Stability for Queues with Time Varying Rates

Abstract

An M(t)/M(t)/1 queue or M/M/1 queue with time varying rates, may alternate through periods of underloading, overloading, and critical loading. We analyze this model by using a general asymptotic method called uniform acceleration, which we will show is the appropriate time-varying analogue to steady state analysis. Applying this method to the transition probabilities of the queue length process, we obtain necessary and sufficient conditions for underloading which we will show is the time-varying analogue to steady state stability.

Using the theory of strong approximations, we can also apply a similar asymptotic analysis directly to the random sample paths of the queueing process. In obtaining a functional strong law of large numbers and central limit theorem for the M(t)/M(t)/1 queue, we obtain a rigorous basis for the fluid and diffusion approximations that are used to analyze this system. Moreover, the will be many candidates for the time-varying analogue to heavy traffic limit processes. The results are presented to suggest new methods for the asymptotic analysis of nonstationary, continuous time Markov chains.