Analysis and Optimal Design of a Multiserver, Multiqueue System With Finite Waiting Space in Each Queue

The purpose of this paper is to present an analysis of a particular type of multiserver, muitiqueue system in which each queue has a finite number of waiting positions and the waiting positions are not vacated until service is completed. In particular, the steady-state distribution of states are derived, and expressions for the probability of loss for each queue and the average delay are given. It is shown that the queuing model described is representative of systems characterized by a finite number of trunk groups that carry calls to a group of attendants who then perform some service for the caller. (One such system is used in the directory assistance service provided by the telephone company.) Results are given to illustrate the effects of varying the number of servers and number of positions in each queue. Finally, a design procedure to determine the minimum-cost configura595 tions for such systems under various grade-of-service constraints is developed. This procedure ignores such effects as day-to-day variation, noncoincidence of peak loads among incoming trunk groups, and retrials of blocked calls, which should be investigated in the development of procedures for traffic engineering and administration. However, as in most cases, it is difficult, if not impossible, to obtain analytical results with these effects included. This paper should provide useful insight that can later be incorporated in a complete traffic engineering procedure. The system analyzed consists of I input queues each with a finite number of waiting positions, Ni (i = 1, · · I), which have full access to M servers.