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. The partial differential equation (PDE) and boundary conditions satisfied by the diffusion approximation to the joint distribution of the queue lengths are derived. These equations are then solved to obtain an explicit expression for the diffusion approximation, and an alternate representation establishes that it is nonnegative. Moreover, the tail behavior of the joint and marginal distributions is investigated.
