Transient Analysis of an M/M/1 Queue with Reneging, Catastrophes, Server Failures and Repairs

An M / M / 1 queue with reneging, catastrophes, server failures and repairs is considered. The arrivals follow a Poisson process and the servers serve according to an exponential distribution. On arrival a customer decides to join the queue and after joining the queue if a customer has to wait for the service longer than his expectation, he may renege. Explicit expression for the time-dependent probabilities of the system size is obtained in terms of the modified Bessel function of first kind by making use of Laplace transform and probability generating function techniques. The system queue length and failure distribution for steady state are derived. Additionally, time-dependent mean and variance are obtained. A numerical example is presented to study the behavior of the system.