Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.

A single product perishing inventory model with demand interaction

Abstract: The paper describes a single perishing product inventory model in which items deteriorate in two phases and then perish. An independent demand takes place at constant rates for items in both phases. A demand for an item in Phase I not satisfied may be satisfied by an item in Phase II, based on a probability measure. Demand for items in Phase II during stock-out is lost. The re-ordering policy is an adjustable (S, s) policy with the lead-time following an arbitrary distribution. Identifying the underlying stochastic process as a renewal process, the probability distribution of the inventory level at any arbitrary point in time is obtained. The expressions for the mean stationary rates of lost demand, substituted demand, perished units and scrapped units are also derived. A numerical example is considered to highlight the results obtained.

Reliability analysis of a one-unit system with finite vacations

Abstract: This paper introduces finite vacations from a reliability theory viewpoint and deals with a one-unit repairable system with a repairman who takes finite vacations. A total probability decomposition method for analyzing the reliability problems of the system is presented with the help of the renewal process theory. With the decomposition method, two key reliability characteristics of the system i.e. the availability and mean failure number of the system are discussed under some assumptions. It is important that the steady-state availability and steady-state failure frequency of the system are obtained. Furthermore, three special cases of the model have showed that the vacation number taken by the repairman affects the performance of the system and the results presented in this paper are more general than the existing results in some literatures.

A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions

Abstract: We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution. Keywords : Extreme value theory; Poisson process; regular variation; renewal process; ruin probability; shot noise process; subexponential distribution.

Availability Analysis for the Quasi-Renewal Process

Abstract: Historically, the behaviors of repairable systems were usually modeled under the assumption that repair implied system renewal. Availability functions were then constructed using renewal functions. Often, equipment is not renewed by repair, and for equipment that is not renewed, existing models fail to capture the key features of their behavior-ongoing degradation. More recently, nonrenewal models have been proposed to reflect the fact that equipment is usually not as good as new following maintenance. A wide variety of such models have been defined. They are usually called imperfect repair models. These models have the advantage that they are more realistic but they are also more complicated; therefore, analytical results for the models have been limited. In this paper, one of the nonrenewal models is analyzed, and an approach for obtaining a detailed measure of equipment performance, the point availability, is presented. The ultimate point availability function must be approximated numerically. Nevertheless, the analysis does lead to the time-dependent measure for a variety of possible distribution models. This paper contains two contributions to the study of repairable equipment performance. First, the models analyzed include both stochastic equipment deterioration and stochastically degrading repair performance over multiple operating intervals. Second, the analytical approach to obtaining the point availability function and its approximation is based on the combined analysis of operation-time- and downtime-based formulations of the system availability. This analytical approach to availability computation has not been used previously and is quite effective.

Functions of discrete probability measures: Rates of convergence in the renewal theorem

Abstract: Let ν and τ be finite measures on the set of integers such that the Fourier transform of ν is an analytic function of the τ transform. The central result shows how ν may be approximated by linear combinations of convolution powers of τ. Applications are given to renewal theory, infinitely divisible measures and age-dependent branching processes.