  In this paper we propose a highly accurate approximate performance analysis
of a heterogeneous server system with a processor sharing service discipline
and a general job-size distribution under a generalized join the shortest queue
(GJSQ) routing protocol. The GJSQ routing protocol is a natural extension of
the well-known join the shortest queue routing policy that takes into account
the non-identical service rates in addition to the number of jobs at each
server. The performance metrics that are of interest here are the equilibrium
distribution and the mean and standard deviation of the number of jobs at each
server. We show that the latter metrics are near-insensitive to the job-size
distribution using simulation experiments. By applying a single queue
approximation we model each server as a single server queue with a
state-dependent arrival process, independent of other servers in the system,
and derive the distribution of the number of jobs at the server. These
state-dependent arrival rates are intended to capture the inherent correlation
between servers in the original system and behave in a rather atypical way.
