Dispatching Policies for a Transportation Route

Suppose that a given number n of vehicles (buses, trains, etc.) maybe dispatched at any times from some source to serve passengers along a route. The arrival rate of passengers is some given (nonconstant) smooth function of time, typically having one or more peaks. One wishes to choose the dispatch times t_j, j = 1, …, n in such a way as to minimize the total waiting time of all passengers. It is shown that if the capacity of the vehicles is sufficiently large to serve all waiting passengers and n is large, then the optimal flow rate of vehicles (reciprocal of the headways) and the number of passengers served per vehicle, both vary with time approximately as the square root of the arrival rate of passengers. If the vehicles have limited capacity, their dispatch schedule will be distorted so that certain vehicles are dispatched as soon as they are full.