Infinite Horizon Average Optimality of the N-network Queueing Model in the Halfin-Whitt Regime

TL;DR

This paper analyzes the optimal control of a two-class, two-server queueing network in the Halfin-Whitt regime, establishing asymptotic optimality and proposing a state-dependent priority policy for efficient management.

Contribution

It proves asymptotic optimality for three control objectives in the N-network queueing model under the Halfin-Whitt regime and introduces a simple priority scheduling policy ensuring ergodicity.

Findings

Asymptotic optimality of the control policies is established.

A simple state-dependent priority policy ensures geometric ergodicity.

Convergence of the diffusion-scaled process to the controlled diffusion limit is demonstrated.

Abstract

We study the infinite horizon optimal control problem for N-network queueing systems, which consist of two customer classes and two server pools, under average (ergodic) criteria in the Halfin-Whitt regime. We consider three control objectives: 1) minimizing the queueing (and idleness) cost, 2) minimizing the queueing cost while imposing a constraint on idleness at each server pool, and 3) minimizing the queueing cost while requiring fairness on idleness. The running costs can be any nonnegative convex functions having at most polynomial growth. For all three problems we establish asymptotic optimality, namely, the convergence of the value functions of the diffusion-scaled state process to the corresponding values of the controlled diffusion limit. We also present a simple state-dependent priority scheduling policy under which the diffusion-scaled state process is geometrically ergodic in the Halfin-Whitt regime, and some results on convergence of mean empirical measures which facilitate the proofs.