneck model is the simplest structural model of congestion. Travellers arrive at a single bottleneck, which has a fixed capacity per time unit. Travellers are served in the order in which they arrive. A queue builds up whenever the rate of arrivals is greater than the capacity and the queue is discharged when the rate of arrivals is less than capacity. The shape of the peak is endogenous, being the sum of individual scheduling decisions. The model is solved assuming an equilibrium whereby no commuter can reduce her scheduling costs by changing departure time.

Following Arnott, de Palma and Lindsey, this model is extended to take into account the empirically important fact that capacity and demand are random from the perspective of users. This introduces uncertainty into the individual scheduling choices.

The contribution of the current paper is that I derive the expected marginal and total congestion costs and compare to the case with fixed capacity and demand.

Using stylised values for scheduling costs relative to the value of time, I find that randomness of capacity and demand increases congestions cost by up to 50 percent relative to the deterministic case. The bound is general for any distribution of random capacity and demand.


Association for European Transport