Dynamic user equilibrium has received considerable theoretical attention for morning peak-period travel but very little for the evening peak. In an attempt to redress this imbalance, morning and evening travel are characterized and compared by using Vickrey's bottleneck model. To focus ideas, it is assumed that morning and evening travel differ in just one respect: scheduling preferences for the morning are defined in terms of arrival time at work, whereas preferences for the evening are defined in terms of departure time from work. Sufficient conditions are identified for the existence and uniqueness of a deterministic dynamic user equilibrium in terms of departure times for the morning and evening peaks. These conditions, which go well beyond previous work, involve relatively general assumptions about the schedule delay cost functions for morning and evening and essentially no restrictions on the degree of heterogeneity in trip-timing preferences of travelers. Plausibility of the conditions is examined in light of the limited empirical evidence. A numerical example is developed at length to illustrate the importance of traveler heterogeneity and the extent of differences between morning and evening in the time pattern of departures and aggregate travel costs.
Abstract