Skip to main content
SpringerLink
Log in

Large deviations without principle: join the shortest queue

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

We develop a methodology for studying “large deviations type” questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a large class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are routed to the (weighted) shortest queue. It is not known whether the large deviations principle holds for this system. Using the tools developed here we derive large deviations type estimates for the most likely behavior, the most likely path to overflow and the probability of overflow. The analysis applies to any finite number of queues. We show via a counterexample that this system may exhibit unexpected behavior

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Atar R, Dupuis P (1999) Large deviations and queueing networks: methods for rate identification. Stoch Proc Appl 84:255–296

    Article  MATH  MathSciNet  Google Scholar 

  • Bertsimas D, Paschalides L, Tsitsiklis JN (1999) Large deviations analysis of the generalized processer sharing policy. Queueing Syst 32:319–349

    Article  MATH  Google Scholar 

  • Chang C-S (1995) Sample path large deviations and intree networks. Queueing Syst 20:7–36

    Article  MATH  Google Scholar 

  • Dembo A, Zeitouni O (1996) Large deviations techniques and applications, 2nd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Dupuis P, Ellis RE (1995a) The large deviations principle for a general class of queueing systems I. Trans Am Math Soc 347:2689–2751

    Article  MATH  MathSciNet  Google Scholar 

  • Dupuis P, Ellis RE (1995b) A weak convergence approach to the theory of large deviations. Wiley, New York

    Google Scholar 

  • Foley RD, McDonald D (2001) Join the shortest queue: stability and exact asymptotics. Ann Appl Prob 11:569–607

    MATH  MathSciNet  Google Scholar 

  • Ganesh AJ, O’Connell N (2002) A large deviation principle with queueing applications. Stoch Stoch Rep 73:25–35

    Article  MATH  MathSciNet  Google Scholar 

  • Ganesh AJ, O’Connell N, Wischik D (2004) Big queues. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Ignatiouk-Robert I (2005) Large deviations for processes with discontinuous statistics. Ann Prob (to appear)

  • Mandjes M (1999) Rare event analysis of the state frequencies of a large number of markov chains. Stoch Models 15:577–592

    Article  MATH  MathSciNet  Google Scholar 

  • Mandjes M, Ridder A (1999) Optimal trajectory in a queue fed by a large number of sources. Queueing Syst 31:137–170

    Article  MATH  MathSciNet  Google Scholar 

  • Mandjes M, Weiss A (2003) Sample path large deviations of a multiple time-scale queueing model (preprint)

  • McDonald D (1996) Overloading parallel servers when arrivals join the shorest queue. In: Glasserman P, Sigman K, Yao D (eds) Stochastic networks: stability and rare events. Springer, Berlin Heidelberg New York, pp 169–196

    Google Scholar 

  • Puhalskii AA, Whitt W (1998) Functional large deviation principles for waiting and departure processes. Prob Eng Info Sci 12:479–507

    Article  MATH  MathSciNet  Google Scholar 

  • Ridder A, Shwartz A (2005) Large deviations methods and the join-the-shortest-queue model. Report, available (JSQreport) from http://www.ee.technion.ac.il/~adam/PAPERS

  • Shwartz A, Weiss A (1995) Large deviations for performance analysis: queues, communication and computing. Chapman Hall, London

    Google Scholar 

  • Shwartz A, Weiss A (1999) Multiple time scales in Markovian ATM models I. Formal calculations. In: CC PUB 267. Electrical Engineering, Technion

  • Shwartz A, Weiss A (2005) Large Deviations with diminishing rates. Math Oper Res (to appear)

  • Stolyar AL (2003) Control of end-to-end delay tails in a multiclass network: LWDF discipline optimality. Ann Appl Prob 13:1151–1206

    Article  MATH  MathSciNet  Google Scholar 

  • Stolyar AL, Ramanan K (2001) Largest weighted delay first scheduling: large deviations and optimality. Ann Appl Prob 11:1–48

    Article  MATH  MathSciNet  Google Scholar 

  • Turner S (2000) Large deviations for join the shorter queue. In: McDonald D, Turner S (eds) Analysis of communication networks: call centres, traffic and performance. The Fields Institute, Toronto

    Google Scholar 

  • Wischik DJ (2001) Sample path large deviations for queues with many inputs. Ann Appl Prob 11:379–404

    Article  MATH  MathSciNet  Google Scholar 

  • Weiss A (1986) A new technique for analyzing large traffic systems. Adv Appl Prob 18:506–532

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vrije Universiteit, Amsterdam, The Netherlands

    Ad Ridder

  2. Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel

    Adam Shwartz

Authors
  1. Ad Ridder
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adam Shwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ad Ridder.

Additional information

Work of the first author was performed in part while visiting the Technion. Work of the second author was performed in part while visiting the Vrije Universiteit, Amsterdam, and was supported in part by Fund for the promotion of research at the Technion.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ridder, A., Shwartz, A. Large deviations without principle: join the shortest queue. Math Meth Oper Res 62, 467–483 (2005). https://doi.org/10.1007/s00186-005-0037-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-005-0037-1

Keywords

Search

Navigation