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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 randomsizes arrive at the system according to a Poisson process with rate $N\lambda$. Upon each arrival, a small number of servers from each type issampled uniformly at random. The job is then assigned to one of the sampledservers based on a selection rule. We propose two schemes, each correspondingto a specific selection rule that aims at reducing the mean sojourn time ofjobs in the system.  We first show that both methods achieve the maximal stability region. We thenanalyze the system operating under the proposed schemes as $N \to \infty$ whichcorresponds to the mean field. Our results show that asymptotic independenceamong servers holds even when $M$ is finite and exchangeability holds onlywithin servers of the same type. We further establish the existence anduniqueness of stationary solution of the mean field and show that the taildistribution of server occupancy decays doubly exponentially for each servertype. When the estimates of arrival rates are not available, the proposedschemes offer simpler alternatives to achieving lower mean sojourn time ofjobs, as shown by our numerical studies.

Randomized Assignment of Jobs to Servers in Heterogeneous Clusters of Shared Servers for Low Delay