IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i14p1709-d598046.html

Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals

Author

Listed:
  • Freek Verdonck

    (SMACS Research Group, Department of Telecommunications and Information Processing (TELIN), Ghent University (UGent), Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium)

  • Herwig Bruneel

    (SMACS Research Group, Department of Telecommunications and Information Processing (TELIN), Ghent University (UGent), Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium)

  • Sabine Wittevrongel

    (SMACS Research Group, Department of Telecommunications and Information Processing (TELIN), Ghent University (UGent), Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium)

Abstract

In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay.

Suggested Citation

  • Freek Verdonck & Herwig Bruneel & Sabine Wittevrongel, 2021. "Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals," Mathematics, MDPI, vol. 9(14), pages 1-17, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1709-:d:598046
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/14/1709/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/14/1709/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michiel Muynck & Herwig Bruneel & Sabine Wittevrongel, 2020. "Analysis of a queue with general service demands and correlated service capacities," Annals of Operations Research, Springer, vol. 293(1), pages 73-99, October.
    2. Rouba Ibrahim & Ward Whitt, 2011. "Wait-Time Predictors for Customer Service Systems with Time-Varying Demand and Capacity," Operations Research, INFORMS, vol. 59(5), pages 1106-1118, October.
    3. Bruneel, Herwig & Steyaert, Bart & Desmet, Emmanuel & Petit, Guido H., 1994. "Analytic derivation of tail probabilities for queue lengths and waiting times in ATM multiserver queues," European Journal of Operational Research, Elsevier, vol. 76(3), pages 563-572, August.
    4. 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.
    5. Mohan L. Chaudhry & James J. Kim & Abhijit D. Banik, 2019. "Analytically Simple and Computationally Efficient Results for the GI X / Geo / c Queues," Journal of Probability and Statistics, Hindawi, vol. 2019, pages 1-18, September.
    6. Laevens, Koenraad & Bruneel, Herwig, 1995. "Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers," European Journal of Operational Research, Elsevier, vol. 85(1), pages 161-177, August.
    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. Freek Verdonck & Herwig Bruneel & Sabine Wittevrongel, 2022. "Delay analysis of a discrete-time single-server queue with an occasional extra server," Annals of Operations Research, Springer, vol. 310(2), pages 551-575, March.
    2. Michiel De Muynck & Herwig Bruneel & Sabine Wittevrongel, 2023. "Analysis of a Queue with General Service Demands and Multiple Servers with Variable Service Capacities," Mathematics, MDPI, vol. 11(4), pages 1-21, February.
    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. KM Rashmi & S. K. Samanta, 2024. "Analysis of Two-Way Communication in $$M_1^X, M_2/G_1,G_2/1$$ M 1 X , M 2 / G 1 , G 2 / 1 Retrial Queue Under the Constant Retrial Policy," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-32, December.
    5. Divya Velayudhan Nair & Achyutha Krishnamoorthy & Agassi Melikov & Sevinj Aliyeva, 2021. "MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback," Mathematics, MDPI, vol. 9(15), pages 1-26, July.
    6. Zerui Wu & Ran Liu & Xu Sun, 2026. "Near-Optimal Pricing and Resource Allocation in a Large-Scale Service System," Operations Research, INFORMS, vol. 74(2), pages 632-650, March.
    7. S. Pradhan & U.C. Gupta & S.K. Samanta, 2016. "Queue-length distribution of a batch service queue with random capacity and batch size dependent service: M / G r Y / 1 $M/{G^{Y}_{r}}/1$," OPSEARCH, Springer;Operational Research Society of India, vol. 53(2), pages 329-343, June.
    8. 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.
    9. Siddharth Prakash Singh & Mohammad Delasay & Alan Scheller‐Wolf, 2023. "Real‐time delay announcement under competition," Production and Operations Management, Production and Operations Management Society, vol. 32(3), pages 863-881, March.
    10. Yuting Yuan, 2026. "Case Article—Chick-fil-A Drive-Through: Managing Congestion with Discrete Event Simulation," INFORMS Transactions on Education, INFORMS, vol. 26(3), pages 200-206, May.
    11. Bart Steyaert & Sabine Wittevrongel & Herwig Bruneel, 2017. "Characterisation of the output process of a discrete-time GI / D / 1 queue, and its application to network performance," Annals of Operations Research, Springer, vol. 252(1), pages 175-190, May.
    12. 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.
    13. Yina Lu & Andrés Musalem & Marcelo Olivares & Ariel Schilkrut, 2013. "Measuring the Effect of Queues on Customer Purchases," Management Science, INFORMS, vol. 59(8), pages 1743-1763, August.
    14. Claeys, Dieter & Walraevens, Joris & Laevens, Koenraad & Bruneel, Herwig, 2010. "A queueing model for general group screening policies and dynamic item arrivals," European Journal of Operational Research, Elsevier, vol. 207(2), pages 827-835, December.
    15. Nooshin Salari & Sheng Liu & Zuo-Jun Max Shen, 2022. "Real-Time Delivery Time Forecasting and Promising in Online Retailing: When Will Your Package Arrive?," Manufacturing & Service Operations Management, INFORMS, vol. 24(3), pages 1421-1436, May.
    16. Mitchell M Kracman, 2025. "A stochastic jump model applied to collaborative queue-based high energy laser defense," The Journal of Defense Modeling and Simulation, , vol. 22(2), pages 187-195, April.
    17. Pedram Sahba & Bariş Balciog̃lu & Dragan Banjevic, 2013. "Analysis of the finite‐source multiclass priority queue with an unreliable server and setup time," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(4), pages 331-342, June.
    18. Ramendra Singh Chauhan & N. Selvaraju, 2025. "Strategic behavior and social optimization in Markovian working vacation queues with delayed observations," Operational Research, Springer, vol. 25(2), pages 1-64, June.
    19. Najiya Fatma & Varun Ramamohan, 2023. "Patient diversion using real-time delay predictions across healthcare facility networks," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 45(2), pages 437-476, June.
    20. Miaomiao Yu & Yinghui Tang, 2022. "Analysis of a renewal batch arrival queue with a fault-tolerant server using shift operator method," Operational Research, Springer, vol. 22(3), pages 2831-2858, July.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:14:p:1709-:d:598046. 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 The email address of this maintainer does not seem to be valid anymore. Please ask MDPI Indexing Manager to update the entry or send us the correct address (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.