Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs (Q656443)

A single-server infinite capacity queue with balking, catastrophes and server failure and repairs is studied, where all governing random variables are exponential. If there are not less than k customers in the system, a new customer joins the queue with probability \(\beta\), and balks with probability \(1-\beta\). A catastrophe occurs (regardless of the status of the server) and, if the server is busy, all customers in the system are destroyed immediately. It means that a failure occurs, and then repair of server is started. The number of customers in the system is a Markov process, and, after standard routine calculations, the author finds both time-dependent and steady-state distributions of the process. Some special case are also included.