A Product-Form Solution for a Two-Class $$Geo^{Geo}/D/1$$ Queue with Random Routing and Randomly Alternating Service | SpringerLink
Skip to main content
Springer Nature Link
Log in

A Product-Form Solution for a Two-Class \(Geo^{Geo}/D/1\) Queue with Random Routing and Randomly Alternating Service

  • Conference paper
  • First Online:
Performance Evaluation Methodologies and Tools (VALUETOOLS 2022)

  • 256 Accesses

Abstract

We analyze a discrete-time queueing system, consisting of two queues and a single server. The server randomly distributes its time between the two queues. Service times of any customer of either queue are deterministically equal to 1 time slot. In general, the joint analysis of such a two-queue system turns out to be very hard. In this paper, we assume that the total number of arrivals into the system constitutes a series of i.i.d. random variables with common geometric distribution. Each arriving customer is routed probabilistically to a queue. By means of a state-of-the-art approach, we obtain a closed-form expression of the steady-state joint PGF of the number of customers present (“system contents”) in both queues, at the beginning of a random slot. We find that the joint PGF is of product form, which proves that the system contents in both queues are independent. We provide an additional intuitive stochastic explanation for this remarkable result. We discuss several model extensions using the stochastic analysis.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 8579
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 10724
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Bruneel, H.: A general model for the behaviour of infinite buffers with periodic service opportunities. Eur. J. Oper. Res. 16(1), 98–106 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruneel, H., Kim, B.G.: Discrete-Time Models for Communication Systems Including ATM. Kluwer Academic Publisher, Boston (1993)

    Book  Google Scholar 

  3. Bruneel, H., Wittevrongel, S., Claeys, D., Walraevens, J.: Discrete-time queues with variable service capacity: a basic model and its analysis. Ann. Oper. Res. 239(2), 359–380 (2016). https://doi.org/10.1007/s10479-013-1428-y

    Article  MathSciNet  MATH  Google Scholar 

  4. Al Hanbali, A., de Haan, R., Boucherie, R.J., van Ommeren, J.: Time-limited polling systems with batch arrivals and phase-type service times. Ann. Oper. Res. 198(1), 57–82 (2012). https://doi.org/10.1007/s10479-011-0846-y

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen, J.W., Boxma, O.J.: Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  6. Cohen, J.W.: Boundary value problems in queueing theory. Queueing Syst. 3(2), 97–128 (1988). https://doi.org/10.1007/BF01189045

    Article  MathSciNet  MATH  Google Scholar 

  7. De Clercq, S., Laevens, K., Steyaert, B., Bruneel, H.: A multi-class discrete-time queueing system under the FCFS service discipline. Ann. Oper. Res. 202(1), 59–73 (2013). https://doi.org/10.1007/s10479-011-1051-8

    Article  MathSciNet  MATH  Google Scholar 

  8. de Haan, R., Boucherie, R.J., van Ommeren, J.: A polling model with an autonomous server. Queueing Syst. 62(3), 279–308 (2009). https://doi.org/10.1007/s11134-009-9131-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Devos, A., Walraevens, J., Fiems, D., Bruneel, H.: Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline. Ann. Oper. Res. 310(2), 477–503 (2022). https://doi.org/10.1007/s10479-020-03776-5

    Article  MathSciNet  MATH  Google Scholar 

  10. Devos, A., Walraevens, J., Fiems, D., Bruneel, H.: Analysis of a discrete-time two-class randomly alternating service model with Bernoulli arrivals. Queueing Syst. 96, 133–152 (2020). https://doi.org/10.1007/s11134-020-09663-x

    Article  MathSciNet  MATH  Google Scholar 

  11. Dvir, N., Hassin, R., Yechiali, U.: Strategic behaviour in a tandem queue with alternating server. Queueing Syst. 96, 205–244 (2020). https://doi.org/10.1007/s11134-020-09665-9

    Article  MathSciNet  MATH  Google Scholar 

  12. Fayolle, G., Iasnogorodski, R.: Two coupled processors: the reduction to a Riemann-Hilbert problem. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 47(3), 325–351 (1979). https://doi.org/10.1007/BF00535168

    Article  MathSciNet  MATH  Google Scholar 

  13. Fayolle, G., Malyshev, V.A., Iasnogorodski, R.: Random Walks in the Quarter-Plane. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-642-60001-2

    Book  MATH  Google Scholar 

  14. Halfin, S.: Batch delays versus customer delays. Bell Syst. Tech. J. 62(7), 2011–2015 (1983)

    Article  Google Scholar 

  15. Konheim, A.G., Meilijson, I., Melkman, A.: Processor-sharing of two parallel lines. J. Appl. Probab. 18(4), 952–956 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Saxena, M., Boxma, O., Kapodistria, S., Queija, R.: Two queues with random time-limited polling. Probab. Math. Stat. 37(2), 257–289 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Telecommunications and Information Processing (EA07), Ghent University - UGent, Sint-Pietersnieuwstraat 41, 9000, Gent, Belgium

    Arnaud Devos, Michiel De Muynck, Herwig Bruneel & Joris Walraevens

Authors
  1. Arnaud Devos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michiel De Muynck
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Herwig Bruneel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joris Walraevens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnaud Devos .

Editor information

Editors and Affiliations

  1. University of Iceland, Reykjavik, Iceland

    Esa Hyytiä

  2. Indian Institute of Technology Bombay, Mumbai, India

    Veeraruna Kavitha

Rights and permissions

Reprints and permissions

Copyright information

© 2023 ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Devos, A., De Muynck, M., Bruneel, H., Walraevens, J. (2023). A Product-Form Solution for a Two-Class \(Geo^{Geo}/D/1\) Queue with Random Routing and Randomly Alternating Service. In: Hyytiä, E., Kavitha, V. (eds) Performance Evaluation Methodologies and Tools. VALUETOOLS 2022. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 482. Springer, Cham. https://doi.org/10.1007/978-3-031-31234-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-31234-2_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-31233-5

  • Online ISBN: 978-3-031-31234-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation