Explicit Steady-State Approximations for Parallel Server Systems with Heterogeneous Servers

TL;DR

This paper analyzes the steady-state behavior of heterogeneous parallel-server systems under load balancing, proving workload distribution convergence and robustness of the WWTA policy without requiring a unique static plan.

Contribution

It establishes a strong form of state-space collapse under relaxed conditions, showing workload independence from local scheduling and negligible impact of non-basic activities.

Findings

Workload converges to exponential distribution with explicit parameters.

Relaxed CRP condition replaces the need for a unique static allocation.

WWTA is robust, with negligible routing to non-basic activities.

Abstract

We study the steady-state performance of parallel-server systems under an immediate routing architecture with two sources of heterogeneity: servers and job classes, subject to compatibility constraints. We focus on the weighted-workload-task-allocation (WWTA) policy, a load-balancing scheme known to be throughput-optimal for such systems. Under a relaxed complete-resource-pooling (CRP) condition, we prove a "strong form" of state-space collapse in heavy traffic and that the scaled workload of each server converges in distribution to an exponential random variable, whose parameter is explicitly given by system primitives. Our analysis yields three main insights. First, the conventional heavy-traffic requirement of a unique static allocation plan can be dropped; a relaxed CRP condition suffices. Second, the limiting workload distribution is shown to be independent of local scheduling policy on server side, allowing substantial flexibility. Third, the inefficient (non-basic) activities prescribed by static allocation plan is proved to receive an asymptotically negligible fraction of routing and service, even though WWTA has no prior knowledge of which activities are basic, highlighting its robustness to changing arrival rates.