Insensitive Generalized Semi-Markov Schemes with Point Process Input

Abstract

Where sequences of i.i.d. random variables are used for modelling successive repair limes, life times, service times, etc, it can happen that stationary probabilities are found for the—suitably defined—system state, which depend only on the means of those variables, not on any further characteristics of their distributions. The well-known Erlang loss model provides an example. For a large class of models, Jansen, Koenig, and Nawrotzki (Jansen, U., D. Koenig, K. Nawrotzki. 1979. A criterion of insensitivity for a class of queuing systems with random marked point processes. Math. Operationsforsch. Statist. 10 379–403.) have shown that, whenever such insensitivity prevails towards an i.i.d. sequence or—equivalently—a renewal process, then replacing this process by an arbitrary point process with the same intensity (when stationary) still does not change the stationary distribution of the system state. We extend their model and prove the analogous result by way of a different technique of general interest.