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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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Journal ArticleDOI
TL;DR: The economic and economic-statistical designs of an X¯ control chart for two-identical unit series systems with condition-based maintenance is studied and optimization models have been developed to find the optimal control chart parameters for minimizing the average maintenance costs.

56 citations

16 Jan 2007
TL;DR: In this article, a non-Markovian renewal process with a waiting time distribution described by the MittagLeffler function is analyzed, which plays a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power-asymptotic waiting time.
Abstract: It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the MittagLeffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power-asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.

56 citations

Journal ArticleDOI
TL;DR: A probabilistic renewal method is used to explicitly calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets and shows that both models have the same splitting probabilities, and that increasing the resetting rate r increases (reduces) the splitting probability for proximal (distal) targets.
Abstract: We show how certain active transport processes in living cells can be modeled in terms of a directed search process with stochastic resetting and delays. Two particular examples are the motor-driven intracellular transport of vesicles to synaptic targets in the axons and dendrites of neurons, and the cytonemebased transport of morphogen to target cells during embryonic development. In both cases, the restart of the search process following reset has a finite duration with two components: a finite return time and a refractory period. We use a probabilistic renewal method to explicitly calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets. We consider two different search scenarios: bounded search on the interval [0, L], where L is the length of the array, with a refractory boundary at x = 0 and a reflecting boundary at x = L (model A), and partially bounded search on the half-line (model B). In the latter case there is a non-zero probability of failure to find a target in the absence of resetting. We show that both models have the same splitting probabilities, and that increasing the resetting rate r increases (reduces) the splitting probability for proximal (distal) targets. On the other hand the MFPTs for model A are monotonically increasing functions of r, whereas the MFPTs of model B are non-monotonic with a minimum at an optimal resetting rate. We also formulate multiple rounds of search-and-capture events as a G/M/∞ queue and use this to calculate the steady-state accumulation of resources in the targets. Directed search process with stochastic resetting and delays 2

56 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of two-dimensional renewal counting processes Nxy and Nx y, and the moments of the minimum of two correlated normal random variables.
Abstract: In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nxy and the two-dimensional renewal function Nx y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended. TWO-DIMENSIONAL RENEWAL THEORY; RENEWAL FUNCITIONS; ASYMPTOTIC DISTRIBUTIONS; MINIMUM OF TWO CORRELATED NORMAL RANDOM VARIABLES

56 citations

Patent
Amit Garg1, Jayant R. Kalagnanam1
14 Jul 1997
TL;DR: In this article, the renewal function of the truncated normal distribution can be characterized by two parameters: (1) its coefficient of variation, and (2) the point at which the function needs to be evaluated.
Abstract: A computer implemented process is provided for fast and accurate evaluation of the performance characteristics of the periodic-review (s,S) inventory policy with complete back ordering. This policy has an underlying stochastic process that is a renewal process. The method provides a novel computer implementation of a fast and accurate way to compute approximations of the renewal function. In order to overcome the computational problems in evaluating renewal functions numerically, an approximation scheme has been devised whereby the renewal function of the truncated normal distribution can be characterized by two parameters: (1) its coefficient of variation, and (2) the point at which the function needs to be evaluated. This approximation is derived in two stages. In the first stage, a class of rational polynomial approximations are developed to the renewal function, called Pade approximants. In the second stage, polynomial expressions are derived for each coefficient of the Pade approximants in terms of the coefficient of variation of the distribution.

55 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202327
202260
202173
202083
201973
201886