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

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  • 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
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    References listed on IDEAS

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    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.
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    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.
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    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.

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