Skip to main content
SpringerLink
Log in

The non-preemptive ‘Join the Shortest Queue–Serve the Longest Queue’ service system with or without switch-over times

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

Abstract

A 2-queue system with a single-server operating according to the combined ‘Join the Shortest Queue–Serve the Longest Queue’ regime is analyzed. Both cases, with or without server’s switch-over times, are investigated under the non-preemptive discipline. Instead of dealing with a state space comprised of two un-bounded dimensions, a non-conventional formulation is constructed, leading to an alternative two-dimensional state space, where only one dimension is infinite. As a result, the system is defined as a quasi birth and death process and is analyzed via both the probability generating functions method and the matrix geometric formulation. Consequently, the system’s two-dimensional probability mass function is derived, from which the system’s performance measures, such as mean queue sizes, mean sojourn times, fraction of time the server resides in each queue, correlation coefficient between the queue sizes, and the probability mass function of the difference between the queue sizes, are obtained. Extensive numerical results for various values of the system’s parameters are presented, as well as a comparison between the current non-preemptive model and its twin system of preemptive service regime. One of the conclusions is that, depending on the variability of the various parameters, the preemptive regime is not necessarily more efficient than the non-preemptive one. Finally, economic issues are discussed and numerical comparisons are presented, showing the advantages and disadvantages of each regime.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Adan IJ, Wessels J, Zijm W (1991a) Analysis of the asymmetric shortest queue problem. Queueing Syst 8(1):1–58

    Article  MathSciNet  Google Scholar 

  • Adan IJ, Wessels J, Zijm W (1991b) Analysis of the asymmetric shortest queue problem with threshold jockeying. Commun Stat Stoch Models 7(4):615–627

    Article  MathSciNet  Google Scholar 

  • Adan IJ, Boxma OJ, Kapodistria S, Kulkarni VG (2016) The shorter queue polling model. Ann Oper Res 241(1–2):167–200

    Article  MathSciNet  Google Scholar 

  • Armony M, Perel E, Perel N, Yechiali U (2019) Exact analysis for multiserver queueing systems with cross selling. Ann Oper Res 274(1):75–100

    Article  MathSciNet  Google Scholar 

  • Artalejo JR, Gómez-Corral A (2008) Retrial queueing systems: a computational approach. Springer, Berlin

    Book  Google Scholar 

  • Avrachenkov K, Nain P, Yechiali U (2014) A retrial system with two input streams and two orbit queues. Queueing Syst 77(1):1–31

    Article  MathSciNet  Google Scholar 

  • Boon MA, Van der Mei R, Winands EM (2011) Applications of polling systems. Surv Oper Res Manag Sci 16(2):67–82

    Google Scholar 

  • Braverman A (2020) Steady-state analysis of the join-the-shortest-queue model in the Halfin–Whitt regime. Math Oper Res 45(3):1069–1103

    Article  MathSciNet  Google Scholar 

  • Bright L, Taylor PG (1995) Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stoch Model 11(3):497–525

    MathSciNet  Google Scholar 

  • Browne S, Yechiali U (1989) Dynamic priority rules for cyclic-type queues. Adv Appl Probab 21(2):432–450

    Article  MathSciNet  Google Scholar 

  • Cohen JW (1987) A two-queue, one-server model with priority for the longer queue. Queueing Syst 2(3):261–283

    Article  MathSciNet  Google Scholar 

  • Cohen JW (1998) Analysis of the asymmetrical shortest two-server queueing model. Int J Stoch Anal 11(2):115–162

    MathSciNet  Google Scholar 

  • Conway R, Maxwell W, Miller L (2003) Theory of scheduling. Dover books on computer science series. Dover, New York

    Google Scholar 

  • Curtiss D (1918) Recent extentions of descartes’ rule of signs. Ann Math 19:251–278

    Article  MathSciNet  Google Scholar 

  • Dawson DA, Tang J, Zhao YQ et al (2019) Performance analysis of joining the shortest queue model among a large number of queues. Asia-Pac J Oper Res 36(04):1–23

    Article  MathSciNet  Google Scholar 

  • Dimitriou I (2021) Analysis of the symmetric join the shortest orbit queue. Oper Res Lett 49(1):23–29

    Article  MathSciNet  Google Scholar 

  • Eschenfeldt P, Gamarnik D (2018) Join the shortest queue with many servers. the heavy-traffic asymptotics. Math Oper Res 43(3):867–886

    Article  MathSciNet  Google Scholar 

  • Flatto L (1989) The longer queue model. Probab Eng Inf Sci 3(4):537–559

    Article  Google Scholar 

  • Halfin S (1985) The shortest queue problem. J Appl Probab 22(4):865–878

    Article  MathSciNet  Google Scholar 

  • Hanukov G, Yechiali U (2021) Explicit solutions for continuous-time QBD processes by using relations between matrix geometric analysis and the probability generating functions method. Probab Eng Inf Sci 35:565–580

    Article  MathSciNet  Google Scholar 

  • Harchol-Balter M (2013) Performance modeling and design of computer systems: queueing theory in action. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hordijk A, Koole G (1990) On the optimality of the generalized shortest queue policy. Probab Eng Inf Sci 4(4):477–487

    Article  Google Scholar 

  • Jolles A, Perel E, Yechiali U (2018) Alternating server with non-zero switch-over times and opposite-queue threshold-based switching policy. Perform Eval 126:22–38

    Article  Google Scholar 

  • Kella O, Yechiali U (1988) Priorities in m/g/1 queue with server vacations. Naval Res Logist 35(1):23–34

    Article  MathSciNet  Google Scholar 

  • Knessl C, Yao H (2013) On the nonsymmetric longer queue model: joint distribution, asymptotic properties, and heavy traffic limits. Adv Oper Res. https://doi.org/10.1155/2013/680539

    Article  MathSciNet  Google Scholar 

  • Latouche G, Ramaswami V (1999) Introduction to matrix analytic methods in stochastic modeling. SIAM, Philadelphia

    Book  Google Scholar 

  • Maguluri ST, Hajek B, Srikant R (2014) The stability of longest-queue-first scheduling with variable packet sizes. IEEE Trans Autom Control 59(8):2295–2300

    Article  MathSciNet  Google Scholar 

  • Neuts MF (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. The Johns Hopkins University Press, Baltimore

    Google Scholar 

  • Paz N, Yechiali U (2014) An M/M/1 queue in random environment with disasters. Asia-Pac J Oper Res 31(03):1450016

    Article  MathSciNet  Google Scholar 

  • Pedarsani R, Walrand J (2016) Stability of multiclass queueing networks under longest-queue and longest-dominating-queue scheduling. J Appl Probab 53(2):421–433

    Article  MathSciNet  Google Scholar 

  • Perel E, Yechiali U (2013a) On customers acting as servers. Asia-Pac J Oper Res 30(05):1350019

    Article  MathSciNet  Google Scholar 

  • Perel N, Yechiali U (2013b) The Israeli queue with priorities. Stoch Model 29(3):353–379

    Article  MathSciNet  Google Scholar 

  • Perel E, Yechiali U (2017) Two-queue polling systems with switching policy based on the queue that is not being served. Stoch Model 33(3):430–450

    Article  MathSciNet  Google Scholar 

  • Perel E, Perel N, Yechiali U (2020) A polling system with “join the shortest-serve the longest’’ policy. Comput Oper Res 114:104809

    Article  MathSciNet  Google Scholar 

  • Perel E, Perel N, Yechiali U (2022) A 3-queue polling system with join the shortest-serve the longest policy. Indag Math 34:1101–1120

    Article  MathSciNet  Google Scholar 

  • Phung-Duc T (2017) Exact solutions for m/m/c/setup queues. Telecommun Syst 64(2):309–324

    Article  Google Scholar 

  • Takagi H (1986) Analysis of polling systems. MIT Press, Cambridge

    Google Scholar 

  • van Houtum G-J, Adan IJ, Der wal J (1997) The symmetric longest queue system. Stoch Model 13(1):105–120

    MathSciNet  Google Scholar 

  • van Houtum G-J, Adan IJ, Wessels J, Zijm WH (2001) Performance analysis of parallel identical machines with a generalized shortest queue arrival mechanism. OR-Spektrum 23(3):411–427

    Article  MathSciNet  Google Scholar 

  • Winston W (1977) Optimality of the shortest line discipline. J Appl Probab 14(1):181–189

    Article  MathSciNet  Google Scholar 

  • Yao H, Knessl C (2005) On the infinite server shortest queue problem: symmetric case. Stoch Model 21(1):101–132

    Article  MathSciNet  Google Scholar 

  • Yao H, Knessl C (2006) On the infinite server shortest queue problem: non-symmetric case. Queueing Syst 52(2):157–177

    Article  MathSciNet  Google Scholar 

  • Yechiali U (1993) Analysis and control of polling systems. In: Donatiello L, Nelson R (eds) Performance evaluation of computer and communication systems. Springer, Berlin, pp 630–650

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Industrial Engineering and Management, Afeka Tel Aviv Academic College of Engineering, Tel Aviv, Israel

    Efrat Perel & Nir Perel

  2. Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Uri Yechiali

Authors
  1. Efrat Perel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nir Perel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Uri Yechiali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nir Perel.

Ethics declarations

Data availability statements

We do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

Conflict of interest

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Perel, E., Perel, N. & Yechiali, U. The non-preemptive ‘Join the Shortest Queue–Serve the Longest Queue’ service system with or without switch-over times. Math Meth Oper Res (2024). https://doi.org/10.1007/s00186-023-00848-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00186-023-00848-7

Keywords

Search

Navigation