Heavy-Traffic Approximation for Head of the Line Processor Sharing for Two Parallel Queues.

In this paper the steady-state behavior of two parallel M/M/1 queues under the head of the line processor-sharing discipline is investigated, when both queues are in heavy traffic. The Poisson arrival rates, and the exponential service rates, for the two queues are in general different. The single server splits its capacity (unevenly, in general) between the jobs at the head of each queue, unless one queue is empty, in which case the full capacity of the server is given to the job at the head of the other queue. An explicit expression is derived for the diffusion approximation to the joint distribution of the queue lengths, with the help of the known generating function for this quantity.