Asymptotic expansions for closed Markovian networks with state-dependent service rates

Abstract

A method is presented for calculating the partition function, and from it, performance measures, for closed Markovian stochastic networks with queuing centers in which the service or processing rate depends on the center's state or load. The analysis on which this method is based is new and a major extension of our earlier work on load-independent queuing networks. The method gives asymptotic expansions for the partition function in powers of 1/ N , where N is a parameter that reflects the size of the network. The expansions are particularly useful for large networks with many classes, each class having many customers. The end result is a decomposition by which expansion coefficients are obtained exactly by linear combinations of partition function values of small network constructs called pseudonetworks. Effectively computable bounds are given for errors arising from the use of a finite number of expansion terms. This method is important because load dependence is at once an essential element of sophisticated network models of computers, computer communications, and switching, teletraffic, and manufacturing systems, and the cause of very intensive computations in conventional techniques. With this method, very large load-dependent networks can be analyzed, whereas previously only small networks were computationally tractable.