IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v423y2022ics0096300322000959.html
   My bibliography  Save this article

Upper bound on the rate of convergence and truncation bound for non-homogeneous birth and death processes on Z

Author

Listed:
  • Satin, Y.A.
  • Razumchik, R.V.
  • Zeifman, A.I.
  • Kovalev, I.A.

Abstract

We consider the well-known problem of the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time birth and death processes on Z with the time–varying and possibly state-dependent intensities. First in the literature upper bounds on the rate of convergence are provided. Upper bounds for the truncation errors are also given. The condition under which a limiting (time-dependent) distribution exists is formulated but relies on the quantities that need to be guessed in each use-case. The developed theory is illustrated by two numerical examples within the queueing theory context.

Suggested Citation

  • Satin, Y.A. & Razumchik, R.V. & Zeifman, A.I. & Kovalev, I.A., 2022. "Upper bound on the rate of convergence and truncation bound for non-homogeneous birth and death processes on Z," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    • Handle: RePEc:eee:apmaco:v:423:y:2022:i:c:s0096300322000959
    DOI: 10.1016/j.amc.2022.127009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322000959
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2022.127009?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. Zeifman, A.I., 1995. "Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 157-173, September.
    2. Zhen Wang & Liwei Liu & Yuanfu Shao & Xudong Chai & Baoxian Chang, 2020. "Equilibrium Joining Strategy in a Batch Transfer Queuing System with Gated Policy," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 75-99, March.
    3. Giorno, Virginia & Nobile, Amelia G., 2020. "On a class of birth-death processes with time-varying intensity functions," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    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. Yacov Satin & Rostislav Razumchik & Ivan Kovalev & Alexander Zeifman, 2023. "Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience," Mathematics, MDPI, vol. 11(9), pages 1-15, April.

    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. André de Palma & Claude Lefèvre, 2018. "Bottleneck models and departure time problems," Working Papers hal-01581519, HAL.
    2. Zeifman, A.I. & Korolev, V.Yu., 2014. "On perturbation bounds for continuous-time Markov chains," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 66-72.
    3. Yacov Satin & Rostislav Razumchik & Ivan Kovalev & Alexander Zeifman, 2023. "Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    4. Giorno, Virginia & Nobile, Amelia G., 2023. "On a time-inhomogeneous diffusion process with discontinuous drift," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    5. Korolev, V.Yu. & Chertok, A.V. & Korchagin, A.Yu. & Zeifman, A.I., 2015. "Modeling high-frequency order flow imbalance by functional limit theorems for two-sided risk processes," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 224-241.
    6. P. Vijaya Laxmi & E. Girija Bhavani, 2024. "Strategic behavior of customers in a second optional service queue with service interruptions," OPSEARCH, Springer;Operational Research Society of India, vol. 61(2), pages 762-784, June.
    7. Zeifman, A.I. & Korolev, V.Yu. & Satin, Ya.A. & Kiseleva, K.M., 2018. "Lower bounds for the rate of convergence for continuous-time inhomogeneous Markov chains with a finite state space," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 84-90.
    8. Yacov Satin & Alexander Zeifman & Alexander Sipin & Sherif I. Ammar & Janos Sztrik, 2020. "On Probability Characteristics for a Class of Queueing Models with Impatient Customers," Mathematics, MDPI, vol. 8(4), pages 1-15, April.
    9. Zeifman, A.I. & Korolev, V. Yu., 2015. "Two-sided bounds on the rate of convergence for continuous-time finite inhomogeneous Markov chains," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 30-36.
    10. Giorno, Virginia & Nobile, Amelia G., 2022. "On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    11. Zeifman, A. & Satin, Y. & Kiseleva, K. & Korolev, V. & Panfilova, T., 2019. "On limiting characteristics for a non-stationary two-processor heterogeneous system," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 48-65.
    12. Alexander Zeifman & Yacov Satin & Ksenia Kiseleva & Victor Korolev, 2019. "On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process," Mathematics, MDPI, vol. 7(5), pages 1-10, May.
    13. Erik Doorn, 2011. "Rate of convergence to stationarity of the system M/M/N/N+R," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 336-350, December.
    14. Ayane Nakamura & Tuan Phung-Duc, 2023. "Equilibrium Analysis for Batch Service Queueing Systems with Strategic Choice of Batch Size," Mathematics, MDPI, vol. 11(18), pages 1-22, September.
    15. Guodong Pang & Andrey Sarantsev & Yuri Suhov, 2022. "Birth and death processes in interactive random environments," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 269-307, October.
    16. Alexander Zeifman & Victor Korolev & Yacov Satin, 2020. "Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains," Mathematics, MDPI, vol. 8(2), pages 1-25, February.
    17. Granovsky, Boris L. & Zeifman, Alexander I., 1997. "The decay function of nonhomogeneous birth-death processes, with application to mean-field models," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 105-120, December.

    More about this item

    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:eee:apmaco:v:423:y:2022:i:c:s0096300322000959. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.