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

TL;DR

This paper introduces two job assignment schemes in heterogeneous multi-server systems that achieve maximal stability and significantly reduce mean sojourn time, with proven asymptotic independence and doubly exponential tail decay.

Contribution

The paper proposes two novel job assignment schemes for heterogeneous server clusters that achieve optimal stability and analyze their performance in the mean field limit.

Findings

Both schemes achieve maximal stability region.

Server occupancy tail decays doubly exponentially.

Schemes perform well even without precise arrival rate estimates.

Abstract

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.