Moments-Based Approximation to the Renewal Function

Kambo, N.S.; Rangan, A.; Hadji, Ehsan Moghimi

doi:10.1080/03610926.2010.533231

Cited by 17 publications

(10 citation statements)

References 25 publications

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“…This is because the renewal equation given in (1) is not amenable for explicit solution. Kambo et al (2012) developed a method to approximate the univariate renewal function. This approximation is given in terms of the moments of order one to three of the process's distribution.…”

Section: Approximation To the Laplace Transform Of The Bivariate Renementioning

confidence: 99%

“…Li and Lou (2005) found upper and lower bounds for the solutions of the Markov renewal equations in the onedimensional case. Recently, Kambo et al (2012) proposed an approximation method for the renewal density function, based on the first three moments of the distribution function, if these exist. Their method is an interesting one because they do not use prior knowledge of the distribution functions.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of the Bivariate Renewal Function

Arunachalam

Calvache

2014

Communications in Statistics - Simulation and Computation

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Recently, Kambo and his co-researchers (2012) proposed a method of approximation for evaluating the one-dimensional renewal function based on the first three moments. Their method is simple and elegant, which gives exact values for well-known distributions. In this article, we propose an analogous method for the evaluation of bivariate renewal function based on the first two moments of the variables and their joint moment. The proposed method yields exact results for certain widely used bivariate distributions like bivariate exponential distribution, bivariate Weibull distributions, and bivariate Pareto distributions. An illustrative example in the form of a two-dimensional warranty problem is considered and comparisons of our method are made with the results of other models.

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Section: Approximation To the Laplace Transform Of The Bivariate Renementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximation of the Bivariate Renewal Function

Arunachalam

Calvache

2014

Communications in Statistics - Simulation and Computation

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“…However, good approximations are available for the evaluation of M D (t) (Deligonul, 1985;Xie, 1989;Kambo et al, 2011). Thus, the total expected post warranty cost under replacement strategy is given by…”

Section: In Our Case G(t) Is Specified Bymentioning

confidence: 99%

Optimal burn-in, warranty, and maintenance decisions in system design

Hadji

Rangan

2012

IJMOM

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Failure rate curves of many deteriorating systems are typically bathtub shaped. Bathtub failure curves based on failure rate functions show three distinct phases namely, infant mortality period, useful period, and wear out period. Existing reliability models have been developed purely from the consumer or manufacturer's perspective thereby delinking the third phase from the first two phases. The objective of the present work is to discuss several optimality issues based on the total average cost of the system over its entire useful life that encompasses the cost incurred on the system by the consumer as well as the manufacturer.Reference to this paper should be made as follows: Hadji, E.M. and Rangan, A. (2012) 'Optimal burn-in, warranty, and maintenance decisions in system design', Int.

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“…9 In contrast, the literature on approximations to renewal functions is rich. In this sequel, we refer to a few interesting contributions only; approximations, [10][11][12][13][14][15] Páde approximations, 16 power series expansions, 13 Riemann-Stieltjes integration methods 17,18 and bounds. 4,12,19,20 Efficacy of an availability function approximation can generally be gauged by the successful applicability of the approximations to various input distributions of interest as well as accuracy and convergence speed.…”

Section: Introductionmentioning

confidence: 99%

Some Useful Approximations for the Availability Function

Arunachalam

Calvache

Tansu

2015

Int. J. Rel. Qual. Saf. Eng.

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Availability function which forms an important part of reliability analysis is expressed in terms of an integral equation. The analytical solution of such an equation is possible only in very simple cases and hence approximations are the only tools available; very few such approximations are available in the literature. This paper proposes three useful approximations, two of which are based only on the first few moments of the underlying distributions and do not require their functional forms. The third approximation uses the Riemannian sum to approximate the integral equation. Numerical illustrations based on test cases are provided to show the efficacy of the approximations. As an application, the problem of an opportunistic channel access scheme in a communication network is used to test the approximations.

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Moments-Based Approximation to the Renewal Function

Cited by 17 publications

References 25 publications

Approximation of the Bivariate Renewal Function

Approximation of the Bivariate Renewal Function

Optimal burn-in, warranty, and maintenance decisions in system design

Some Useful Approximations for the Availability Function

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