Necessary condition for null controllability in many-server heavy traffic

TL;DR

This paper explores the relationship between throughput sub-optimality and null controllability in many-server queueing networks, establishing conditions under which TSO is both necessary and sufficient for null controllability in specific system configurations.

Contribution

It proves that TSO is both necessary and sufficient for null controllability in systems with two customer classes or vice-versa, extending previous results to more general structures.

Findings

TSO implies null controllability in certain queueing systems.

Under specific conditions, TSO is also necessary for null controllability.

The paper extends the understanding of resource allocation effects in many-server queues.

Abstract

Throughput sub-optimality (TSO), introduced in Atar and Shaikhet [Ann. Appl. Probab. 19 (2009) 521-555] for static fluid models of parallel queueing networks, corresponds to the existence of a resource allocation, under which the total service rate becomes greater than the total arrival rate. As shown in Atar, Mandelbaum and Shaikhet [Ann. Appl. Probab. 16 (2006) 1764-1804] and Atar and Shaikhet (2009), in the many server Halfin-Whitt regime, TSO implies null controllability (NC), the existence of a routing policy under which, for every finite $T$, the measure of the set of times prior to $T$, at which at least one customer is in the buffer, converges to zero in probability at the scaling limit. The present paper investigates the question whether the converse relation is also true and TSO is both sufficient and necessary for the NC behavior. In what follows we do get the affirmation for systems with either two customer classes (and possibly more service pools) or vice-versa and state a condition on the underlying static fluid model that allows the extension of the result to general structures.