Exact stationary solution for a fluid queue driven by an <i>M/M/</i>1 queue with disaster and subsequent repair

Abstract

This paper deals with the stationary analysis of a fluid queuing model driven by an M/M/1 queue subject to disaster and subsequent repair. Further, arrivals are allowed to join the background queuing model during the period of repair at a slower rate as compared to the arrivals during regular busy period of the server. Such a model was analysed earlier by Ammar (2014), however the model formulation and hence the main results are found to be incorrect. In this paper, the assumptions are suitably modified to ensure correctness, detailed mathematical analysis is carried out to find an explicit analytical expression for the buffer content distribution. The underlying system of differential difference equations that govern the process are solved using Laplace transform and generating function methodologies. The closed form expressions for the joint steady state probabilities of the state of the background queuing model and the content of the buffer are obtained in terms of modified Bessel function of the first kind.