The drift method was recently developed to study queuing systems in steady state. It was used successfully to obtain bounds on the moments of the scaled queue lengths that are asymptotically tight in heavy traffic and in a wide variety of systems, including generalized switches, input-queued switches, bandwidth-sharing networks, and so on. In this paper, we develop the use of transform techniques for heavy-traffic analysis, with a special focus on the use of moment-generating functions. This approach simplifies the proofs of the drift method and provides a new perspective on the drift method. We present a general framework and then use the moment-generating function method to obtain the stationary distribution of scaled queue lengths in heavy traffic in queuing systems that satisfy the complete resource pooling condition. In particular, we study load balancing systems and generalized switches under general settings.

We study the heavy-traffic limit of the generalized switch operating under MaxWeight, without assuming that the complete resource pooling condition is satisfied and allowing for correlated arrivals. The main contribution of this paper is the steady-state mean of linear combinations of queue lengths in heavy traffic. We showcase the generality of our result by presenting various stochastic networks as corollaries, each of which is a contribution by itself. In particular, we study the input-queued switch with correlated arrivals, and we show that, if the state space collapses to a full-dimensional subspace, the correlation among the arrival processes does not matter in heavy traffic. We exemplify this last case with a parallel-server system, an [Formula: see text]-system, and an ad hoc wireless network. Whereas these results are obtained using the drift method, we additionally present a negative result showing a limitation of the drift method. We show that it is not possible to obtain the individual queue lengths using the drift method with polynomial test functions. We do this by presenting an alternate view of the drift method in terms of a system of linear equations, and we use this system of equations to obtain bounds on arbitrary linear combinations of the queue lengths.

Stochastic Processing Networks that model wired and wireless networks, and other queueing systems, have been studied in heavytraffic limit under the so-called Complete Resource Pooling (CRP) condition. 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 steadystate mean of certain linear combinations of queue lengths in the heavy-traffic limit under MaxWeight scheduling algorithm. We use 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.

Motivated by application in wireless networks, cloud computing, data centers etc, Stochastic Processing Networks have been studied in the literature under various asymptotic regimes. In the heavy-traffic regime, the steady state mean queue length is proved to be O( 1ǫ ) where ǫ is the heavytraffic parameter, that goes to zero in the limit. The focus of this paper is on obtaining queue length bounds on prelimit systems, thus establishing the rate of convergence to the heavy traffic. In particular, we study the generalized switch model operating under the MaxWeight algorithm, and we show that the mean queue length of the prelimit system is only O log 1 ǫ away from its heavy-traffic limit. We do this even when the so called complete resource pooling (CRP) condition is not satisfied. When the CRP condition is satisfied, in addition, we show that the MaxWeight algorithm is within O log 1 ǫ of the optimal. Finally, we obtain similar results in load balancing systems operating under the join the shortest queue routing algorithm.

Motivated by applications to wireless networks, cloud computing, data centers, etc., stochastic processing networks have been studied in the literature under various asymptotic regimes. In the heavy traffic regime, the steady-state mean queue length is proved to be $\Theta({1}/{\epsilon})$ , where $\epsilon$ is the heavy traffic parameter (which goes to zero in the limit). The focus of this paper is on obtaining queue length bounds on pre-limit systems, thus establishing the rate of convergence to heavy traffic. For the generalized switch, operating under the MaxWeight algorithm, we show that the mean queue length is within $\textrm{O}({\log}({1}/{\epsilon}))$ of its heavy traffic limit. This result holds regardless of the complete resource pooling (CRP) condition being satisfied. Furthermore, when the CRP condition is satisfied, we show that the mean queue length under the MaxWeight algorithm is within $\textrm{O}({\log}({1}/{\epsilon}))$ of the optimal scheduling policy. Finally, we obtain similar results for the rate of convergence to heavy traffic of the total queue length in load balancing systems operating under the ‘join the shortest queue’ routeing algorithm.