Convergence properties of many parallel servers under power-of-D load balancing

TL;DR

This paper analyzes the convergence and independence properties of large-scale parallel server systems under power-of-D load balancing, providing quantitative estimates and proving asymptotic insensitivity for various scheduling disciplines.

Contribution

It introduces the clan of ancestors technique to quantify queue dependence and proves propagation of chaos and asymptotic insensitivity in large systems with general service times.

Findings

Quantitative estimates on queue correlations

Propagation of chaos established for Markovian arrivals

Asymptotic insensitivity shown for a wide class of disciplines

Abstract

We consider a system of N queues with decentralized load balancing such as power-of-D strategies(where D may depend on N) and generic scheduling disciplines. To measure the dependence of the queues, we use the clan of ancestors, a technique coming from interacting particle systems. Relying in that analysis we prove quantitative estimates on the queues correlations implying propagation of chaos for systems with Markovian arrivals and general service time distribution. This solves the conjecture posed by Bramsom et. al. in [*] concerning the asymptotic independence of the servers in the case of processor sharing policy. We then proceed to prove asymptotic insensitivity in the stationary regime for a wide class of scheduling disciplines and obtain speed of convergence estimates for light tailed service distribution. [*] M. BRAMSON, Y. LU AND B. PRABHAKAR, Asymptotic independence of queues under randomized load balancing, Queueing Syst., 71:247-292, 2012.