Asymptotic Analysis of Two Coupled Queues with Vastly Different Arrival Rates and Finite Customer Capacities

We consider two coupled queues, with each having a finite capacity of customers. When both queues are nonempty they evolve independently, but when one becomes empty the service rate in the other changes. Such a model corresponds to a generalized processor sharing (GPS) discipline. We study the joint distribution p(m, n) of finding (m, n) customers in the (first, second) queue, in the steady state. We study the problem in an asymptotic limit of “heavy traffic,” where also the arrival rate to the second queue is assumed to be small relative to that of the first. The capacity of the first queue is scaled to be large, while that of the second queue is held constant. We consider several different scalings, and in each case obtain limiting differential and/or difference equation for p(m, n), and these we explicitly solve. We show that our asymptotic approximations are quite accurate numerically. This work supplements previous investigations into this GPS model, which assumed infinite capacities/buffers. The present model corresponds to a random walk in a lattice rectangle, where p(m, n) satisfies a different boundary condition on each edge.