Topic
Renewal theory
About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.
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TL;DR: In this article, a model of a safety inspection process is proposed for the expected consequence of inspections over a finite time horizon, and a pragmatic procedure for formulating objective functions which may be optimized to determine the optimal inspection intervals.
Abstract: In this paper a model of a safety inspection process is proposed for the expected consequence of inspections over a finite time horizon. A single dominant failure mode is modelled, which has considerable safety or risk consequences assumed measurable either in cost terms or in terms of the probability of failure over the time horizon. The model established extends earlier work assuming an infinite time horizon, and uses the concept of delay time and asymptotic results from the theory of renewal and renewal reward processes. The paper establishes a pragmatic procedure for formulating objective functions which may be optimized to determine the optimal inspection intervals. Merits of both the exact and asymptotic formulations of these objective functions for possible use in the inspection optimization process are considered. Although the procedure for developing an objective function over a finite time zone inspection process assumes perfect inspection, it can be generalized to the imperfect inspection case. Because of the intractability of the mathematics, it is suggested that when optimizing an inspection process over a finite time zone, an asymptotic formulation of the objective function should be optimized, and this solution then checked and if necessary refined, using simulation calculation. A numerical example illustrates the performance of the basic periodic inspection policy over different time horizons using the asymptotic solution. The results are compared with simulations performed to estimate the exact expected cost measure. It is shown that the simpler asymptotic solution is satisfactory in the case considered, especially when the time horizon is relatively long. © 1997 John Wiley & Sons, Ltd.
30 citations
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TL;DR: In this paper, a superposition of renewal process is used to model the effect of imperfect maintenance, a different modelling approach than the more common use of a virtual age process.
30 citations
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TL;DR: In this article, the authors extended the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes and applied the key superposition renewal theorem to the study of renewal superpositions.
Abstract: This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of
30 citations
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TL;DR: This paper provides a compound variation of the Newsboy problem, instead of demand simply being known as to its distribution, here demand is generated by customers arriving at different points in time requiring amounts of varying size.
30 citations
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TL;DR: In this article, a non-trivial extension to a stochastic instantaneous interest rate is presented for Erlang claims number processes, and for the Ho-Lee-Merton and Vasicek interest rate models.
Abstract: Formulas have been obtained for the moments of the discounted aggregate claims process, for a constant instantaneous interest rate, and for a claims number process that is an ordinary or a delayed renewal process. In this paper, we present explicit formulas on the first two moments and the joint moment of this risk process, for a non-trivial extension to a stochastic instantaneous interest rate. Examples are given for Erlang claims number processes, and for the Ho–Lee–Merton and the Vasicek interest rate models.
30 citations