Applying a New Device in the Optimization of Exponential Queuing Systems

We consider the problem of controlling M/M/c queuing systems. By providing a new definition of the time of transition, we enlarge the standard set of decision epochs and obtain a preferred version of the n-period problem in which the times between transitions are exponential random variables with constant parameter. Using this new device, we are able to utilize the inductive approach in a manner characteristic of inventory theory. The efficacy of the approach is demonstrated by successfully finding the form of an optimal policy for three distinct models that have appeared in the literature, namely, those of (i) Miller and Cramer, (ii) Crabill and Sabeti, and (iii) Low of particular note is our analysis of the Miller-Cramer model, in which we show that a policy optimal for all sufficiently small discount factors can be obtained from the usual average cost functional equation without recourse to further computation.