A transportation system is operated with one vehicle andNservice points. At one terminal of the system passengers arrive in groups only at discrete points in time; at the other service points, riders enter steadily at constant rates. Vehicle travel times between adjacent service points are random variables. The problem addressed in this paper is that of devising the operating strategy which minimizes average waiting time for passengers of the system. Using different kinds of probabilistic analysis, we obtain optimal policy statements both when the discrete‐time arrivals occur at fixed instants of equal spacing, and when these arrivals occur in Poisson fashion. Some specific applications are discusse