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The M/M/k with Deterministic Setup Times

Published: 19 June 2023 Publication History

Abstract

Capacity management, whether it involves servers in a data center, or human staff in a call center, or doctors in a hospital, is largely about balancing a resource-delay tradeoff. On the one hand, one would like to turn off servers when not in use (or send home staff that are idle) to save on resources. On the other hand, one wants to avoid the considerable setup time required to turn an off server back on. This paper aims to understand the delay component of this tradeoff, namely, what is the effect of setup time on average delay in a multi-server system?
Surprisingly little is known about the effect of setup times on delay. While there has been some work on studying the M/M/k with Exponentially-distributed setup times, these works provide only iterative methods for computing mean delay, giving little insight as to how delay is affected by k, by load, and by the setup time. Furthermore, setup time in practice is much better modeled by a Deterministic random variable, and, as this paper shows, the effect of a Deterministic setup time is nothing like that of an Exponentially-distributed setup time.
This paper provides the first analysis of the M/M/k with Deterministic setup times. We prove a lower bound on the effect of setup on delay. Our result is a simple algebraic formula which provides insight into how delay scales with the input parameters. For more details, see the full paper.

References

[1]
Anshul Gandhi, Sherwin Doroudi, Mor Harchol-Balter, and Alan Scheller-Wolf. 2013. Exact analysis of the M/M/k/setup class of Markov chains via Recursive Renewal Reward. In Queueing Syst. 153--166.
[2]
Anshul Gandhi, Varun Gupta, Mor Harchol-Balter, and Michael Kozuch. 2010a. Optimality Analysis of Energy-Peformance Trade-off for Server Farm Management. In PERFORMANCE 2010 Conference. IFIP WG 7.3 International Symposium on Computer Modeling, Measurement and Evaluation. Namur, Belgium.
[3]
Anshul Gandhi and Mor Harchol-Balter. 2011. How Data Center Size Impacts the Effectiveness of Dynamic Power Management. In 49th Annual Allerton Conference on Communication, Control, and Computing. Urbana-Champaign, IL, 1164--1169.
[4]
Anshul Gandhi, Mor Harchol-Balter, and Ivo Adan. 2010b. Server farms with setup costs. Performance Evaluation, Vol. 67, 11 (2010), 1123--1138.
[5]
Anshul Gandhi, Mor Harchol-Balter, Ram Raghunathan, and Michael A Kozuch. 2012. AutoScale: Dynamic, Robust Capacity Management for Multi-Tier Data Centers. ACM Trans. Comput. Syst., Vol. 30, 4 (2012), 1--26.
[6]
Ming Mao and Marty Humphrey. 2012. A Performance Study on the VM Startup Time in the Cloud. In IEEE Int. Conf. Cloud Computing (CLOUD). Honolulu, HI, 423--430.
[7]
Jamol Pender and Tuan Phung-Duc. 2016. A Law of Large Numbers for M/M/c/Delayoff-Setup Queues with Nonstationary Arrivals. In Int. Conf. on Analytical and Stochastic Modeling Techniques and Applications. Springer, Cardiff, UK, 253--268.
[8]
Tuan Phung-Duc. 2017. Exact solutions for M/M/c/Setup queues. Telecommun. Syst., Vol. 64, 2 (2017), 309--324.
[9]
Jalani K Williams, Mor Harchol-Balter, and Weina Wang. 2022. The M/M/k with Deterministic Setup Times. Proc. ACM SIGMETRICS Int. Conf. Measurement and Modeling of Computer Systems, Vol. 6, 3 (2022), 1--45.

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Published In

cover image ACM Conferences
SIGMETRICS '23: Abstract Proceedings of the 2023 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems
June 2023
123 pages
ISBN:9798400700743
DOI:10.1145/3578338
  • cover image ACM SIGMETRICS Performance Evaluation Review
    ACM SIGMETRICS Performance Evaluation Review  Volume 51, Issue 1
    SIGMETRICS '23
    June 2023
    108 pages
    ISSN:0163-5999
    DOI:10.1145/3606376
    Issue’s Table of Contents
Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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Publication History

Published: 19 June 2023

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Author Tags

  1. capacity provisioning
  2. deterministic setup times
  3. exceptional first service
  4. large-system scaling
  5. m/m/k/setup

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