Repairable queue with non-exponential service time and variable breakdown rates

Consider a single server queue in which the service station may breakdown according to a Poisson process with rates γ in busy time and γ’ in idle time respectively. After a breakdown, the service station will be repaired immediately and the repair time is assumed to have an exponential distribution with rate δ. Suppose the arrival time has an exponential distribution with rate λ, and the probability density function g(t) and the cumulative distribution function G(t) of the service time are such that the rate g(t)/[1 – G(t)] tends to a constant as t tends to infinity. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution of the system.