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Explicit Solutions for Coupled Parallel Queues

Author

Listed:
  • Herwig Bruneel

    (SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, 9000 Ghent, Belgium)

  • Arnaud Devos

    (SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, 9000 Ghent, Belgium)

Abstract

We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete . In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, but the numbers of arrivals into both queues within a slot may be mutually dependent. Their joint probability generating function ( pgf ) is indicated as A ( z 1 , z 2 ) and characterizes the whole model. In general, determining the steady-state joint probability mass function ( pmf ) u ( m , n ) , m , n ≥ 0 or the corresponding joint pgf U ( z 1 , z 2 ) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A ( z 1 , z 2 ) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A ( z 1 , z 2 ) , for which we can explicitly determine the system-content pgf U ( z 1 , z 2 ) . We find that, for arrival pgfs of this class, U ( z 1 , z 2 ) has a denominator that is a product, say r 1 ( z 1 ) r 2 ( z 2 ) , of two univariate functions. This property allows a straightforward inversion of U ( z 1 , z 2 ) , resulting in a pmf u ( m , n ) which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases.

Suggested Citation

  • Herwig Bruneel & Arnaud Devos, 2024. "Explicit Solutions for Coupled Parallel Queues," Mathematics, MDPI, vol. 12(15), pages 1-31, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2345-:d:1443972
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    References listed on IDEAS

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    1. J. W. Cohen, 1998. "Analysis of the asymmetrical shortest two-server queueing model," International Journal of Stochastic Analysis, Hindawi, vol. 11, pages 1-48, January.
    2. Leonard Kleinrock & Hanoch Levy, 1988. "The Analysis of Random Polling Systems," Operations Research, INFORMS, vol. 36(5), pages 716-732, October.
    3. Herwig Bruneel, 2022. "Some thoughts on the analysis of coupled queues," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 185-187, April.
    4. Joris Walraevens & Thomas Giel & Stijn Vuyst & Sabine Wittevrongel, 2022. "Asymptotics of waiting time distributions in the accumulating priority queue," Queueing Systems: Theory and Applications, Springer, vol. 101(3), pages 221-244, August.
    5. J. P. C. Blanc, 1992. "The Power-Series Algorithm Applied to the Shortest-Queue Model," Operations Research, INFORMS, vol. 40(1), pages 157-167, February.
    6. Joris Walraevens & Bart Steyaert & Herwig Bruneel, 2006. "A preemptive repeat priority queue with resampling: Performance analysis," Annals of Operations Research, Springer, vol. 146(1), pages 189-202, September.
    7. Vladimir Vishnevsky & Olga Semenova, 2021. "Polling Systems and Their Application to Telecommunication Networks," Mathematics, MDPI, vol. 9(2), pages 1-30, January.
    8. Walraevens, Joris & Steyaert, Bart & Bruneel, Herwig, 2008. "Analysis of a discrete-time preemptive resume priority buffer," European Journal of Operational Research, Elsevier, vol. 186(1), pages 182-201, April.
    9. Tom Maertens & Joris Walraevens & Herwig Bruneel, 2008. "Performance comparison of several priority schemes with priority jumps," Annals of Operations Research, Springer, vol. 162(1), pages 109-125, September.
    10. Ivo J. B. F. Adan & Onno J. Boxma & Stella Kapodistria & Vidyadhar G. Kulkarni, 2016. "The shorter queue polling model," Annals of Operations Research, Springer, vol. 241(1), pages 167-200, June.
    11. Ahmad Hanbali & Roland Haan & Richard Boucherie & Jan-Kees Ommeren, 2012. "Time-limited polling systems with batch arrivals and phase-type service times," Annals of Operations Research, Springer, vol. 198(1), pages 57-82, September.
    12. Sofian De Clercq & Joris Walraevens, 2020. "Delay analysis of a two-class priority queue with external arrivals and correlated arrivals from another node," Annals of Operations Research, Springer, vol. 293(1), pages 57-72, October.
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