# Many-Server Queues with Random Service Rates in the Halfin-Whitt Regime: A Measure-Valued Process Approach

## TL;DR

This paper develops a measure-valued process framework to analyze many-server queues with heterogeneous, randomly distributed service rates in heavy traffic, revealing new convergence behaviors and fairness dynamics.

## Contribution

It introduces a novel measure-valued approach and a new convergence concept for analyzing heavy traffic limits in heterogeneous many-server queues.

## Key findings

- Fairness processes do not converge in the usual topology
- A new convergence notion based on shifted processes is proposed
- Martingales are used to identify limiting fairness processes

## Abstract

We consider many-server queueing systems with heterogeneous exponential servers and renewal arrivals. The service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy traffic limit of these queues in random environment using probability measure-valued stochastic processes. We introduce the measure-valued fairness process which denotes the proportion of cumulative idleness experienced by servers whose rates fall in a Borel subset of the support of the service rates. It can be shown that these fairness processes do not converge in the usual Skorokhod-$J_1$ topology, hence we introduce a new notion of convergence based on shifted versions of these processes. We also introduce some useful martingales to identify limiting fairness processes under different routing policies.