Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms

TL;DR

This paper demonstrates that the LB-zero load balancing algorithm maintains zero-waiting performance across a broad class of service time distributions, specifically Coxian, using a novel analytical technique called ISSP.

Contribution

It introduces the ISSP method to prove large-system insensitivity of LB-zero for Coxian service times, extending zero-waiting results beyond exponential distributions.

Findings

LB-zero achieves asymptotic zero-waiting for Coxian service times.

The ISSP technique effectively analyzes complex stochastic systems.

Simulation confirms the theoretical insensitivity results.

Abstract

This paper studies the sensitivity (or insensitivity) of a class of load balancing algorithms that achieve asymptotic zero-waiting in the sub-Halfin-Whitt regime, named LB-zero. Most existing results on zero-waiting load balancing algorithms assume the service time distribution is exponential. This paper establishes the {\em large-system insensitivity} of LB-zero for jobs whose service time follows a Coxian distribution with a finite number of phases. This result suggests that LB-zero achieves asymptotic zero-waiting for a large class of service time distributions, which is confirmed in our simulations. To prove this result, this paper develops a new technique, called "Iterative State-Space Peeling" (or ISSP for short). ISSP first identifies an iterative relation between the upper and lower bounds on the queue states and then proves that the system lives near the fixed point of the iterative bounds with a high probability. Based on ISSP, the steady-state distribution of the system is further analyzed by applying Stein's method in the neighborhood of the fixed point. ISSP, like state-space collapse in heavy-traffic analysis, is a general approach that may be used to study other complex stochastic systems.