Abstract
Open AccessOpen Access licenseAboutSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail Go to SectionOpen AccessOpen Access license HomeStochastic SystemsVol. 6, No. 1 Randomized Assignment of Jobs to Servers in Heterogeneous Clusters of Shared Servers for Low DelayArpan Mukhopadhyay, A. Karthik, Ravi R. MazumdarArpan Mukhopadhyay, A. Karthik, Ravi R. MazumdarPublished Online:17 Oct 2016https://doi.org/10.1287/15-SSY179AbstractWe consider the problem of assignning jobs to servers in a multi-server system consisting of N parallel processor sharing servers, categorized into M (≪ N) different types according to their processing capacities or speeds. Jobs of random sizes arrive at the system according to a Poisson process with rate Nλ. Upon each arrival, some servers of each type are sampled uniformly at random. The job is then assigned to one of the sampled servers based on their states. We propose two schemes, which differ in the metric for choosing the destination server for each arriving job. Our aim is to reduce the mean sojourn time of the jobs in the system.It is shown that the proposed schemes achieve the maximal stability region, without requiring the knowledge of the system parameters. The performance of the system operating under the proposed schemes is analyzed in the limit as N → ∞. This gives rise to a mean field limit. The mean field is shown to have a unique, globally asymptotically stable equilibrium point which approximates the stationary distribution of load at each server. Asymptotic independence among the servers is established using a notion of intra-type exchangeability which generalizes the usual notion of exchangeability. It is further shown that the tail distribution of server occupancies decays doubly exponentially for each server type. Numerical evidence shows that at high load the proposed schemes perform at least as well as other schemes that require more knowledge of the system parameters. Previous Back to Top Next FiguresReferencesRelatedInformation Volume 6, Issue 1June 2016Pages 1-250 Article Information Metrics Information Received:February 01, 2015Published Online:October 17, 2016 Copyright © 2016, The author(s)Cite asArpan Mukhopadhyay, A. Karthik, Ravi R. Mazumdar (2016) Randomized Assignment of Jobs to Servers in Heterogeneous Clusters of Shared Servers for Low Delay. Stochastic Systems 6(1):90-131. https://doi.org/10.1287/15-SSY179 KeywordsProcessor sharingpower-of-twomean fieldasymptotic independencestabilitypropagation of chaosPDF download
Highlights
Consider a stream of jobs arriving at a multi-server system consisting of a large number of parallel processor sharing servers
We propose schemes which do not require the knowledge of the system parameters or the arrival rate of jobs for their operation and yet achieve the maximal stability region
This paper focuses on the design and analysis of randomized job assignment schemes which achieve the maximal stability region for heterogeneous processor sharing systems without requiring the knowledge of system parameters and yet yield smaller delays than randomized state independent schemes
Summary
Consider a stream of jobs arriving at a multi-server system consisting of a large number of parallel processor sharing servers. The optimal routing probabilities are given by Theorem 1 of [1] In this scheme, no communication is required between the job dispatcher and the servers as the job assignment decisions are made independently of the state of the servers. We see that hybrid SQ(d) scheme results in a smaller mean sojourn time of jobs than that in Scheme 1 and Scheme 2, for smaller values of λ This is because, in the hybrid SQ(2) scheme, the routing probabilities are chosen optimally based on the arrival rate λ. Γi Ci j∈J γj Cj for each server type i ∈ J This choice of routing probabilities ensures that all arrival rates in the maximal stability region can be supported by the system operating under either the state independent scheme or the Hybrid SQ(d) scheme.
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