Renewal theory

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

Regenerative compositions in the case of slow variation: A renewal theory approach

Abstract: A regenerative composition structure is a sequence of ordered partitions derived from the range of a subordinator by a natural sampling procedure. In this paper, we extend previous studies on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the Levy measure of the subordinator has a property of slow variation at $0$. Using tools from the renewal theory the limit laws for $K_n$ are obtained in terms of integrals involving the Brownian motion or stable processes. In other words, the limit laws are either normal or other stable distributions, depending on the behavior of the tail of Levy measure at $\infty$. Similar results are also derived for the number of singleton blocks.

Maintenance Policies for Multicomponent Systems: An Overview

Abstract: We present an overview of some recent developments in the area of mathematical modelling of maintenance decisions for multicomponent systems. We do not claim to be complete, but rather we expose some ideas both in modelling and in solution procedures which turned out to be useful in understanding and supporting complex maintenance management decision problems. The mathematical tools that are used mainly stem from applied probability theory, renewal theory, and Markov decision theory.

Estimating the Cost of a Warranty

Abstract: The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.

Generalized fractional poisson process and related stochastic dynamics

Abstract: We survey the 'generalized fractional Poisson process' (GFPP). The GFPP is a renewal process generalizing Laskin's fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 0 and a parameter characterizing the time scale. The GFPP involves Prab-hakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems. MSC 2010 : Primary 60K05, 33E12, 26A33; Secondary 60J60, 65R10, 60K40