Decomposability in queues with background states

A symmetric queue is known to have a nice property, the so-called insensitivity. In this paper, the authors generalize this for a single node queue with Poisson arrivals and background state, which changes at completion instants of lifetimes as well as at the arrival and departure instants. They study this problem by using the decomposability property of the joint stationary distribution of the queue length and supplementary variables, which implies the insensitivity. The authors formulate a Markov process representing the state of the queue as an RGSMP (reallocatable generalized semi-Markov process), and give necessary and sufficient conditions for the decomposability. They then establish general criteria to be sufficient for the queue to possess the property. Various symmetric-like queues with background states, including continuous time versions of moving server queues, are shown to have the decomposability.