Many-Server Queues with Random Service Rates: A Unified Framework Based on Measure-Valued Processes

We consider many-server queueing systems with heterogeneous exponential servers, for which the service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy-traffic diffusion limits of these queues using 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 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. To demonstrate the power of our framework, we show how it can be used to prove diffusion limits for parallel server systems with within-pool heterogeneity.