Analysis of a 2-state discrete-time queue with stochastic state-period lengths and state-dependent server availability and arrivals

Abstract

In this work we study a discrete-time multiserver queueing system with an infinite storage capacity and deterministic service times equal to 1 slot. Specific to the model under study is that the system is assumed to be in one of two different states (state-1 or state-2) and that both the distribution of the number of available servers and the arrival process depend on the system state. State changes can only occur at slot boundaries and mark the beginning and end of state-1-periods and state-2-periods. The lengths of these state-1-periods and state-2-periods, expressed as a number of slots, are assumed to be two independent sets of independent and identically distributed stochastic variables. The number of available servers during a slot is a stochastic variable with a distribution that is completely determined by the system state during that slot. Likewise, the distribution of the number of arrivals during a slot only depends on the system state during that slot. The only restrictions we put on the distributions of the state-1-periods, state-2-periods and number of available servers is that they have rational probability generating functions (pgfs), and that during each slot at least one server is available. For the considered queueing system we present a method to determine the pgf of the steady-state system content at various observation instants. Several numerical examples demonstrate the possibilities of this model.