Limit theorems for Markov-modulated queues

This thesis considers queueing systems affected by a random environment. The behaviour of these queues is studied under specific asymptotic regimes. Embedding a queueing system in a random environment is a way to add flexibility to a model. This flexibility comes at the cost of increased complexity, in that the already stochastic arrival and service processes are also assumed to have randomly fluctuating parameters governed by the external environment. Here, scalings are applied to impose either a central limit theorem (CLT) type of scaling or a heavy traffic scaling.

After speeding up the environment that modulates the Poisson arrivals to an infinite server queue, the arrival process is shown to be asymptotically Poisson with a uniform rate, see Chapter 2. By also speeding up the arrival rates, the scaled and centered queue length converges to a normally distributed random variable. The results are extended in Chapter 3 to a multi-dimensional CLT for an M/G/1 queue with Markov-modulation. Chapter 4 contains a functional CLT for the queue length with modulated arrivals using the martingale CLT to prove weak convergence to an OU process, where the environment moves either faster or slower than the arrival process. In Chapter 5, assuming a fairly general class of service disciplines, it is shown that the workload of an M/G/∞ queue with modulated service capacity converges to an exponentially distributed random variable in heavy traffic. A special case of the queue is analysed under the discriminatory processor sharing discipline.