Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward

Anshul Gandhi, Sherwin Doroudi, Mor Harchol-Balter, Alan Scheller-Wolf

Research output: Chapter in Book/Report/Conference proceedingConference contribution

31 Scopus citations

Abstract

The M/M/k/setup model, where there is a penalty for turning servers on, is common in data centers, call centers and manufacturing systems. Setup costs take the form of a time delay, and sometimes there is additionally a power penalty, as in the case of data centers. While the M/M/1/setup was exactly analyzed in 1964, no exact analysis exists to date for the M/M/k/setup with k > 1. In this paper we provide the first exact, closed-form analysis for the M/M/k/setup and some of its important variants including systems in which idle servers delay for a period of time before turning off or can be put to sleep. Our analysis is made possible by our development of a new technique, Recursive Renewal Reward (RRR), for solving Markov chains with a repeating structure. RRR uses ideas from renewal reward theory and busy period analysis to obtain closed-form expressions for metrics of interest such as the transform of time in system and the transform of power consumed by the system. The simplicity, intuitiveness, and versatility of RRR makes it useful for analyzing Markov chains far beyond the M/M/k/setup. In general, RRR should be used to reduce the analysis of any 2-dimensional Markov chain which is infinite in at most one dimension and repeating to the problem of solving a system of polynomial equations. In the case where all transitions in the repeating portion of the Markov chain are skip-free and all up/down arrows are unidirectional, the resulting system of equations will yield a closed-form solution.

Original languageEnglish (US)
Title of host publicationSIGMETRICS 2013 - Proceedings of the 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems
Pages153-166
Number of pages14
Edition1 SPEC. ISS.
DOIs
StatePublished - Jul 22 2013
Event2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS 2013 - Pittsburgh, PA, United States
Duration: Jun 17 2013Jun 21 2013

Publication series

NamePerformance Evaluation Review
Number1 SPEC. ISS.
Volume41
ISSN (Print)0163-5999

Conference

Conference2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS 2013
Country/TerritoryUnited States
CityPittsburgh, PA
Period6/17/136/21/13

Keywords

  • Performance
  • Queueing theory
  • Resource allocation

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