Large number of queues in tandem: Scaling properties under back-pressure algorithm

TL;DR

This paper analyzes how queues in a tandem system scale under a back-pressure algorithm, revealing a phase transition at a critical load of 1/4 where queues switch from bounded to unbounded growth as the number of queues increases.

Contribution

It establishes the scaling behavior of tandem queues under back-pressure control, linking the critical load to the maximum flux of a related particle system.

Findings

Queues remain bounded if load is below 1/4

Queues grow unbounded if load exceeds 1/4

Critical load corresponds to maximum flux in particle system

Abstract

We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of next queue n+1. The question addressed is how steady-state queues scale as N goes to infinity. We show that the answer depends on whether lambda is below or above the critical value 1/4: in the former case queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.