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.
In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay.
In this work we look at a discrete-time multiserver queueing system where the number of available servers is distributed according to one of two geometrics. The arrival process is assumed to be general independent, the service times deterministically equal to one slot and the buffer capacity infinite. The queueing system resides in one of two states and the number of available servers follows a geometric distribution with parameter determined by the system state. At the end of a slot there is a fixed probability that the system evolves from one state to the other, with this probability depending on the current system state only, resulting in geometrically distributed sojourn times. We obtain the probability generating function (pgf) of the system content of an arbitrary slot in steady-state, as well as the pgf of the system content at the beginning of an arbitrary slot with a given state. Furthermore we obtain an approximation of the distribution of the delay a customer experiences in the proposed queueing system. This approximation is validated by simulation and the results are illustrated with a numerical example.
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