Heavy Traffic Queue Length Behavior in a Switch Under the MaxWeight Algorithm

R. Srikant,Siva Theja Maguluri

doi:10.1287/15-ssy193

Abstract

We consider a switch operating under the MaxWeight scheduling algorithm, under any traffic pattern such that all the ports are loaded. This system is interesting to study since the queue lengths exhibit a multi-dimensional state-space collapse in the heavy-traffic regime. We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state, under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions, we show that the heavy-traffic scaled queue length is given by [Formula: see text], where σ is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by [Formula: see text]. The result shows that the heavy-traffic scaled queue length has optimal scaling with respect to n, thus settling one version of an open conjecture; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast.

Highlights

Consider a collection of queues arranged in the form of an n × n matrix
The following constraints are imposed on the service process of the queueing system: (a) at most one queue can be served in each time slot in each row of the matrix, (b) at most one queue can be served in each time slot in each column of the matrix, and (c) when a queue is served, at most one packet can be removed from the queue
We show that under MaxWeight policy, the limiting average queue length is within a factor of less than 2 of the universal lower bound and MaxWeight has optimal average queue length scaling

Summary

Introduction

Consider a collection of queues arranged in the form of an n × n matrix. The queues are assumed to operate in discrete-time and jobs arriving to the queues will be called packets. Heavy traffic optimality of MaxWeight algorithm for generalized switches is shown in [3, 7] when a single input or output port is saturated or in other words when approaching an arrival rate vector on a facet of the capacity region. We will prove the following lower bound on the average queue lengths, which is valid under any scheduling policy.

Results

Conclusion