Insensitivity in stochastic models

Abstract

A stochastic model is said to be insensitive if its stationary distribution depends on one or more of its constituent lifetime distributions only through the mean. Insensitivity is usually associated with partial balance in the corresponding Markovianmodel when all lifetimes are taken to be exponential, and a product-form stationary distribution of the Markov chain, constructed by supplementing the state by information on the progress of generally-distributed lifetimes. In this chapter I shall discuss insensitivity by presenting a detailed analysis of the canonical insensitive queueing model, the Erlang loss system, from two different directions, as a queue and as a Generalised Semi-Markov Process (GSMP). I shall then show how the underlying ideas extend to insensitive queueing network models and finish off with a discussion of the few known non-standard insensitive systems which are not associated with partial balance or a product-form supplemented stationary distribution.