Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case

We consider two coupled queues with a generalized processor sharing service discipline. The second queue has a much smaller Poisson arrival rate than the first queue, while the customer service times are of comparable magnitude. The processor sharing server devotes most of its resources to the first queue, except when it is empty. The fraction of resources devoted to the second queue is small, of the same order as the ratio of the arrival rates. We assume that the primary queue is heavily loaded and that the secondary queue is critically loaded. If we let the small arrival rate to the secondary queue be O ( ε ) , where 0 ≤ ε ≪ 1 , then in this asymptotic limit the number of customers in the first queue will be large, of order O ( ε - 1 ) , while that in the second queue will be somewhat smaller, of order O ( ε - 1 / 2 ) . We obtain a two-dimensional diffusion approximation for this model and explicitly solve for the joint steady state probability distribution of the numbers of customers in the two queues. This work complements that in (Morrison, 2010), which the second queue was assumed to be heavily or lightly loaded, leading to mean queue lengths that were O ( ε - 1 ) or O ( 1 ) , respectively.