The stability of storage models with shot noise input

Abstract

We examine the existence of limiting behavior, or stability, for storage models with shot noise input and general release rules. The shot noise feature of the input process allows the individual inputs to gradually enter the store.We first show that a store under the unit release rule is stable if and only if the traffic intensity is less than one; this extends the classic result of Prabhu (1980) to the case of shot noise input. The stability of the unit release rule store and various stochastic orderings are then used to derive a sufficient condition for a store with a general release rule to be stable. Finally, we show that when restricted to a compact state space, our storage model is always stable.An important component of the paper is the methodology employed: coupling and stochastic monotonicity play a key role in analyzing the non-Markov processes encountered.