  We consider the job assignment problem in a multi-server system consisting of
$N$ parallel processor sharing servers, categorized into $M$ ($\ll N$)
different types according to their processing capacity or speed. Jobs of random
sizes arrive at the system according to a Poisson process with rate $N
\lambda$. Upon each arrival, a small number of servers from each type is
sampled uniformly at random. The job is then assigned to one of the sampled
servers based on a selection rule. We propose two schemes, each corresponding
to a specific selection rule that aims at reducing the mean sojourn time of
jobs in the system.
  We first show that both methods achieve the maximal stability region. We then
analyze the system operating under the proposed schemes as $N \to \infty$ which
corresponds to the mean field. Our results show that asymptotic independence
among servers holds even when $M$ is finite and exchangeability holds only
within servers of the same type. We further establish the existence and
uniqueness of stationary solution of the mean field and show that the tail
distribution of server occupancy decays doubly exponentially for each server
type. When the estimates of arrival rates are not available, the proposed
schemes offer simpler alternatives to achieving lower mean sojourn time of
jobs, as shown by our numerical studies.

Distributed, Parallel, and Cluster Computing;Performance;Systems and Control;Probability;Applications