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Ergodic and light traffic properties of a complex repairable system

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  • Igor Kovalenko

Abstract

A repairable system is composed of components ofI types. A component can be loaded, put on standby, queued or repaired. The repair facility is here assumed to be a queueing system of a rather general structure though interruption of repairs is not allowed. Typei components possess a lifetime distributionA i (t) and repair time distributionB t (t). The lifetime of componentj is exhausted with a state-dependent rate α j (t). A Markov process Z(t) with supplementary variables is built to investigate the system behaviour. An ergodic result, Theorem 1, is established under a set of conditions convenient for light traffic analysis. In Theorems 2 to 6, a light traffic limit is derived for the joint steady state distribution of supplementary variables. Applying these results, Theorems 7 to 10 derive light traffic properties of a busy period-measured random variable. Essentially, the concepts of light traffic equivalence due to Daley and Rolski (1992) and Asmussen (1992) are used. The asymptotic (light traffic) insensitivity of busy period and steady state parameters to the form ofA i (t) [given their means and (in some cases) values of density functions for smallt], is observed under some analytic conditions. Copyright Physica-Verlag 1997

Suggested Citation

  • Igor Kovalenko, 1997. "Ergodic and light traffic properties of a complex repairable system," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(3), pages 387-409, October.
    • Handle: RePEc:spr:mathme:v:45:y:1997:i:3:p:387-409
    DOI: 10.1007/BF01194787
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    References listed on IDEAS

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    1. Daley, D. J. & Rolski, T., 1994. "Light traffic approximations in general stationary single-server queues," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 141-158, January.
    2. Martin I. Reiman & Burton Simon, 1989. "Open Queueing Systems in Light Traffic," Mathematics of Operations Research, INFORMS, vol. 14(1), pages 26-59, February.
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