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Renewal theory
About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.
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TL;DR: Applying a matrix analytic approach, fluid flow techniques and martingales, methods are developed to obtain explicit formulas for the cost functionals (setup, holding, production and lost demand costs) in the discounted case and under the long-run average criterion.
Abstract: We study the performance of a reflected fluid production/inventory model operating in a stochastic environment that is modulated by a finite state continuous time Markov chain. The process alternates between ON and OFF periods. The ON period is switched to OFF when the content level reaches a predetermined level q and returns to ON when it drops to 0. The ON/OFF periods generate an alternative renewal process. Applying a matrix analytic approach, fluid flow techniques and martingales, we develop methods to obtain explicit formulas for the cost functionals (setup, holding, production and lost demand costs) in the discounted case and under the long-run average criterion. Numerical examples present the trade-off between the holding cost and the loss cost and show that the total cost appears to be a convex function of q.
15 citations
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TL;DR: This paper provides a detailed derivation of the original Bellman Harris process, and introduces generalisations, that allow for time-varying reproduction numbers and the accounting of exogenous events, such as importations.
Abstract: Renewal processes are a popular approach used in modelling infectious disease outbreaks. In a renewal process, previous infections give rise to future infections. However, while this formulation seems sensible, its application to infectious disease can be difficult to justify from first principles. It has been shown from the seminal work of Bellman and Harris that the renewal equation arises as the expectation of an age-dependent branching process. In this paper we provide a detailed derivation of the original Bellman Harris process. We introduce generalisations, that allow for time-varying reproduction numbers and the accounting of exogenous events, such as importations. We show how inference on the renewal equation is easy to accomplish within a Bayesian hierarchical framework. Using off the shelf MCMC packages, we fit to South Korea COVID-19 case data to estimate reproduction numbers and importations. Our derivation provides the mathematical fundamentals and assumptions underpinning the use of the renewal equation for modelling outbreaks.
15 citations
TL;DR: In this article, a generalization of the decreasing failure rate (DFR) concept is introduced, based on the following principle: if there have been many points of occurrence recently, then we will soon experience another one.
15 citations
TL;DR: In this article, the authors considered a model of directed polymers interacting with a one-dimensional defect line on which random charges are placed and showed that the quenched correlation length of the two-point function does not diverge faster than 1/F.
Abstract: We consider models of directed polymers interacting with a one-dimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on $Z$ and gives a random (site-dependent) reward or penalty to the occurrence of a renewal at any given point of $Z$. These models are known to undergo a delocalization-localization transition, and the free energy $F$ vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length $\xi$, defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than $1/F$. We prove also an exponentially decaying upper bound for the disorder-averaged two-point function, with a good control of the sub-exponential prefactor. We discuss how, in the particular case where disorder is absent, this result can be seen as a refinement of the classical renewal theorem, for a specific class of renewal sequences.
15 citations
TL;DR: This publication presents useful computational techniques to determine the probabilistic characteristics of a renewal process that focus on continuous-time renewal processes and their approximations with discrete-time processes.
15 citations