  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.
