Studies on approximations to renewal functions and their applications

ABSTRACT: Renewal equations and renewal type equations are frequently encountered in several applications when regenerative arguments are used in the modeling. These equations which are Volterra type integral equations contain the renewal function in the kernel which is a key tool in renewal processes. Analytical solutions of the renewal equations are possible only for a very few cases. Although several approximations for the renewal function are available, the use of any method depends on the characteristics of the underlying distribution function such as skewness, kurtosis, modes, and singularities. Further, all the approximations proposed so far presuppose the knowledge of the underlying distribution function. This thesis proposes nonparametric approximations to several renewal functions based only on the first few moments of the distribution function. The renewal functions considered are onedimensional renewal function, g-renewal function, and two-dimensional renewal function. The approximations are compared with the actual values wherever available and with benchmark approximations. Examples from areas such as reliability, queuing theory, and warranty models are developed to illustrate the efficacy of the approximations. Keywords: Renewal Function, G-renewal Function, Two-dimensional Renewal Function, Moments Based Approximation, General Repair, Renewing Warranty. …………………………………………………………………………………………………………………………