Skip to main content

Stability analysis of parallel server systems under longest queue first

Abstract

We consider the stability of parallel server systems under the longest queue first (LQF) rule. We show that when the underlying graph of a parallel server system is a tree, the standard nominal traffic condition is sufficient for the stability of that system under LQF when interarrival and service times have general distributions. Then we consider a special parallel server system, which is known as the X-model, whose underlying graph is not a tree. We provide additional “drift” conditions for the stability and transience of these queueing systems with exponential interarrival and service times. Drift conditions depend in general on the stationary distribution of an induced Markov chain that is derived from the underlying queueing system. We illustrate our results with examples and simulation experiments. We also demonstrate that the stability of the LQF depends on the tie-breaking rule used and that it can be unstable even under arbitrary low loads.

This is a preview of subscription content, access via your institution.

References

  • Bell SL, Williams RJ (2001) Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy. Ann Appl Probab 11: 608–649

    MathSciNet  MATH  Article  Google Scholar 

  • Bell SL, Williams RJ (2005) Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: asymptotic optimality of a threshold policy. Electron J Probab 10: 1044–1115

    MathSciNet  Google Scholar 

  • Bramson M (2008) Stability of queueing networks. Probab Surv 5: 165–345

    MathSciNet  Google Scholar 

  • Chen H, Yao DD (2001) Fundamentals of queueing networks : performance, asymptotics, and optimization. Springer, New York

    MATH  Google Scholar 

  • Coffman EG, Puhalskii AA, Reiman MI (1995) Polling systems with zero switchover times: a heavy-traffic averaging principle. Ann Appl Probab 5: 681–719

    MathSciNet  MATH  Article  Google Scholar 

  • Dai JG (1995) On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann Appl Probab 5: 49–77

    MathSciNet  MATH  Article  Google Scholar 

  • Dai JG (1996) A fluid-limit model criterion for instability of multiclass queueing networks. Ann Appl Probab 6: 751–757

    MathSciNet  MATH  Article  Google Scholar 

  • Dai JG (1999) Stability of fluid and stochastic processing networks. MaPhySto Miscellanea Publication, No. 9

  • Dai JG, Meyn SP (1995) Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans Automat Contr 40: 1889–1904

    MathSciNet  MATH  Article  Google Scholar 

  • Dimakis A, Walrand J (2006) Sufficient conditions for stability of longest-queue-first scheduling: second-order properties using fluid limits. Adv Appl Probab 38: 505–521

    MathSciNet  MATH  Article  Google Scholar 

  • Fayolle G, Malyshev VA, Menshikov MV (1995) Topics in the constructive theory of countable Markov chains. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Foss S, Kovalevskii A (1999) A stability criterion via fluid limits and its application to a polling system. Queueing Syst Theory Appl 32: 131–168

    MathSciNet  MATH  Article  Google Scholar 

  • Hunt PJ, Kurtz TG (1994) Large loss networks. Stoch Process Appl 53:363–378, 10

    Google Scholar 

  • Kumar S, Giaccone P, Leonardi E (2002) Rate stability of stable marriage scheduling algorithms in input-queued switches. In: 40th annual allerton conference on computers, communication, and control. University of Illinois, Urbana-Champaign

  • Meyn S (1995) Transience of multiclass queueing networks via fluid limit models. Ann Appl Probab 5: 946–957

    MathSciNet  MATH  Article  Google Scholar 

  • Perry O, Whitt W (2010) A fluid limit for an overloaded X-model via an averaging principle. Technical report, Columbia University

  • Stolyar AL, Ramakrishnan KK (1999) The stability of a flow merge point with non-interleaving cut-through scheduling disciplines. In: INFOCOM ’99, vol 3, pp 1231–1238

  • Stolyar AL, Yudovina E (2010) Systems with large flexible server pools: instability of “natural” load balancing. Technical report, Bell Labs

  • Tezcan T (2010) Stabillity of N-model systems under static priority policies. Technical report, University of Rochester

  • Tezcan T, Dai JG (2010) Dynamic control of N-systems with many servers: asymptotic optimality of a static priority policy in heavy traffic. Oper Res 58: 94–110

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tolga Tezcan.

Additional information

Research supported by NSF Grant CMMI-0954126.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baharian, G., Tezcan, T. Stability analysis of parallel server systems under longest queue first. Math Meth Oper Res 74, 257–279 (2011). https://doi.org/10.1007/s00186-011-0362-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-011-0362-5

Keywords

  • Stability
  • Longest queue first
  • Parallel server systems
  • Fluid model