Time to Buffer Overflow in a Finite-Capacity Queueing Model with Setup and Closedown Times

Abstract

A single-channel queueing model with finite buffer capacity, Poisson arrivals and generally distributed processing times is investigated. According to frequent energy saving requirements, after each busy period the service station is being switched off during a randomly distributed closedown time. Similarly, the first processing in each busy period is preceded by a random setup time, during which the service process is suspended and the machine is being switched on and achieves full readiness for the processing. A system of Volterra-type integral equations for the distribution of the time to the first buffer overflow, conditioned by the initial level of buffer saturation, is built, by applying the idea of embedded Markov chain and continuous version of total probability law. Using the linear algebraic approach, the solution of the corresponding system written for Laplace transforms is obtained explicitly.