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Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals
Mathematics  (IF2.258),  Pub Date : 2021-07-20, DOI: 10.3390/math9141709
Freek Verdonck, Herwig Bruneel, Sabine Wittevrongel

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.