An Extension of Moore's Result for Closed Queuing Networks

In this communication, Moore's result for the normalization constant of a closed queuing network of exponential servers is extended to the case of nondistinct traffic intensities. We then show that this result can be applied to various more general closed and semiclosed queuing networks with a product-form solution. Introduction Substantial progress has been made recently in extending the scope of Jackson's product-form solution for queuing networks [ 1 51. At the same time, a number of very efficient computational algorithms [ 3, 681 have been developed for evaluating the normalization constant and marginal queue length statistics for such networks. These algorithms are all based upon some recursive scheme. For the special case of a closed network of exponential servers with distinct traffic intensities, Moore has shown an explicit solution for the normalization constant using the partial fraction method [9]. In this note, we extend Moore's result to the case of nondistinct traffic intensities and show that it can be applied to various more general closed and semiclosed queuing networks with the product-form solution. We note, however, that the partial fraction method is less versatile than recursive techniques. I t also requires the summation of terms with alternating signs, and thus may be subject to round-off errors in some cases [ 31. Assumptions We outline here the scope of queuing networks for which the results in this communication are applicable. Consider a queuing network with M , service stations and R classes of customers. (See [ 21 .) At the completion of a service request, a customer may change its class membership and proceed to another service station (or leave the network) according to fixed transition probabilities. Four types of service stations may be considered: 1 ) a single server, first-come-first-served service discipline (FCFS) ; 2 ) a single server, processor-sharing service discipline (PS) ; 3) a single server, last-come-first-served preemptive-resume service discipline (LCFS) ; and 4) no queuing, arbitrarily many servers (IS). In an FCFS ser384 vice station, all customers have the same negative S. S. LAM exponential service time distribution with a fixed service rate. In PS, LCFS, or IS service stations, each class of customers may have its own general service time distribution which has a rational Laplace transform. Let ni, 1 5 i 5 M,, denote the number of customers at service station i. We assume that the underlying Markov chain of customer transitions is irreducible. The arrival process of customers to the network is a Poisson process with an arrival rate dependent on the instantaneous number n' of customers in the network at time t . Moreover, if n' = N,, a departing customer is immediately replaced; if nt = N,, all arrivals are lost. Thus, n' is constrained: N, 5 n' 5 N,. In particular, for a closed queuing network N, = N,. Below, we first present our results for closed queuing networks and then indicate how to extend these results to the more general case of semiclosed networks (N, < N,) [7]. Results for closed queuing networks Let there be M service stations of types FCFS, PS, and LCFS labeled by { 1, 2 , . . ., M } , and M , M service stations of type IS labeled by { M + 1 , . . ., M s } . Suppose pir, 1 5 i 5 M , and 1 5 Y 5 R , are the relative traffic intensities [3] of class r customers at service station i . Define pi 6 2 pi,., 1 5 i 5 Ms. The equilibrium joint queue length distribution is R