Join-Idle-Queue with Service Elasticity: Large-Scale Asymptotics of a Non-monotone System

TL;DR

This paper proves the stability and optimal steady-state behavior of a large-scale load balancing system with elastic service capacity, addressing key questions left open in prior work by developing novel proof techniques.

Contribution

It introduces a new method using induction and weak monotonicity to establish stability and convergence for a non-monotone, large-scale load balancing system with infinite buffers.

Findings

System is stable under subcritical load for large N

Steady-state distributions converge to the optimal as N increases

New proof technique applicable to non-monotone systems

Abstract

We consider the model of a token-based joint auto-scaling and load balancing strategy, proposed in a recent paper by Mukherjee, Dhara, Borst, and van Leeuwaarden (SIGMETRICS '17, arXiv:1703.08373), which offers an efficient scalable implementation and yet achieves asymptotically optimal steady-state delay performance and energy consumption as the number of servers $N\to\infty$. In the above work, the asymptotic results are obtained under the assumption that the queues have fixed-size finite buffers, and therefore the fundamental question of stability of the proposed scheme with infinite buffers was left open. In this paper, we address this fundamental stability question. The system stability under the usual subcritical load assumption is not automatic. Moreover, the stability may not even hold for all $N$. The key challenge stems from the fact that the process lacks monotonicity, which has been the powerful primary tool for establishing stability in load balancing models. We develop a novel method to prove that the subcritically loaded system is stable for large enough $N$, and establish convergence of steady-state distributions to the optimal one, as $N \to \infty$. The method goes beyond the state of the art techniques -- it uses an induction-based idea and a "weak monotonicity" property of the model; this technique is of independent interest and may have broader applicability.