Skip to main content
SpringerLink
Log in
Book cover

Advances in Queueing Theory and Network Applications pp 213–226Cite as

Explicit Probability Density Function for the Length of a Busy Period in an M/M/1/K Queue

  • Chapter

Abstract

A new closed-form explicit expression is derived for the probability den sity function of the length of a busy period starting with i customers in an M/M/1/K queue, where K is the capacity of the system. The density function is given as a weighted sum of K negative exponential distributions with coefficients calculated from K distinct zeros of a polynomial that involves Chebyshev polynomials of the second kind. The mean and second moment of the busy period are also shown ex plicitly. In addition, the symmetric results for the first passage time from state i to state K are presented. We also consider the regeneration cycle of state i.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

References

  1. H. G. Perros and T. Altiok, Queueing Network With Blocking. Amsterdam: Elsevier Science, 1989.

    Google Scholar 

  2. L. Takñcs, Introduction to the Theory of Queues. London: Oxford University Press, 1962.

    Google Scholar 

  3. A. I. Ismailov, The distribution of the busy period in a certain queueing model, Doklady Akademii Nauk UzSSrR, no. 5, pp. 3–4, 1970 (in Russian).

    MathSciNet  Google Scholar 

  4. H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory. New York: Academic Press, 1982.

    MATH  Google Scholar 

  5. O. P. Sharma and B. Shobha, On the busy period of an M/M/1/N queue, Journal of Combinatorics, Information and System Sciences, vol. 11, nos. 2–4, pp. 110–114, 1986.

    MATH  MathSciNet  Google Scholar 

  6. O. P. Sharma, Markovian Queues. Chichester, England: Ellis Horwood Limited, 1990.

    MATH  Google Scholar 

  7. W. Stadje, The busy periods of some queueing systems, Stochastic Processes and Their Ap plications, vol. 55, no. 1, pp. 159–167, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  8. K. K. J. Kinateder and E.-Y. Lee, A new approach to the busy period of the M/M/1 queue, Queueing Systems: Theory and Applications, vol. 35, issues 1–4, pp. 105–115, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. L. Saaty, Elements of Queueing Theory with Applications. New York: McGraw-Hill, 1961. Republished by New York: Dover, 1983.

    MATH  Google Scholar 

  10. W. Ledermann and G. E. H. Reuter, Spectral theory for the differential equations of simple birth and death processes, Philosophical Transactions of the Royal Society of London, vol. 246, pp. 321–369, 1954.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki, 305–8573, Japan

    Hideaki Takagi

  2. Damietta Faculty of Science, Mathematics Department, New Damietta, Egypt

    Ahmed M. K. Tarabia

Authors
  1. Hideaki Takagi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed M. K. Tarabia
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Intelligence and Informatics, Konan University, Kobe, 658-8501, Japan

    Wuyi Yue

  2. Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Yataka Takahashi

  3. Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki, 305-8573, Japan

    Hideaki Takagi

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Takagi, H., Tarabia, A.M.K. (2009). Explicit Probability Density Function for the Length of a Busy Period in an M/M/1/K Queue. In: Yue, W., Takahashi, Y., Takagi, H. (eds) Advances in Queueing Theory and Network Applications. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09703-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-0-387-09703-9_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-09702-2

  • Online ISBN: 978-0-387-09703-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation