Transient analysis of an M/M/1/N queue with balking, catastrophes, server failures and repairs

An M/M/1/N queue with balking, catastrophes, server failures and repairs is considered. The arrivals follow a Poisson distribution and the servers serve according to an exponential distribution. On arrival, a customer either decides to join the queue or may balk the queue based on number of customers in the system. The explicit expressions for the time-dependent system size probabilities are obtained in terms of the modified Bessel function of first kind.