Scheduling arrivals to queues for minimum average blocking: The S(n)/M/C/C system

Given a finite time interval [0, T], a fixed number of customers N, a stochastic service system with C servers, and an initial occupancy distribution Π(0), we show how to calculate a schedule for the customer arrival times to minimize average blocking. We have in mind a facility user who infrequently receives a burst of N items to submit where N is a random variable. If such a “bursty” user can tolerate the delay associated scheduling items over an interval of duration T, then substantial improvements in blocking performance can be had over random scheduling (N Poisson arrivals on [0, T]) and periodic scheduling (constant interarrivals). The matrix exponential method used herein can be applied to any stochastic service system whose state transitions can be modeled by linear differential equations and for which an arrival changes the system state in a predictable way. However, we here concentrate on the tractable C-server exponential service system.