Two-dimensional Renewal Function Approximation

Abstract

Two-dimensional renewal functions, which are naturally extensions of one-dimensional renewal functions, have wide applicability in areas where two random variables are needed to characterize the underlying process. These functions satisfy the renewal equation, which is not amenable for analytical solutions. This paper proposes a simple approximation for the computation of the two-dimensional renewal function based only on the first two moments and the correlation coefficient of the variables. The approximation yields exact values of renewal function for bivariate exponential distribution function. Illustrations are presented to compare our approximation with that of Iskandar (1991) who provided a computational procedure which requires the use of the bivariate distribution function of the two variables. A two-dimensional warranty model is used to illustrate the approximation.

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…”

On an Interrupted Bivariate Renewal Process and Its Applications

“…Analogously, calculation of the cumulative distribution function H(t) by the formula (7) requires considerable computation time. This time can be decreased if the density h(t) is presented by a linear combination of two easily calculated densities d1(t) and d2(t) with a coefficient 𝜒𝜒, 0 < 𝜒𝜒 < 1, namely:…”

“…A simple approximation for the two-dimensional renewal function, based only on the first two moments of the variables and their correlation coefficient, is considered in paper [5]. Many investigations about various approximations are described in papers [6][7][8][9][10].…”

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.…”