The Distribution of Queuing Network States at Input and Output Instants

TLDR: In this paper, the authors studied the relation between the distribution of the system state embedded at those moments and the steady-state (or random point) distribution, and showed that if a job class belongs to an open subchain, the state distributions at input pomts, output points, and random points are identical, and if the job classes belong to a closed sub chain, the distribution at input and output points ts the same as the steady state distribution of a network with one less job in that subchain.

Abstract: Queuing networks are studied at selected points in the steady state, namely, at the moments when jobs of a given class arrive into a given node (either from the outside or from other nodes) and at the moments when jobs of a given class leave a given node (either for the outside or for other nodes). The processes defined by these points are known to be, in general, non-Potsson, interdependent, and serially correlated; therefore the relation between the distribution of the system state embedded at those moments and the steady-state (or random point) distribution is not obvious a priori. For a large class of networks having product-form equihbrium distribnttons it is shown that (a) if the given job class belongs to an open subchain, the state distributions at input pomts, output points, and random points are identical, and (b) if the job class belongs to a closed subchain, the distribution at input and output points ts the same as the steady-state distribution of a network with one less job in that subchain.