Abstract
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.
Highlights
IntroductionIn the early 20th century, the Danish mathematician Agner Erlang used a mathematical model to describe a telephone switch (at the time, an office where workers manually connected phone lines), he became the founder of the field of queueing theory
When, in the early 20th century, the Danish mathematician Agner Erlang used a mathematical model to describe a telephone switch, he became the founder of the field of queueing theory
In [14], a continuous-time queueing system is treated where N servers are subject to breakdowns and repair; the results are the distributions of the queue length and waiting times
Summary
In the early 20th century, the Danish mathematician Agner Erlang used a mathematical model to describe a telephone switch (at the time, an office where workers manually connected phone lines), he became the founder of the field of queueing theory. The authors of [15] obtain the queue length distribution and the delay distribution for a discrete-time model with general service demands and correlated service capacities. In their model, the service time of a customer depends on both its service demand and the service capacity. We consider a discrete-time multiserver queueing system that is special in the way that it allows correlation in both the slot-to-slot server availability and the arrival process. A stochastic number of servers is available, and a stochastic number of new customers arrive These are both i.i.d. within each state, but these distributions can be different for each state.
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