A closed BCMP queuing network consisting of R job classes (chains), K+1 single-server, fixed-rate nodes, and M/sub j/ class j jobs (j=1, 2, . . ., R) is considered. Asymptotic expansions are constructed for the partition function under assumptions (1) K>>1, (2) M/sub j/>>1 for each j, and (3) K/M/sub j/=O(1). Analytic expressions for performance measures such as the mean queue length are also given. The approach employs the ray method and the method of matched asymptotic expansions. Numerical comparisons illustrate the accuracy of the approximations.>

In this paper, a new asymptotic method is developed for analyzing closed BCMP queuing networks with a single class (chain) consisting of a large number of customers, a single infinite server queue, and a large number of single server queues with fixed (state-independent) service rates. Asymptotic approximations are computed for the normalization constant (partition function) starting directly from a recursion relation of Buzen. The approach of the authors employs the ray method of geometrical optics and the method of matched asymptotic expansions. The method is applicable when the servers have nearly equal relative utilizations or can be divided into classes with nearly equal relative utilizations. Numerical comparisons are given that illustrate the accuracy of the asymptotic approximations.

A closed multiple access system composed of M sources and a single service facility is analyzed. The closed network is modeled as a finite source ${M / G / 1}$ queueing system. We consider systems with a large number of sources (i.e. $M \gg 1$) and we assume that the mean time required to process an individual request is short $(O({1 / M}))$. We then construct asymptotic approximations to the stationary distribution of the number of requests in the service facility by using the method of matched asymptotic expansions. We give formulas for the first and second moments of the number of requests, for all traffic intensities.

We consider two parallel, infinite capacity,M/G/1queues characterized by (U_{1}(t), U_{2}(t)) withU_{j}(t)denoting the unfinished work (buffer content) in queuej. A new arrival is assigned to the queue with the smaller buffer content. We construct formal (as opposed to rigorous) asymptotic approximations to the Joint stationary distribution of the Markov process (U_{1}(t), U_{2}(t)), treating separately the asymptotic limits of heavy traffic, light traffic, and large buffer contents. In heavy traffic, the stochastic processesU_{1}(t) + U_{2}(t)andU_{2}(t) - U_{1}(t)become independent, with the distribution ofU_{1}(t) + U_{2}(t)identical to the heavy traffic waiting time distribution in the standardM/G/2queue, and the distribution ofU_{2}(t) - U_{1}(t)closely related to the tail of the service time density. In light traffic, we obtain a formal expansion of the stationary distribution in powers of the arrival rate.

We consider two parallelM/M/1queueing systems where a new arrival (customer, job, message) joins the shorter of the two queues. Such problems arise naturally in computer communications and packet switched data networks. An asymptotic approach is developed to obtain approximations to the steady-state joint distribution of the number of customers in the two systems. We first analyze the case where the two queueing systems are identical and then consider the case when the two servers work at different rates. Our results are shown to agree with the expansions of known exact solutions, when such solutions are available, and to yield new approximations when such solutions are not available.