Asymptotic Expansion of the Stationary Probability Distribution for States of a Closed Queueing Network with a Demand Transmission Channel

A closed queueing network with multiserver nodes and several finite sources is considered where each source has its own transition matrix. Network nodes serve demands of different types. Each node has its own number of demand types. From a finite source to an arbitrary network node, from one node to another, and from an arbitrary node back to the “original” source, demands are transmitted through a multiline demand transmission channel. The service discipline at the network nodes and the discipline of choosing a demand for sending to the channel are random. Demand generation times at the sources, service times at the nodes, and transmission times in the channel are random variables, which have exponential distribution with parameters depending on the aggregate network state. For such a network, the stationary probability distribution of states is not representable in a multiplicative form. For the case where the intensity of the demand transmission in a channel is much greater than intensities of demand generation at the sources and service intensities at the nodes, a method for the asymptotic expansion of the stationary distribution is proposed and an algorithm for the computation of coefficients for arbitrarily many expansion terms is constructed.