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We consider many-server queueing systems with heterogeneous exponentialservers and renewal arrivals. The service rate of each server is a randomvariable drawn from a given distribution. We develop a framework for analyzingthe heavy traffic limit of these queues in random environment using probabilitymeasure-valued stochastic processes. We introduce the measure-valued fairnessprocess which denotes the proportion of cumulative idleness experienced byservers 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 usualSkorokhod-$J_1$ topology, hence we introduce a new notion of convergence basedon shifted versions of these processes. We also introduce some usefulmartingales to identify limiting fairness processes under different routingpolicies.

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