This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 -G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained.

Pathway structure determination in complex stochastic networks with non-exponential dwell times J. Chem. Phys. 140, 184102 (2014) Abstract. 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.