The morning commute problem with heterogeneous travellers: the case of continuously distributed parameters

We study the morning commute problem with a heterogeneous travelling population whose early/late arrival penalty parameters are continuously distributed. Following Arnott et al.'s (1988) [Schedule delay and departure time decisions with heterogeneous commuters, Transportation Research Record, 1197, 56-67] model, where the ratio of the value of early schedule delay (VESD) over the value of late schedule delay (VLSD) is assumed to be constant across the population, we first derive the user-optimal travel profiles and the corresponding total travel time (TTT) for a one-to-one network with a single route. We show that although every commuter is better off if the bottleneck capacity is enlarged, commuters with high values of early arrival penalty (EAP) benefit more than those with low values. In addition, the homogeneity assumption overestimates the queuing delay and thus the TTT. Then we extend our analysis to a network with two routes, a freeway and an arterial road (AR) connecting a single origin-destination (O-D) pair. In this case, we find that the travellers who weigh schedule delay more will first shift to the AR, prompted by an increase in total demand. However, the critical travel demand at which travellers start to use the AR remains the same if the common EAP for homogeneous users is equal to the expected value of EAP distribution for heterogeneous users. Interestingly, there exists a critical value of EAP parameter, such that all the travellers whose EAP is less than it choose the freeway, and those whose EAP is larger than it choose the AR. Sensitivity analysis on TTT is performed with respect to freeway capacity, AR capacity and free-flow travel time on the AR. The results indicate that enlarging freeway (AR) capacity will always reduce the TTT of the whole network and the TTT of the AR (freeway), and increase the demand share of the freeway (AR). We also show that every commuter is better off if either the freeway capacity or the AR capacity is enlarged. Finally, we provide numerical examples to demonstrate the considerable differences in flow patterns and network performances between a homogeneous and a heterogeneous population.