IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i14p2214-d1435734.html
   My bibliography  Save this article

Stability of Queueing Systems with Impatience, Balking and Non-Persistence of Customers

Author

Listed:
  • Alexander N. Dudin

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

  • Sergey A. Dudin

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

  • Valentina I. Klimenok

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

  • Olga S. Dudina

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

Abstract

The operation of many queueing systems is adequately described by the structured multidimensional continuous-time Markov chains. The most well-studied classes of such chains are level-independent Quasi-Birth-and-Death processes, G I / M / 1 type and M / G / 1 type Markov chains, generators of which have the block tri-diagonal, lower- and upper-Hessenberg structure, respectively. All these classes assume that the matrices of transition rates are quasi-Toeplitz. This property greatly simplifies their analysis but makes them inappropriate for the study of many important systems, e.g., retrial queues with a retrial rate depending on the number of customers in orbit, queues with impatient customers, etc. The importance of such systems attracts significant interest to their analysis. However, in the literature, there is a methodological gap relating to the ergodicity condition of the corresponding Markov chains. To fulfill this gap and facilitate the analysis of a wide range of such systems, we show that under non-restrictive assumptions, the following hold true: (i) if the customers can balk or are impatient or non-persistent, then the Markov chain describing the behavior of the system belongs to the class of asymptotically quasi-Toeplitz Markov chains; (ii) this chain is ergodic; (iii) known algorithms can be applied for the calculation of the stationary distribution of the corresponding queueing system.

Suggested Citation

  • Alexander N. Dudin & Sergey A. Dudin & Valentina I. Klimenok & Olga S. Dudina, 2024. "Stability of Queueing Systems with Impatience, Balking and Non-Persistence of Customers," Mathematics, MDPI, vol. 12(14), pages 1-16, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:14:p:2214-:d:1435734
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/14/2214/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/14/2214/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Che Kim & Vilena Mushko & Alexander Dudin, 2012. "Computation of the steady state distribution for multi-server retrial queues with phase type service process," Annals of Operations Research, Springer, vol. 201(1), pages 307-323, December.
    2. Jeongsim Kim & Bara Kim, 2016. "A survey of retrial queueing systems," Annals of Operations Research, Springer, vol. 247(1), pages 3-36, December.
    3. S. Subba Rao, 1967. "Queuing with balking and reneging in M|G|1 systems," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 12(1), pages 173-188, December.
    4. A. Gómez-Corral, 2006. "A bibliographical guide to the analysis of retrial queues through matrix analytic techniques," Annals of Operations Research, Springer, vol. 141(1), pages 163-191, January.
    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. Alexander Dudin & Achyutha Krishnamoorthy & Sergei Dudin & Olga Dudina, 2024. "Queueing system with control by admission of retrial requests depending on the number of busy servers and state of the underlying process of Markov arrival process of primary requests," Annals of Operations Research, Springer, vol. 335(1), pages 135-150, April.
    2. Chesoong Kim & Valentina Klimenok & Alexander Dudin, 2014. "A G/M/1 retrial queue with constant retrial rate," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 509-529, July.
    3. Ioannis Dimitriou, 2016. "A queueing model with two classes of retrial customers and paired services," Annals of Operations Research, Springer, vol. 238(1), pages 123-143, March.
    4. Se Won Lee & Bara Kim & Jeongsim Kim, 2022. "Analysis of the waiting time distribution in M/G/1 retrial queues with two way communication," Annals of Operations Research, Springer, vol. 310(2), pages 505-518, March.
    5. 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.
    6. Samira Taleb & Amar Aissani, 2016. "Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers," Annals of Operations Research, Springer, vol. 247(1), pages 291-317, December.
    7. K. R. Rejitha & K. P. Jose, 2018. "A stochastic inventory system with two modes of service and retrial of customers," OPSEARCH, Springer;Operational Research Society of India, vol. 55(1), pages 134-149, March.
    8. Anatoly Nazarov & János Sztrik & Anna Kvach & Ádám Tóth, 2020. "Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs," Annals of Operations Research, Springer, vol. 288(1), pages 417-434, May.
    9. Bin Liu & Jie Min & Yiqiang Q. Zhao, 2023. "Refined tail asymptotic properties for the $$M^X/G/1$$ M X / G / 1 retrial queue," Queueing Systems: Theory and Applications, Springer, vol. 104(1), pages 65-105, June.
    10. Raina Raj & Vidyottama Jain, 2023. "Resource optimization in $$\textit{MMAP[2]/PH[2]/S}$$ MMAP [ 2 ] / PH [ 2 ] / S priority queueing model with threshold $$\textit{PH}$$ PH retrial times and the preemptive resume policy," Annals of Operations Research, Springer, vol. 331(2), pages 1119-1148, December.
    11. Lamia Lakaour & Djamil Aïssani & Karima Adel-Aissanou & Kamel Barkaoui, 2019. "M/M/1 Retrial Queue with Collisions and Transmission Errors," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1395-1406, December.
    12. Sergei A. Dudin & Olga S. Dudina & Olga I. Kostyukova, 2023. "Analysis of a Queuing System with Possibility of Waiting Customers Jockeying between Two Groups of Servers," Mathematics, MDPI, vol. 11(6), pages 1-21, March.
    13. Yang, Dong-Yuh & Tsao, Chih-Lung, 2019. "Reliability and availability analysis of standby systems with working vacations and retrial of failed components," Reliability Engineering and System Safety, Elsevier, vol. 182(C), pages 46-55.
    14. Anatoly Nazarov & János Sztrik & Anna Kvach & Ádám Tóth, 2022. "Asymptotic Analysis of Finite-Source M/GI/1 Retrial Queueing Systems with Collisions and Server Subject to Breakdowns and Repairs," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1503-1518, September.
    15. Radhika Agarwal & Divya Agarwal & Shweta Upadhyaya & Izhar Ahmad, 2023. "Optimization of a stochastic model having erratic server with immediate or delayed repair," Annals of Operations Research, Springer, vol. 331(2), pages 605-628, December.
    16. Yoshiaki Inoue & Onno Boxma & David Perry & Shelley Zacks, 2018. "Analysis of $$\hbox {M}^{\mathrm {x}}/\hbox {G}/1$$ M x / G / 1 queues with impatient customers," Queueing Systems: Theory and Applications, Springer, vol. 89(3), pages 303-350, August.
    17. B. Krishna Kumar & R. Sankar & R. Navaneetha Krishnan & R. Rukmani, 2022. "Performance Analysis of Multi-processor Two-Stage Tandem Call Center Retrial Queues with Non-Reliable Processors," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 95-142, March.
    18. Wang, Kuo-Hsiung & Wu, Chia-Huang & Yen, Tseng-Chang, 2022. "Comparative cost-benefit analysis of four retrial systems with preventive maintenance and unreliable service station," Reliability Engineering and System Safety, Elsevier, vol. 221(C).
    19. V. Saravanan & V. Poongothai & P. Godhandaraman, 2023. "Admission control policy of a two heterogeneous server finite capacity retrial queueing system with maintenance activity," OPSEARCH, Springer;Operational Research Society of India, vol. 60(4), pages 1902-1925, December.
    20. 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.

    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:12:y:2024:i:14:p:2214-:d:1435734. 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.