IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v229y2013i2p391-403.html
   My bibliography  Save this article

Separable solutions for Markov processes in random environments

Author

Listed:
  • Balsamo, Simonetta
  • Marin, Andrea

Abstract

In this paper we address the problem of efficiently deriving the steady-state distribution for a continuous time Markov chain (CTMC) S evolving in a random environment E. The process underlying E is also a CTMC. S is called Markov modulated process. Markov modulated processes have been widely studied in literature since they are applicable when an environment influences the behaviour of a system. For instance, this is the case of a wireless link, whose quality may depend on the state of some random factors such as the intensity of the noise in the environment. In this paper we study the class of Markov modulated processes which exhibits separable, product-form stationary distribution. We show that several models that have been proposed in literature can be studied applying the Extended Reversed Compound Agent Theorem (ERCAT), and also new product-forms are derived. We also address the problem of the necessity of ERCAT for product-forms and show a meaningful example of product-form not derivable via ERCAT.

Suggested Citation

  • Balsamo, Simonetta & Marin, Andrea, 2013. "Separable solutions for Markov processes in random environments," European Journal of Operational Research, Elsevier, vol. 229(2), pages 391-403.
    • Handle: RePEc:eee:ejores:v:229:y:2013:i:2:p:391-403
    DOI: 10.1016/j.ejor.2013.03.011
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221713002191
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2013.03.011?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. Jing Jia & Sunderesh S. Heragu, 2009. "Solving Semi-Open Queuing Networks," Operations Research, INFORMS, vol. 57(2), pages 391-401, April.
    2. G.Sh. Tsitsiashvili & M.A. Osipova & N.V. Koliev & D. Baum, 2002. "A Product Theorem for Markov Chains with Application to PF-Queueing Networks," Annals of Operations Research, Springer, vol. 113(1), pages 141-154, July.
    3. Fukunari, Miki & Malmborg, Charles J., 2009. "A network queuing approach for evaluation of performance measures in autonomous vehicle storage and retrieval systems," European Journal of Operational Research, Elsevier, vol. 193(1), pages 152-167, February.
    4. Richard J. Boucherie & Nico M. van Dijk, 2000. "On a Queueing Network Model for Cellular Mobile Telecommunications Networks," Operations Research, INFORMS, vol. 48(1), pages 38-49, February.
    5. Villén-Altamirano, José, 2010. "Importance functions for restart simulation of general Jackson networks," European Journal of Operational Research, Elsevier, vol. 203(1), pages 156-165, May.
    6. George, David K. & Xia, Cathy H., 2011. "Fleet-sizing and service availability for a vehicle rental system via closed queueing networks," European Journal of Operational Research, Elsevier, vol. 211(1), pages 198-207, May.
    7. Gelenbe, Erol & Shachnai, Hadas, 2000. "On G-networks and resource allocation in multimedia systems," European Journal of Operational Research, Elsevier, vol. 126(2), pages 308-318, October.
    8. Erol Gelenbe & Catherine Rosenberg, 1990. "Queues with Slowly Varying Arrival and Service Processes," Management Science, INFORMS, vol. 36(8), pages 928-937, August.
    9. Gelenbe, Erol & Labed, Ali, 1998. "G-networks with multiple classes of signals and positive customers," European Journal of Operational Research, Elsevier, vol. 108(2), pages 293-305, July.
    10. Economou, A. & Fakinos, D., 1999. "Product form stationary distributions for the MGk group-arrival group-departure loss system under a general acceptance policy," European Journal of Operational Research, Elsevier, vol. 112(1), pages 196-206, January.
    11. James R. Jackson, 1963. "Jobshop-Like Queueing Systems," Management Science, INFORMS, vol. 10(1), pages 131-142, October.
    12. Osorio, Carolina & Bierlaire, Michel, 2009. "An analytic finite capacity queueing network model capturing the propagation of congestion and blocking," European Journal of Operational Research, Elsevier, vol. 196(3), pages 996-1007, August.
    13. Artalejo, J. R., 2000. "G-networks: A versatile approach for work removal in queueing networks," European Journal of Operational Research, Elsevier, vol. 126(2), pages 233-249, October.
    14. Gelenbe, Erol, 2000. "The first decade of G-networks," European Journal of Operational Research, Elsevier, vol. 126(2), pages 231-232, October.
    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. Simonetta Balsamo & Gian-Luca Dei Rossi & Andrea Marin, 2016. "Lumping and reversed processes in cooperating automata," Annals of Operations Research, Springer, vol. 239(2), pages 695-722, 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. Jean-Michel Fourneau & Erol Gelenbe, 2017. "G-Networks with Adders," Future Internet, MDPI, vol. 9(3), pages 1-7, July.
    2. Flötteröd, G. & Osorio, C., 2017. "Stochastic network link transmission model," Transportation Research Part B: Methodological, Elsevier, vol. 102(C), pages 180-209.
    3. Leon Yang Chu & Zhixi Wan & Dongyuan Zhan, 2018. "Harnessing the Double-edged Sword via Routing: Information Provision on Ride-hailing Platforms," Working Papers 18-04, NET Institute.
    4. Tappia, Elena & Roy, Debjit & Melacini, Marco & De Koster, René, 2019. "Integrated storage-order picking systems: Technology, performance models, and design insights," European Journal of Operational Research, Elsevier, vol. 274(3), pages 947-965.
    5. Wang, Jinting & Liu, Bin & Li, Jianghua, 2008. "Transient analysis of an M/G/1 retrial queue subject to disasters and server failures," European Journal of Operational Research, Elsevier, vol. 189(3), pages 1118-1132, September.
    6. Osorio, Carolina & Wang, Carter, 2017. "On the analytical approximation of joint aggregate queue-length distributions for traffic networks: A stationary finite capacity Markovian network approach," Transportation Research Part B: Methodological, Elsevier, vol. 95(C), pages 305-339.
    7. Bipan Zou & René De Koster & Xianhao Xu, 2018. "Operating Policies in Robotic Compact Storage and Retrieval Systems," Transportation Science, INFORMS, vol. 52(4), pages 788-811, August.
    8. Ivan Atencia & José Luis Galán-García, 2021. "Sojourn Times in a Queueing System with Breakdowns and General Retrial Times," Mathematics, MDPI, vol. 9(22), pages 1-25, November.
    9. Bi, Huibo & Shang, Wen-Long & Chen, Yanyan & Wang, Kezhi & Yu, Qing & Sui, Yi, 2021. "GIS aided sustainable urban road management with a unifying queueing and neural network model," Applied Energy, Elsevier, vol. 291(C).
    10. Pavel Bocharov & Ciro D’Apice & Alexandre Pechinkin, 2006. "Product form solution for exponential G-networks with dependent service and completion of service of killed customers," Computational Management Science, Springer, vol. 3(3), pages 177-192, July.
    11. Atencia, I., 2017. "A Geo/G/1 retrial queueing system with priority services," European Journal of Operational Research, Elsevier, vol. 256(1), pages 178-186.
    12. Atencia-Mc.Killop, Ivan & Galán-García, José L. & Aguilera-Venegas, Gabriel & Rodríguez-Cielos, Pedro & Galán-García, MÁngeles, 2018. "A Geo[X]/G[X]/1 retrial queueing system with removal work and total renewal discipline," Applied Mathematics and Computation, Elsevier, vol. 319(C), pages 245-253.
    13. J. P. van der Gaast & M. B. M. de Koster & I. J. B. F. Adan, 2018. "Conveyor Merges in Zone Picking Systems: A Tractable and Accurate Approximate Model," Service Science, INFORMS, vol. 52(6), pages 1428-1443, December.
    14. Zou, Bipan & Xu, Xianhao & Gong, Yeming (Yale) & De Koster, René, 2018. "Evaluating battery charging and swapping strategies in a robotic mobile fulfillment system," European Journal of Operational Research, Elsevier, vol. 267(2), pages 733-753.
    15. Kaveh Azadeh & René De Koster & Debjit Roy, 2019. "Robotized and Automated Warehouse Systems: Review and Recent Developments," Transportation Science, INFORMS, vol. 53(4), pages 917-945, July.
    16. Jelmer P. van der Gaast & René B. M. de Koster & Ivo J. B. F. Adan & Jacques A. C. Resing, 2020. "Capacity Analysis of Sequential Zone Picking Systems," Operations Research, INFORMS, vol. 68(1), pages 161-179, January.
    17. Bipan Zou & Xianhao Xu & Yeming Gong & René de Koster, 2016. "Modeling parallel movement of lifts and vehicles in tier-captive vehicle-based warehousing systems," Post-Print hal-01892897, HAL.
    18. Bipan Zou & Yeming (Yale) Gong & Xianhao Xu & Zhe Yuan, 2017. "Assignment rules in robotic mobile fulfilment systems for online retailers," International Journal of Production Research, Taylor & Francis Journals, vol. 55(20), pages 6175-6192, October.
    19. Zou, B. & de Koster, M.B.M. & Xu, X., 2016. "Evaluating dedicated and shared storage policies in robot-based compact storage and retrieval systems," ERIM Report Series Research in Management ERS-2016-010-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    20. Ule, Aljaz & Boucherie, Richard J., 2003. "On the distribution of calls in a wireless network driven by fluid traffic," European Journal of Operational Research, Elsevier, vol. 147(1), pages 146-155, May.

    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:ejores:v:229:y:2013:i:2:p:391-403. 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: http://www.elsevier.com/locate/eor .

    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.