Cyclic Queuing Systems with Restricted Length Queues

This paper is concerned with an analysis of the stochastic behavior of a system of tandem queuing stations in which capacity restrictions are imposed on the queue lengths. The closed, cyclic systems that we consider are shown to be stochastically equivalent to open systems in which the number of customers is a random variable. A concept of duality is introduced on the basis of the simple observation that the sequential movement of customers through the stages generates a counter-sequential motion of “holes.” The implications of the duality relation are discussed at some length. The differential-difference equations for the time-dependent stochastic structure of the system are derived, and the remainder of the paper is devoted to the solution of the equilibrium equations for several special systems. First, we analyze completely systems with only two stages. The well-known results for a finite capacity queue appear as a special case. Next, systems are considered for which the number of customers is so small that there is no possibility of blocking. Then, by a duality argument, an analysis is carried out for systems with a very large number of customers in which the blocking effect dominates. Finally, we compare two extreme systems. In the one system there is no blocking and customers may queue at each stage. The other system has unit capacity at each of the M stages so that the distribution of customers is determined solely by the effect of blocking.