# Infinite horizon asymptotic average optimality for large-scale parallel server networks

## TL;DR

This paper investigates optimal control strategies for large-scale parallel server networks in the Halfin-Whitt regime, establishing asymptotic optimality and proposing stabilizing policies with exponential ergodicity.

## Contribution

It introduces a new framework for asymptotic optimality in multi-class, multi-pool server networks and develops state-dependent policies ensuring stability and optimality.

## Key findings

- Optimal values converge to ergodic control solutions.
- State-dependent policies stabilize the diffusion process.
- Exponential ergodicity under certain abandonment conditions.

## Abstract

We study infinite-horizon asymptotic average optimality for parallel server network with multiple classes of jobs and multiple server pools in the Halfin-Whitt regime. Three control formulations are considered: 1) minimizing the queueing and idleness cost, 2) minimizing the queueing cost under a constraints on idleness at each server pool, and 3) fairly allocating the idle servers among different server pools. For the third problem, we consider a class of bounded-queue, bounded-state (BQBS) stable networks, in which any moment of the state is bounded by that of the queue only (for both the limiting diffusion and diffusion-scaled state processes). We show that the optimal values for the diffusion-scaled state processes converge to the corresponding values of the ergodic control problems for the limiting diffusion. We present a family of state-dependent Markov balanced saturation policies (BSPs) that stabilize the controlled diffusion-scaled state processes. It is shown that under these policies, the diffusion-scaled state process is exponentially ergodic, provided that at least one class of jobs has a positive abandonment rate. We also establish useful moment bounds, and study the ergodic properties of the diffusion-scaled state processes, which play a crucial role in proving the asymptotic optimality.