IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i17p2134-d627749.html
   My bibliography  Save this article

Analysis of Multi-Server Queue with Self-Sustained Servers

Author

Listed:
  • Alexander Dudin

    (Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus
    Applied Mathematics and Communications Technology Institute, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia)

  • Olga Dudina

    (Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus)

  • Sergei Dudin

    (Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus)

  • Konstantin Samouylov

    (Applied Mathematics and Communications Technology Institute, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia)

Abstract

A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.

Suggested Citation

  • Alexander Dudin & Olga Dudina & Sergei Dudin & Konstantin Samouylov, 2021. "Analysis of Multi-Server Queue with Self-Sustained Servers," Mathematics, MDPI, vol. 9(17), pages 1-18, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2134-:d:627749
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/17/2134/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/17/2134/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hiroyuki Masuyama, 2019. "Correction to: A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 201-202, June.
    2. Che Kim & Moon Lee & Alexander Dudin & Valentina Klimenok, 2008. "Multi-server queueing systems with cooperation of the servers," Annals of Operations Research, Springer, vol. 162(1), pages 57-68, September.
    3. A. Krishnamoorthy & P. Pramod & S. Chakravarthy, 2014. "Queues with interruptions: a survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 290-320, April.
    4. Dudin, Alexander & Kim, Chesoong & Dudin, Sergey & Dudina, Olga, 2015. "Priority retrial queueing model operating in random environment with varying number and reservation of servers," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 674-690.
    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, April.
    6. Tetsuya Takine, 2016. "Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/ $$\infty $$ ∞ and BMAP/M/c+M queues," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 49-77, October.
    7. Marcel F. Neuts & David M. Lucantoni, 1979. "A Markovian Queue with N Servers Subject to Breakdowns and Repairs," Management Science, INFORMS, vol. 25(9), pages 849-861, September.
    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. Li, Hui & Yang, Tao, 2000. "Queues with a variable number of servers," European Journal of Operational Research, Elsevier, vol. 124(3), pages 615-628, August.
    10. Kim, Chesoong & Klimenok, Valentina I. & Orlovsky, Dmitry S., 2008. "The BMAP/PH/N retrial queue with Markovian flow of breakdowns," European Journal of Operational Research, Elsevier, vol. 189(3), pages 1057-1072, September.
    11. Erol Gelenbe, 2015. "Synchronising Energy Harvesting and Data Packets in a Wireless Sensor," Energies, MDPI, vol. 8(1), pages 1-14, January.
    12. Hiroyuki Masuyama, 2019. "A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 173-200, June.
    13. Chesoong Kim & Alexander Dudin & Olga Dudina & Valentina Klimenok, 2020. "Analysis of Queueing System with Non-Preemptive Time Limited Service and Impatient Customers," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 401-432, June.
    14. A. Krishnamoorthy & R. Manikandan & Dhanya Shajin, 2015. "Analysis of a Multiserver Queueing-Inventory System," Advances in Operations Research, Hindawi, vol. 2015, pages 1-16, January.
    15. Chao, Xiuli & Zhao, Yiqiang Q., 1998. "Analysis of multi-server queues with station and server vacations," European Journal of Operational Research, Elsevier, vol. 110(2), pages 392-406, October.
    16. Naishuo Tian & Zhe George Zhang, 2006. "Applications of Vacation Models," International Series in Operations Research & Management Science, in: Vacation Queueing Models Theory and Applications, chapter 0, pages 343-358, Springer.
    17. Kim, Chesoong & Klimenok, V.I. & Dudin, A.N., 2017. "Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 154-172.
    18. Alexander Dudin & Sergei Dudin, 2016. "Analysis of a Priority Queue with Phase-Type Service and Failures," International Journal of Stochastic Analysis, Hindawi, vol. 2016, pages 1-11, July.
    19. Linda Green, 1980. "A Queueing System in Which Customers Require a Random Number of Servers," Operations Research, INFORMS, vol. 28(6), pages 1335-1346, December.
    Full references (including those not matched with items on IDEAS)

    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. 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. Alexander Dudin & Sergei Dudin & Valentina Klimenok & Yuliya Gaidamaka, 2021. "Vacation Queueing Model for Performance Evaluation of Multiple Access Information Transmission Systems without Transmission Interruption," Mathematics, MDPI, vol. 9(13), pages 1-15, June.
    3. 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).
    4. 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.
    5. Kim, Chesoong & Klimenok, V.I. & Dudin, A.N., 2017. "Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 154-172.
    6. Hiroyuki Masuyama, 2022. "Subexponential asymptotics of asymptotically block-Toeplitz and upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 175-217, October.
    7. Alka Choudhary & Srinivas R. Chakravarthy & Dinesh C. Sharma, 2021. "Analysis of MAP / PH /1 Queueing System with Degrading Service Rate and Phase Type Vacation," Mathematics, MDPI, vol. 9(19), pages 1-17, September.
    8. 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.
    9. Yuying Zhang & Dequan Yue & Wuyi Yue, 2022. "A queueing-inventory system with random order size policy and server vacations," Annals of Operations Research, Springer, vol. 310(2), pages 595-620, March.
    10. 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.
    11. Srinivas R. Chakravarthy & Serife Ozkar, 2016. "Crowdsourcing and Stochastic Modeling," Business and Management Research, Business and Management Research, Sciedu Press, vol. 5(2), pages 19-30, June.
    12. 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.
    13. Shunfu Jin & Xiuchen Qie & Wenjuan Zhao & Wuyi Yue & Yutaka Takahashi, 2020. "A clustered virtual machine allocation strategy based on a sleep-mode with wake-up threshold in a cloud environment," Annals of Operations Research, Springer, vol. 293(1), pages 193-212, October.
    14. Amina Angelika Bouchentouf & Abdelhak Guendouzi, 2021. "Single Server Batch Arrival Bernoulli Feedback Queueing System with Waiting Server, K-Variant Vacations and Impatient Customers," SN Operations Research Forum, Springer, vol. 2(1), pages 1-23, March.
    15. F. P. Barbhuiya & U. C. Gupta, 2020. "A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations Under Late and Early Arrival System," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 599-624, June.
    16. Kumar, Anshul & Jain, Madhu, 2023. "Cost Optimization of an Unreliable server queue with two stage service process under hybrid vacation policy," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 259-281.
    17. Igor Kleiner & Esther Frostig & David Perry, 2023. "Busy Periods for Queues Alternating Between Two Modes," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-16, June.
    18. 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.
    19. Dieter Claeys & Stijn De Vuyst, 2019. "Discrete-time modified number- and time-limited vacation queues," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 297-318, April.
    20. Oz, Binyamin & Adan, Ivo & Haviv, Moshe, 2019. "The Mn/Gn/1 queue with vacations and exhaustive service," European Journal of Operational Research, Elsevier, vol. 277(3), pages 945-952.

    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:gam:jmathe:v:9:y:2021:i:17:p:2134-:d:627749. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.