Scheduling a time-shared server to minimize aggregate delay

A 1967 paper by Rangarajan and Oliver [1)1 contains a formulation and analysis of the two problems described below, which pertain to the allocation of servicing times among N incoming streams requiring "processing" of some kind by a single "server." The server might for example be a switching point or a congestion point (e.g., a tunnel entrance) in a transport network, in which case "processing" an item (vehicle) simply means letting it through. Or, the server might be a computer handling reservations from several ticket offices, or exercising real-time control over vehicle movements on several network links, or performing some other tasks on a time·shared basis. The streams are treated as continuous flows. During each service cycle, of duration T, the server handles stream 1 for time Ct , switches (with associated known switch-over or "dead" time) to handle stream 2 for time C2 , etc. The arrivals in each stream are assumed nonrandom, with a known uniform rate (possibly different for different streams). The server's processing rate, when serving a particular stream, is also assumed nonrandom and constant (possibly different for different streams). Each Ci is constrained to be at least large enough so that no queue remains in the i th stream when one of that stream's service periods ends. The two problems formulated and analyzed are these: PROBLEM 1: For given cycle time T, what allocation Ct , C2·, ••• , CN of service times among the various streams is optimal, in the sense of minimizing total waiting time per cycle? PROBLEM 2: What value of the cycle time T will minimize average waiting time? Subsequently Horn [2] showed that the more general case, in which all streams are served equally often (possibly more than once) per cycle, can be reduced to PROBLEM 1. This provides additional reason for offering an alternative analysis, which is more self-contained and (at least to the writer) simpler than that of reference [1]. In addition, a mild generalization will be introduced by permitting the penalties for delay to be different for different streams.