Stochastic Systems

Diffusion limits for shortest remaining processing time queues

H. Christian Gromoll, Łukasz Kruk, and Amber L. Puha

Full-text: Open access

Abstract

We present a heavy traffic analysis for a single server queue with renewal arrivals and generally distributed i.i.d. service times, in which the server employs the Shortest Remaining Processing Time (SRPT) policy. Under typical heavy traffic assumptions, we prove a diffusion limit theorem for a measure-valued state descriptor, from which we conclude a similar theorem for the queue length process. These results allow us to make some observations on the queue length optimality of SRPT. In particular, they provide the sharpest illustration of the well-known tension between queue length optimality and quality of service for this policy.

Article information

Source
Stoch. Syst., Volume 1, Number 1 (2011), 1-16.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1393252122

Digital Object Identifier
doi:10.1214/10-SSY016

Mathematical Reviews number (MathSciNet)
MR2948916

Zentralblatt MATH identifier
1291.60187

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F17: Functional limit theorems; invariance principles
Secondary: 60G57: Random measures 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Heavy traffic queueing shortest remaining processing time diffusion limit

Citation

Gromoll, H. Christian; Kruk, Łukasz; Puha, Amber L. Diffusion limits for shortest remaining processing time queues. Stoch. Syst. 1 (2011), no. 1, 1--16. doi:10.1214/10-SSY016. https://projecteuclid.org/euclid.ssy/1393252122


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References

  • [1] Bansal, N. and Harchol-Balter, M. (2001). Analysis of SRPT scheduling: investigating unfairness. In Proceedings of ACM Sigmetrics 2001 279–290. ACM Press, New York.
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons, Inc., New York.
  • [3] Down, D. and Wu, R. (2006). Multi-layered round robin routing for parallel servers. Queueing Systems 53 177–188.
  • [4] Down, D. G., Gromoll, H. C. and Puha, A. L. (2009). Fluid limits for shortest remaining processing time queues. Mathematics of Operations Research 34 880–911.
  • [5] Down, D. G., Gromoll, H. C. and Puha, A. L. (2009). State-dependent response times via fluid limits in shortest remaining processing time queues. SIGMETRICS Perform. Eval. Rev. 37 75–76.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley & Sons, Inc., New York.
  • [7] Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic I. Advances in Applied Probability 2 150–177.
  • [8] Núñez Queija, R. (2002). Queues with Equally Heavy Sojourn Time and Service Requirement Distributions. Annals of Operations Research 113 101–117.
  • [9] Nuyens, M. and Zwart, B. (2006). A large-deviations analysis of the GI/GI/1 SRPT queue. Queueing Systems 54 85–97.
  • [10] Pavlov, A. V. (1984). A system with Schrage servicing discipline in the case of a high load. Engrg. Cybernetics 21 114–121. Translated from Izv. Akad. Nauk SSSR Tekhn. Kibernet, 6:59–66, 1983 (Russian).
  • [11] Pechinkin, A. V. (1986). Heavy traffic in a system with a discipline of priority servicing for the job with the shortest remaining length with interruption (Russian). Math. Issled. No. 89, Veroyatn. Anal. 97 85–93.
  • [12] Perera, R. (1993). The variance of delay time in queueing system M/G/1 with optimal strategy SRPT. Archiv für Elektronik und Übertragungstechnik 47 110–114.
  • [13] Prohorov, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Theory of Probability and its Applications 1 157–214.
  • [14] Schassberger, R. (1990). The Steady-State Appearance of the M/G/1 Queue under the Discipline of Shortest Remaining Processing Time. Advances in Applied Probability 22 456–479.
  • [15] Schrage, L. E. (1968). A Proof of the Optimality of the Shortest Remaining Processing Time Discipline. Operations Research 16 687–690.
  • [16] Schrage, L. E. and Miller, L. W. (1966). The Queue M/G/1 with the Shortest Remaining Processing Time Discipline. Operations Research 14 670–684.
  • [17] Schreiber, F. (1993). Properties and applications of the optimal queueing strategy SRPT – a survey. Archiv für Elektronik und Übertragungstechnik 47 372–378.
  • [18] Smith, D. R. (1978). A New Proof of the Optimality of the Shortest Remaining Processing Time Discipline. Operations Research 26 197–199.
  • [19] Wierman, A. and Harchol-Balter, M. (2003). Classifying scheduling policies with respect to unfairness in an M/GI/1. In Proceedings of the 2003 ACM SIGMETRICS international conference on Measurement and modeling of computer systems. SIGMETRICS ’03 238–249. ACM, New York, NY, USA.
  • [20] Whitt, W. (1971). Weak Convergence Theorems for Priority Queues: Preemptive-Resume Discipline. Journal of Applied Probability 8 74–94.