Renewal theory

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

Implicit Renewal Theory and Power Tails on Trees

Abstract: We extend Goldie’s (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the powertail asymptotics of the distributions of the solutions R to R d = N=1 Ci Ri + Q, R d

Uniformization and Hybrid Simulation/Analytic Models of Renewal Processes

Abstract: In this paper we demonstrate the use of uniformization in the simulation of renewal processes. Using uniformization, we represent the continuous random variable of interest as the first passage time of a continuous-time stochastic process associated with a Poisson process. We then use this result to develop a hybrid simulation/analytic method to model renewal processes. The estimators obtained from the hybrid simulation/analytic models have lower variance than the variance of the estimators of the traditional simulation models. We also discuss the possible impact of this method on the future of simulation methodology.

Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem

Abstract: We consider the class of graph-directed constructions which are connected and have the property of finite ramification. By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplace operator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of the Laplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling in the eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to do this we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for the eigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.

Moment generating functions of compound renewal sums with discounted claims

Abstract: Leveille & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered. In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.

Inventory systems for perishable commodities with renewal input and poisson output

Abstract: We consider an inventory system to which arrival of items stored is a renewal process, and the demand is a Poisson process. Items stored have finite and fixed lifetimes. The blood-bank model inspired this study. Three models are studied. In the first one, we assume that each demand is for one unit and unsatisfied demands leave the system immediately. Using results on this model one is able to study a model in which arrival of items is Poisson but demands are for several units, and a model in which demands are willing to wait. We compute ergodic limits for the lost demands and the lost items processes and the limiting distribution of the number of items stored. The main tool in this analysis is an analogy to M/G/1 queueing systems with impatient customers.