Skip to main content
SpringerLink
Log in

Transient Analysis of a Multi-server Queuing Model with Discouraged Arrivals and Retention of Reneging Customers

  • Conference paper
  • First Online:
Book cover Analytical and Computational Methods in Probability Theory (ACMPT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10684))

Abstract

In this paper, we study a finite capacity Markovian multi-server queuing system with discouraged arrivals, reneging, and retention of reneging customers. The transient state probabilities of the queuing system are obtained by using a computational technique based on the 4th order Runge- Kutta method. With the help of the transient state probabilities, we develop some important measures of performance of the system, such as time-dependent expected system size, time-dependent expected reneging rate, and time-dependent expected retention rate. The transient behavior of the system size probabilities and the expected system size is also studied. Further, the variations in the expected system size, the expected reneging rate, and the expected retention rate with respect to the probability of retaining a reneging customer are also studied. Finally, the effect of discouraged arrivals in the same model is analyzed.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • 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

References

  • Abdul Rasheed, K.V., Manoharan, M.: Markovian queueing system with discouraged arrivals and self-regulatory servers. Adv. Oper. Res. 2016, 11 (2016). https://doi.org/10.1155/2016/2456135. Article ID 2456135

    MathSciNet  Google Scholar 

  • Abou-El-Ata, M.O., Hariri, A.M.A.: The M/M/c/N queue with balking and reneging. Comput. Oper. Res. 19(8), 713–716 (1992)

    Article  MATH  Google Scholar 

  • Al-Seedy, R.O., El-Sherbiny, A.A., El-Shehawy, S.A., Ammar, S.I.: Transient solution of the M/M/c queue with balking and reneging. Comput. Math. Appl. 57(8), 1280–1285 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Ammar, S.I., El-Sherbiny, A.A., Al-Seedy, R.O.: A matrix approach for the transient solution of an M/M/1/N queue with discouraged arrivals and reneging. Int. J. Comput. Math. 89, 482–491 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Ancker Jr., C.J., Gafarian, A.V.: Some queuing problems with balking and reneging I. Oper. Res. 11, 88–100 (1963a)

    Google Scholar 

  • Ancker Jr., C.J., Gafarian, A.V.: Some queuing problems with balking and reneging II. Oper. Res. 11, 928–937 (1963b)

    Google Scholar 

  • Haight, F.A.: Queuing with reneging. Metrika 2, 186–197 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, R., Sharma, S.K.: An M/M/1/N queuing system with retention of reneged customers. Pak. J. Stat. Oper. Res. 8, 859–866 (2012a)

    Google Scholar 

  • Kumar, R., Sharma, S.K.: A multi-server Markovian queueing system with discouraged arrivals and retention of reneged customers. Int. J. Oper. Res. 9(4), 173–184 (2012b)

    Google Scholar 

  • Kumar, R.: Economic analysis of an M/M/c/N queuing model with balking, reneging and retention of reneged customers. Opsearch 50, 383–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Morse, P.M.: Queues, Inventories and Maintenance. Wiley, New York (1958)

    Google Scholar 

  • Natvig, B.: On the transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Probab. 11, 345–354 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Parthasarathy, P.R., Selvaraju, N.: Transient analysis of a queue where potential customers are discouraged by queue length. Math. Prob. Eng. 7(5), 433–454 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Sharma, O.P., Maheswar, M.V.R.: Transient behavior of a simple queue with discouraged arrivals. Optimization 27, 283–291 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Van Doorn, E.A.: The transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Probab. 18(2), 499–506 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, K.H., Chang, Y.C.: Cost analysis of a finite M/M/R queueing system with balking, reneging and server breakdowns. Math. Methods Oper. Res. 56, 169–180 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, Y., Yue, D., Yue, W.: Optimal performance analysis of an M/M/1/N queue system with balking, reneging and server vacation. Int. J. Pure Appl. Math. 28, 101–115 (2006)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

One of the authors Dr. Rakesh Kumar would like to thank the UGC, New Delhi, India, for financial support given to him for this research work under the Major Research Project vide Letter No. F.-43-434/2014(SR).

Author information

Authors and Affiliations

  1. Department of Mathematics, Shri Mata Vaishno Devi University, Katra, 182320, Jammu and Kashmir, India

    Rakesh Kumar & Sapana Sharma

Authors
  1. Rakesh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sapana Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rakesh Kumar .

Editor information

Editors and Affiliations

  1. Department of Applied Probability and Informatics, RUDN University, Moscow, Russia

    Vladimir V. Rykov

  2. City University of Hong Kong, Hong Kong, Hong Kong

    Nozer D. Singpurwalla

  3. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Andrey M. Zubkov

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kumar, R., Sharma, S. (2017). Transient Analysis of a Multi-server Queuing Model with Discouraged Arrivals and Retention of Reneging Customers. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-71504-9_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71503-2

  • Online ISBN: 978-3-319-71504-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation