Heavy Traffic Queue Length Distribution without Resource Pooling in an Input-Queued Switch

Abstract

This paper studies the heavy-traffic joint distribution of queue lengths of an input-queued switch operating under the MaxWeight scheduling policy. Input-queued switch acts as a representative of SPNs that do not satisfy the so-called complete resource pooling (CRP) condition, and consequently exhibit a multidimensional state space collapse. Except in special cases, only mean queue lengths of such non-CRP systems have been obtained in the literature. In this paper, we develop the transform method to study the steady state distribution of non-CRP systems. The key challenge is in solving an implicit functional equation involving the Laplace transform of the heavy-traffic limiting distribution. We then consider the general n - n input-queued switch that has n2 queues. Under a conjecture on uniqueness of the solution of the functional equation, we obtain an exact joint distribution of the heavy-traffic limiting queue-lengths in terms of a nonlinear transformation of 2n iid exponentials.