Two queues with vastly different arrival rates and processor-sharing factors
01 January 2010
We consider a 2-class queueing system, operating under a generalized processorsharing discipline. The arrival rate to the secondary queue is much smaller than that to the primary queue, while the exponentially distributed service requirements have comparable parameters. The primary queue is assumed to be heavily loaded, so the processor-sharing factor for the secondary queue is assumed to be relatively small. We use singular perturbation analyses in a small parameter measuring the difference in arrival rates, and the closeness of the system to instability. Two different regimes are analyzed, corresponding to a heavily loaded and a lightly loaded secondary queue, respectively. With suitable scaling of variables, lowest order asymptotic approximations to the joint stationary distribution of the numbers of jobs in the two queues are derived, as well as the marginal distributions.