Equilibrium in a dynamic model of congestion with large and small users

Individual users often control a significant share of total traffic flows. Examples include airlines, rail and maritime freight shippers, urban goods delivery companies and passenger transportation network companies. These users have an incentive to internalize the congestion delays their own vehicles impose on each other by adjusting the timing of their trips. We investigate simultaneous trip-timing decisions by large users and small users in a dynamic model of congestion. Unlike previous work, we allow for heterogeneity of trip-timing preferences and for the presence of small users such as individual commuters and fringe airlines. We derive the optimal fleet departure schedule for a large user as a best-response to the aggregate departure rate of other users. We show that when the vehicles in a large user's fleet have a sufficiently dispersed distribution of desired arrival times, there may exist a pure-strategy Nash-equilibrium (PSNE) in which the large user schedules vehicles when there is a queue. This resolves the problem of non-existence of a PSNE identified in Silva et al. (2017) for the case of symmetric large users. We also develop some examples to identify under what conditions a PSNE exists. The examples illustrate how self-internalization of congestion by a large user can affect the nature of equilibrium and the travel costs that it and other users incur.