IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v87y2018i1d10.1007_s00186-017-0606-0.html
   My bibliography  Save this article

Analysis of an M/G/1 queue with vacations and multiple phases of operation

Author

Listed:
  • Jianjun Li

    (Nanjing University of Science and Technology)

  • Liwei Liu

    (Nanjing University of Science and Technology)

  • Tao Jiang

    (Nanjing University of Science and Technology)

Abstract

This paper deals with an M / G / 1 queue with vacations and multiple phases of operation. If there are no customers in the system at the instant of a service completion, a vacation commences, that is, the system moves to vacation phase 0. If none is found waiting at the end of a vacation, the server goes for another vacation. Otherwise, the system jumps from phase 0 to some operative phase i with probability $$q_i$$ q i , $$i = 1,2, \ldots ,n.$$ i = 1 , 2 , … , n . In operative phase i, $$i = 1,2, \ldots ,n$$ i = 1 , 2 , … , n , the server serves customers according to the discipline of FCFS (First-come, first-served). Using the method of supplementary variables, we obtain the stationary system size distribution at arbitrary epoch. The stationary sojourn time distribution of an arbitrary customer is also derived. In addition, the stochastic decomposition property is investigated. Finally, we present some numerical results.

Suggested Citation

  • Jianjun Li & Liwei Liu & Tao Jiang, 2018. "Analysis of an M/G/1 queue with vacations and multiple phases of operation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(1), pages 51-72, February.
    • Handle: RePEc:spr:mathme:v:87:y:2018:i:1:d:10.1007_s00186-017-0606-0
    DOI: 10.1007/s00186-017-0606-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-017-0606-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00186-017-0606-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Baykal-Gürsoy, M. & Xiao, W. & Ozbay, K., 2009. "Modeling traffic flow interrupted by incidents," European Journal of Operational Research, Elsevier, vol. 195(1), pages 127-138, May.
    2. S. W. Fuhrmann & Robert B. Cooper, 1985. "Stochastic Decompositions in the M / G /1 Queue with Generalized Vacations," Operations Research, INFORMS, vol. 33(5), pages 1117-1129, October.
    3. J. George Shanthikumar, 1988. "On Stochastic Decomposition in M / G /1 Type Queues with Generalized Server Vacations," Operations Research, INFORMS, vol. 36(4), pages 566-569, August.
    4. U. Yechiali & P. Naor, 1971. "Queuing Problems with Heterogeneous Arrivals and Service," Operations Research, INFORMS, vol. 19(3), pages 722-734, June.
    5. Naishuo Tian & Zhe George Zhang, 2006. "Vacation Queueing Models Theory and Applications," International Series in Operations Research and Management Science, Springer, number 978-0-387-33723-4, December.
    6. Noam Paz & Uri Yechiali, 2014. "An M/M/1 Queue In Random Environment With Disasters," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 31(03), pages 1-12.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Madhu Jain & Sandeep Kaur & Parminder Singh, 2021. "Supplementary variable technique (SVT) for non-Markovian single server queue with service interruption (QSI)," Operational Research, Springer, vol. 21(4), pages 2203-2246, December.
    2. Sherif I. Ammar & Yousef F. Alharbi & Yiqiang Q. Zhao, 2023. "Analysis of Vacation Fluid M / M /1 Queue in Multi-Phase Random Environment," Mathematics, MDPI, vol. 11(21), pages 1-14, October.
    3. Amir Ahmadi-Javid & Pooya Hoseinpour, 2022. "Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2621-2633, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. B. Krishna Kumar & R. Rukmani & A. Thanikachalam & V. Kanakasabapathi, 2018. "Performance analysis of retrial queue with server subject to two types of breakdowns and repairs," Operational Research, Springer, vol. 18(2), pages 521-559, July.
    2. Sem Borst & Onno Boxma, 2018. "Polling: past, present, and perspective," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 335-369, October.
    3. Priyanka Kalita & Gautam Choudhury & Dharmaraja Selvamuthu, 2020. "Analysis of Single Server Queue with Modified Vacation Policy," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 511-553, June.
    4. Madhu Jain & Sandeep Kaur & Parminder Singh, 2021. "Supplementary variable technique (SVT) for non-Markovian single server queue with service interruption (QSI)," Operational Research, Springer, vol. 21(4), pages 2203-2246, December.
    5. Yi Peng & Jinbiao Wu, 2020. "A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations," Mathematics, MDPI, vol. 8(8), pages 1-12, July.
    6. Zsolt Saffer & Sergey Andreev & Yevgeni Koucheryavy, 2016. "$$M/D^{[y]}/1$$ M / D [ y ] / 1 Periodically gated vacation model and its application to IEEE 802.16 network," Annals of Operations Research, Springer, vol. 239(2), pages 497-520, April.
    7. Atencia, I., 2017. "A Geo/G/1 retrial queueing system with priority services," European Journal of Operational Research, Elsevier, vol. 256(1), pages 178-186.
    8. Chakravarthy, Srinivas R. & Shruti, & Kulshrestha, Rakhee, 2020. "A queueing model with server breakdowns, repairs, vacations, and backup server," Operations Research Perspectives, Elsevier, vol. 7(C).
    9. Tuan Phung-Duc, 2017. "Exact solutions for M/M/c/Setup queues," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 64(2), pages 309-324, February.
    10. Ioannis Dimitriou, 2013. "A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 542-571, October.
    11. Onno Boxma & Dieter Claeys & Lennart Gulikers & Offer Kella, 2015. "A queueing system with vacations after N services," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(8), pages 646-658, December.
    12. Renbin Liu & Attahiru Sule Alfa & Miaomiao Yu, 2019. "Analysis of an ND-policy Geo/G/1 queue and its application to wireless sensor networks," Operational Research, Springer, vol. 19(2), pages 449-477, June.
    13. I. Atencia & G. Bouza & P. Moreno, 2008. "An M [X] /G/1 retrial queue with server breakdowns and constant rate of repeated attempts," Annals of Operations Research, Springer, vol. 157(1), pages 225-243, January.
    14. Qiu, Qinjing & Kawai, Reiichiro, 2022. "A decoupling principle for Markov-modulated chains," Statistics & Probability Letters, Elsevier, vol. 182(C).
    15. Dimitris Bertsimas & José Niño-Mora, 1996. "Optimization of multiclass queueing networks with changeover times via the achievable region method: Part II, the multi-station case," Economics Working Papers 314, Department of Economics and Business, Universitat Pompeu Fabra, revised Aug 1998.
    16. Izak Duenyas & Diwakar Gupta & Tava Lennon Olsen, 1998. "Control of a Single-Server Tandem Queueing System with Setups," Operations Research, INFORMS, vol. 46(2), pages 218-230, April.
    17. B. Krishna Kumar & S. Pavai Madheswari & S. Anantha Lakshmi, 2011. "Queuing system with state-dependent controlled batch arrivals and server under maintenance," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 351-379, December.
    18. Dimitris Bertsimas & José Niño-Mora, 1999. "Optimization of Multiclass Queueing Networks with Changeover Times Via the Achievable Region Approach: Part I, The Single-Station Case," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 306-330, May.
    19. Manickam Vadivukarasi & Kaliappan Kalidass, 2021. "Discussion on the transient behavior of single server Markovian multiple variant vacation queues," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 31(1), pages 123-146.
    20. George C. Mytalas & Michael A. Zazanis, 2022. "Service with a queue and a random capacity cart: random processing batches and E-limited policies," Annals of Operations Research, Springer, vol. 317(1), pages 147-178, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:mathme:v:87:y:2018:i:1:d:10.1007_s00186-017-0606-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.