Fluid and diffusion limits for queues in slowly changing environments

We consider an infinite-capacity s-server queue in a finite-state random environment, where the traffic intensity exceeds 1 in some environment states and the environment states change slowly relative to arrivals and service completions. Queues grow in unstable environment states, so that it is useful to look at the system in the time scale of mean environment-state sojourn times. As the mean environment-state sojourn times grow, the queue-length and workload processes grow. However, with appropriate normalizations, these processes converge to fluid processes and diffusion processes. The diffusion process in a random environment is a refinement of the fluid process in a random environment. We show how the scaling in these limits can help explain numerical results for queues in slowly changing random environments. For that purpose, we apply recently developed numerical-transform-inversion algorithms for the MAP/G/1 queue and the piecewise-stationary Mt/Gt/1 queue