Approximation of the Bivariate Renewal Function

Abstract

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.

The Mean Value Theorem for Integrals Method for Estimating Two-Dimensional Renewal Functions

“…Kambo et al (2012) discuss and propose to approximate the RNF based only on first three moments. Some literature discuss the other approximation of the 2D RNF such as Hadji et al (2015), Arunachalam & Calvache (2015), and Omey et al (2018). ,  Applying MeVTI on the point 𝑥 𝑖−1 , 𝑦 𝑗 −1 at every rectangles 𝑥 𝑖−1 , 𝑥 𝑖 × 𝑦 𝑗 −1 , 𝑦 𝑗 to obtain the estimation of 𝑀 𝑥, 𝑦 , that is given by…”

Two-dimensional warranty: A literature review

“…The computation of 2D renewal function is very difficult because no explicit form exists. Corbu et al, 108 Arunachalam and Calvache, 109 and Hadji et al 110 studied the numerical evaluation of 2D renewal functions, and applied their methods to the 2D warranty cost analysis. Warranty execution.…”

Some Useful Approximations for the Availability Function

“…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.…”