Flexible Load Balancing with Multi-dimensional State-space Collapse: Throughput and Heavy-traffic Delay Optimality

TL;DR

This paper proves that steady-state heavy-traffic delay optimality can be achieved under multi-dimensional state-space collapse in load balancing systems, extending previous results and enabling more flexible policy design.

Contribution

It demonstrates that multi-dimensional state-space collapse suffices for delay optimality, and introduces new load balancing policies leveraging this insight.

Findings

Heavy-traffic delay optimality holds under multi-dimensional state-space collapse.

New load balancing policies can be both throughput and delay optimal.

The methods used are broadly applicable to related systems.

Abstract

Heavy traffic analysis for load balancing policies has relied heavily on the condition of state-space collapse onto a single-dimensional line in previous works. In this paper, via Lyapunov-drift analysis, we rigorously prove that even under a multi-dimensional state-space collapse, steady-state heavy-traffic delay optimality can still be achieved for a general load balancing system. This result directly implies that achieving steady-state heavy-traffic delay optimality simply requires that no server is idling while others are busy at heavy loads, thus complementing and extending the result obtained by diffusion approximations. Further, we explore the greater flexibility provided by allowing a multi-dimensional state-space collapse in designing new load balancing policies that are both throughput optimal and heavy-traffic delay optimal in steady state. This is achieved by overcoming various technical challenges, and the methods used in this paper could be of independent interest.