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Ergodic and light traffic properties of a complex repairable system

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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.

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Abbreviations

LHS, RHS:

lefthand side, righthand side

w.r.t.:

with respect to

(i.i.d.) r.v.:

(independent identically distributed) random variables

A c :

complement event

d.f., p.d.f.:

distribution function, probability density function

m.g.f.:

moment generating function

I A :

indicator function of eventA

Ā(t):

=1−A(t)

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  1. Storm Research Centre, University of North London, 166-220 Holloway Road, N7 8DB, London, UK

    Igor N. Kovalenko

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Kovalenko, I.N. Ergodic and light traffic properties of a complex repairable system. Mathematical Methods of Operations Research 45, 387–409 (1997). https://doi.org/10.1007/BF01194787

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