A Quasi-Reversibility Approach to the Insensitivity of Generalized Semi-Markov Processes

Abstract

This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).