IDEAS home Printed from https://ideas.repec.org/a/wly/navres/v37y1990i5p753-772.html
   My bibliography  Save this article

Diffusion approximation to the G/G/R machine repair problem with warm standby spares

Author

Listed:
  • B. D. Sivazlian
  • K. H. Wang

Abstract

The G/G/R machine repair problem with M operating machines, S warm standby spares, and R repairmen is studied as a diffusion process. The steady‐state equations are formulated as diffusion equations subject to two reflecting barriers. The approximate diffusion parameters of the diffusion equations are obtained (1) under the assumption that the input characteristics of the problem are defined only by their first two moments rather than their probability distribution function, (2) under the assumption of heavy traffic approximation, that is, when queues of failed machines in the repair stage are almost always nonempty, and (3) using well‐known asymptotic results from renewal theory. Expressions for the probability density functions of the number of failed machines in the system are obtained. A study of the derived approximate results, compared to some of the exact results, suggests that the diffusion approach provides a useful method for solving complex machine‐repair problems.

Suggested Citation

  • B. D. Sivazlian & K. H. Wang, 1990. "Diffusion approximation to the G/G/R machine repair problem with warm standby spares," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(5), pages 753-772, October.
    • Handle: RePEc:wly:navres:v:37:y:1990:i:5:p:753-772
    DOI: 10.1002/1520-6750(199010)37:53.0.CO;2-B
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/1520-6750(199010)37:53.0.CO;2-B
    Download Restriction: no
    File URL: https://libkey.io/10.1002/1520-6750(199010)37:53.0.CO;2-B?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
    ---><---

    References listed on IDEAS

    as
    1. Maritas, D. G. & Xirokostas, D. A., 1977. "The M/Ek/r machine interference model steady state equations and numerical solutions," European Journal of Operational Research, Elsevier, vol. 1(2), pages 112-123, March.
    2. Martin I. Reiman, 1984. "Open Queueing Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 441-458, August.
    3. Haryono, & Sivazlian, B.D., 1985. "Analysis of the machine repair problem: a diffusion process approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 27(4), pages 339-364.
    4. Donald P. Gaver & Gerald S. Shedler, 1973. "Processor Utilization in Multiprogramming Systems via Diffusion Approximations," Operations Research, INFORMS, vol. 21(2), pages 569-576, April.
    5. Bunday, B. D. & Scraton, R. E., 1980. "The G/M/r machine interference model," European Journal of Operational Research, Elsevier, vol. 4(6), pages 399-402, June.
    6. Shlomo Halfin & Ward Whitt, 1981. "Heavy-Traffic Limits for Queues with Many Exponential Servers," Operations Research, INFORMS, vol. 29(3), pages 567-588, June.
    7. S. Christian Albright, 1980. "Optimal maintenance‐repair policies for the machine repair problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 27(1), pages 17-27, March.
    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. Srinivas R. Chakravarthy & Atul Agarwal, 2003. "Analysis of a machine repair problem with an unreliable server and phase type repairs and services," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(5), pages 462-480, August.

    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. Josh Reed & Yair Shaki, 2015. "A Fair Policy for the G / GI / N Queue with Multiple Server Pools," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 558-595, March.
    2. Mor Armony & Constantinos Maglaras, 2004. "On Customer Contact Centers with a Call-Back Option: Customer Decisions, Routing Rules, and System Design," Operations Research, INFORMS, vol. 52(2), pages 271-292, April.
    3. Shaler Stidham, 2002. "Analysis, Design, and Control of Queueing Systems," Operations Research, INFORMS, vol. 50(1), pages 197-216, February.
    4. Avishai Mandelbaum & Petar Momčilović, 2017. "Personalized queues: the customer view, via a fluid model of serving least-patient first," Queueing Systems: Theory and Applications, Springer, vol. 87(1), pages 23-53, October.
    5. Amy R. Ward & Mor Armony, 2013. "Blind Fair Routing in Large-Scale Service Systems with Heterogeneous Customers and Servers," Operations Research, INFORMS, vol. 61(1), pages 228-243, February.
    6. Achal Bassamboo & J. Michael Harrison & Assaf Zeevi, 2006. "Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method," Operations Research, INFORMS, vol. 54(3), pages 419-435, June.
    7. Saulius Minkevičius & Igor Katin & Joana Katina & Irina Vinogradova-Zinkevič, 2021. "On Little’s Formula in Multiphase Queues," Mathematics, MDPI, vol. 9(18), pages 1-15, September.
    8. Niyirora, Jerome & Zhuang, Jun, 2017. "Fluid approximations and control of queues in emergency departments," European Journal of Operational Research, Elsevier, vol. 261(3), pages 1110-1124.
    9. Achal Bassamboo & Assaf Zeevi, 2009. "On a Data-Driven Method for Staffing Large Call Centers," Operations Research, INFORMS, vol. 57(3), pages 714-726, June.
    10. Haque, Lani & Armstrong, Michael J., 2007. "A survey of the machine interference problem," European Journal of Operational Research, Elsevier, vol. 179(2), pages 469-482, June.
    11. Ari Arapostathis & Hassan Hmedi & Guodong Pang, 2021. "On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 772-796, May.
    12. van Leeuwaarden, Johan S.H. & Knessl, Charles, 2011. "Transient behavior of the Halfin-Whitt diffusion," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1524-1545, July.
    13. Ramandeep S. Randhawa & Sunil Kumar, 2008. "Usage Restriction and Subscription Services: Operational Benefits with Rational Users," Manufacturing & Service Operations Management, INFORMS, vol. 10(3), pages 429-447, December.
    14. Martino Banchio & Giacomo Mantegazza, 2022. "Artificial Intelligence and Spontaneous Collusion," Papers 2202.05946, arXiv.org, revised Sep 2023.
    15. Zhang, Ping & Serban, Nicoleta, 2007. "Discovery, visualization and performance analysis of enterprise workflow," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2670-2687, February.
    16. Hsiao-Hui Lee & Edieal J. Pinker & Robert A. Shumsky, 2012. "Outsourcing a Two-Level Service Process," Management Science, INFORMS, vol. 58(8), pages 1569-1584, August.
    17. O. Garnet & A. Mandelbaum & M. Reiman, 2002. "Designing a Call Center with Impatient Customers," Manufacturing & Service Operations Management, INFORMS, vol. 4(3), pages 208-227, October.
    18. A. J. E. M. Janssen & J. S. H. van Leeuwaarden & Bert Zwart, 2011. "Refining Square-Root Safety Staffing by Expanding Erlang C," Operations Research, INFORMS, vol. 59(6), pages 1512-1522, December.
    19. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    20. 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.

    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:wly:navres:v:37:y:1990:i:5:p:753-772. 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: Wiley Content Delivery (email available below). General contact details of provider: https://doi.org/10.1002/(ISSN)1520-6750 .

    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.