IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i21p4095-d961673.html
   My bibliography  Save this article

Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model

Author

Listed:
  • Liudmila Zhilyakova

    (V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 65, Profsoyuznaya Street, 117997 Moscow, Russia
    These authors contributed equally to this work.)

  • Vasily Koreshkov

    (V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 65, Profsoyuznaya Street, 117997 Moscow, Russia
    These authors contributed equally to this work.)

  • Nadezhda Chaplinskaia

    (V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 65, Profsoyuznaya Street, 117997 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

The resource network is a non-linear threshold model where vertices exchange resource in infinite discrete time. The model is represented by a directed weighted graph. At each time step, all vertices send their resources along all output edges following one of two rules. For each vertex, the threshold value for changing the operation rule is equal to the total weight of its outgoing edges. If all vertices have resources less than their thresholds, the network is completely described by a homogeneous Markov chain. If at least one of the vertices has a resource above the threshold, the network is described by a non-homogeneous Markov chain. The purpose of this article is to describe and investigate non-homogeneous Markov chains generated by the resource network model. It is proven that they are strongly ergodic. In addition, stochastic matrices of a special form were studied. A number of new properties were revealed for them. The results obtained were generalized to arbitrary stochastic matrices.

Suggested Citation

  • Liudmila Zhilyakova & Vasily Koreshkov & Nadezhda Chaplinskaia, 2022. "Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model," Mathematics, MDPI, vol. 10(21), pages 1-18, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4095-:d:961673
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/21/4095/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/21/4095/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Wang, Xiaoyue & Ning, Ru & Zhao, Xian & Wu, Congshan, 2023. "Reliability assessments for two types of balanced systems with multi-state protective devices," Reliability Engineering and System Safety, Elsevier, vol. 229(C).
    2. Liudmila Zhilyakova, 2021. "Single-Threshold Model Resource Network and Its Double-Threshold Modifications," Mathematics, MDPI, vol. 9(12), pages 1-34, June.
    3. Dhar, Deepak, 1999. "The Abelian sandpile and related models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 263(1), pages 4-25.
    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. Xiaoyue Wang & Jingxuan Wang & Ru Ning & Xi Chen, 2023. "Joint Optimization of Maintenance and Spare Parts Inventory Strategies for Emergency Engineering Equipment Considering Demand Priorities," Mathematics, MDPI, vol. 11(17), pages 1-18, August.
    2. Wei, Xiaohua & Bai, Sijun & Wu, Bei, 2023. "A novel shock-dependent preventive maintenance policy for degraded systems subject to dynamic environments and N-critical shocks," Reliability Engineering and System Safety, Elsevier, vol. 239(C).
    3. Zhao, Xian & Qi, Xin & Wang, Xiaoyue, 2023. "Reliability assessment for coherent systems operating under a generalized mixed shock model with multiple change points of the environment," Reliability Engineering and System Safety, Elsevier, vol. 239(C).
    4. Liudmila Zhilyakova, 2021. "Single-Threshold Model Resource Network and Its Double-Threshold Modifications," Mathematics, MDPI, vol. 9(12), pages 1-34, June.
    5. Tadić, Bosiljka, 2001. "Dynamics of directed graphs: the world-wide Web," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 273-284.
    6. Zhao, Xian & Dong, Bingbing & Wang, Xiaoyue, 2023. "Reliability analysis of a two-dimensional voting system equipped with protective devices considering triggering failures," Reliability Engineering and System Safety, Elsevier, vol. 232(C).
    7. Yangyao, Shi & Xinchen, Zhuang & Tianxiang, Yu & Zijian, Zhang, 2023. "Multi-state balance system reliability research considering load influence," Reliability Engineering and System Safety, Elsevier, vol. 233(C).
    8. Zhao, Xian & Li, Ziyue & Wang, Xiaoyue & Guo, Bin, 2023. "Reliability of performance-based system containing multiple load-sharing subsystems with protective devices considering protection randomness," Reliability Engineering and System Safety, Elsevier, vol. 239(C).
    9. Wu, Congshan & Pan, Rong & Zhao, Xian & Wang, Xiaoyue, 2024. "Designing preventive maintenance for multi-state systems with performance sharing," Reliability Engineering and System Safety, Elsevier, vol. 241(C).
    10. Wang, Siqi & Zhao, Xian & Wu, Congshan & Wang, Xiaoyue, 2023. "Joint optimization of multi-stage component reassignment and preventive maintenance for balanced systems considering imperfect maintenance," Reliability Engineering and System Safety, Elsevier, vol. 237(C).
    11. Hay, M.J. & Schiff, J. & Fisch, N.J., 2017. "On extreme points of the diffusion polytope," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 225-236.

    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:10:y:2022:i:21:p:4095-:d:961673. 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.