Markov-modulated M/G/1 type queue in heavy traffic and its application to time-sharing disciplines

Two variants of a Markov-modulated single server queue are analysed. For the first variant, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. This result is then applied to the second variant analysed in the latter half of the paper; a modulated multiclass system under discriminatory processor sharing (DPS). Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue length of different customer classes undergoes a state-space collapse when subject to heavy-traffic scaling. Using the workload result, the limiting distribution of the queue length vector is shown to be exponential, times a deterministic vector.