Large Scale and Heavy Traffic Asymptotics for Systems with Unreliable Servers

The asymptotic behaviour of the M/M/n queue, with servers subject to independent breakdowns and repairs, is examined in the limit where the number of servers tends to infinity and the repair rate tends to 0, such that their product remains finite. It is shown that the limiting two-dimensional Markov process corresponds to a queue where the number of servers has the same stationary distribution as the number of jobs in an M/M/∞ queue. Hence, the limiting model is referred to as the M/M/[M/M/∞] queue. Its numerical solution is discussed.Next, the behaviour of the M/M/[M/M/∞] queue is analysed in heavy traffic when the traffic intensity approaches 1. The convergence of the (suitably normalized) process of the number of jobs to a diffusion is proved.