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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 paper, the authors characterize a prior that consists of a family of Dirichlet distributions on the space of Markov transition matrices and beta-Stacy processes on distribution functions.
Abstract: Starting from the definitions and the properties of reinforced renewal processes and reinforced Markov renewal processes, we characterize, via exchangeability and de Finetti’s representation theorem, a prior that consists of a family of Dirichlet distributions on the space of Markov transition matrices and beta-Stacy processes on distribution functions. Then, we show that this family is conjugate and give some estimate results.
11 citations
TL;DR: In this paper, the authors discuss the (S - 1,S) stock model where demand follows a renewal process and the lead time is deterministic, and extend this to compound renewal demand and multi-echelon inventory systems.
Abstract: textIn the standard (S - 1,S) stock model, demand follows a Poisson process. It has appeared to many stock analysts that this model causes an abundance of stock in reality. In case demand is caused by failure or is derived from another process, demand typically does not follow a Poisson process. In this paper, we discuss the (S - 1,S) stock model where demand follows a renewal process and the lead time is deterministic. Morover, we will extend this to compound renewal demand and multi-echelon inventory systems. Our goal is to show the severe influence of taking the Poisson process for granted.
11 citations
TL;DR: Stopped random sequences with record regeneration are considered and limit theorems refining the normal approximation are proved in this article, where particular cases of such sequences are stopped random walks, recurrent Markov renewal processes, and certain procedures of sequential estimation.
Abstract: Stopped random sequences with record regeneration are considered and limit theorems refining the normal approximation are proved. Particular cases of such sequences are stopped random walks, recurrent Markov renewal processes, and certain procedures of sequential estimation.
11 citations
01 Jan 2006
TL;DR: Among these models, the G-Renewal Process introduced by Kijima & Sumita (1986) is very attractive, since it covers the intermediate "better-than-old-but-worse- than-new" repair assumption and results in a G-renewal equation, which is a generalization of the well-known ordinary renewal equation.
Abstract: For many years, the most commonly used models for the failure process have been the renewal process (RP) and the nonhomogeneous Poisson process (NHPP). In the framework of the repairable system applications, RP is used to model the situations with restoration to "good-as-new" condition (perfect repair assumption), meanwhile NHPP is applied to the situations with the “same-as-old” restoration (minimal repair assumption). In a sense, these two assumptions can be considered as extreme ones from both theoretical and practical standpoints. In order to avoid this “extremism”, several generalizing models have been introduced in recent years. References include Brown & Proschan (1982), Kijima & Sumita (1986), Filkenstein, (1993), Lindqvist (1999). Among these models, the G-Renewal Process (GRP) introduced by Kijima & Sumita (1986) is very attractive, since it covers the intermediate "better-than-old-but-worse-than-new" repair assumption and results in a G-renewal equation, which is a generalization of the well-known ordinary renewal equation. Unfortunately, a closed form solution of the equation is unavailable, which makes the respective statistical estimation challenging.
11 citations
TL;DR: In this article, the authors prove universal estimates for the expected time to renewal as well as the conditional distribution of the time-to-renewal distribution of a binary renewal process.
Abstract: A binary renewal process is a stochastic process $\{X_n\}$ taking values in $\{0,1\}$ where the lengths of the runs of 1's between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.
11 citations