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In this paper we investigate a bottleneck model in which the capacity of the bottleneck is assumed stochastic and follows a uniform distribution. The commuters' departure time choice is assumed to follow the user equilibrium principle according to mean trip cost. The analytical solution of the proposed model is derived. Both the analytical and numerical results show that the capacity variability would indeed change the commuters' travel behavior by increasing the mean trip cost and lengthening the peak period. We then design congestion pricing schemes within the framework of the new stochastic bottleneck model, for both a time-varying toll and a single-step coarse toll, and prove that the proposed piecewise time-varying toll can effectively cut down, and even eliminate, the queues behind the bottleneck. We also find that the single-step coarse toll could either advance or postpone the earliest departure time. Furthermore, the numerical results show that the proposed pricing schemes can indeed improve the efficiency of the stochastic bottleneck through decreasing the system's total travel cost.

Congestion Behavior and Tolls in a Bottleneck Model with Stochastic Capacity