Heavy-traffic Analysis of the Generalized Switch under Multidimensional State Space Collapse

Stochastic Processing Networks that model wired and wireless networks, and other queueing systems have been studied in heavy-traffic limit in the literature under the so-called Complete Resource Pooling (CRP) condition. Under the CRP condition, these systems behave like a single server queue. When the CRP condition is not satisfied, heavy-traffic results are known only in the special case of an input-queued switch and bandwidth-sharing network. In this paper, we consider a very general queueing system called the 'generalized switch' that includes wireless networks under fading, data center networks, input-queued switch, etc. The primary contribution of this paper is to present the exact value of the steady-state mean of certain linear combinations of queue lengths in the heavy-traffic limit under the MaxWeight scheduling algorithm. We do this using the Drift method, and we also present a negative result that it is not possible to obtain the remaining linear combinations (and consequently all the individual mean queue lengths) using this method. We do this by presenting an alternate view of the Drift method in terms of an (under-determined) system of linear equations. Finally, we use this system of equations to obtain upper and lower bounds on all linear combinations of queue lengths.