Showing papers on "Renewal theory published in 2008"
Journal Article•
TL;DR: In this article, the authors present a framework where the ob-served events are modeled as marked point processes, with marks labeling the types of events, and the emphasis is more on modeling than on statistical inference.
Abstract: We review basic modeling approaches for failure and mainte- nance data from repairable systems. In particular we consider imperfect re- pair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the ob- served events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.
177 citations
Book•
25 Aug 2008
TL;DR: In this article, a semi-Markov extension of the Black-Scholes model is proposed for finance and insurance risk models, as well as generalized non-homogeneous models for Pension Funds and Manpower Management.
Abstract: Probability Tools For Stochastic Modelling.- Renewal Theory and Markov Chains.- Markov Renewal Processes, Semi-Markov Processes and Markov Random Walks.- Discrete Time and Reward Smp and their Numerical Treatment.- Semi-Markov Extensions of the Black-Scholes Model.- Other Semi-Markov Models in Finance and Insurance.- Insurance Risk Models.- Reliability and Credit Risk Models.- Generalised Non-Homogeneous Models for Pension Funds and Manpower Management.
165 citations
TL;DR: It is proved that, like their earlier versions, the two new LZ-based estimators are universally consistent, that is, they converge to the entropy rate for every finite-valued, stationary and ergodic process.
Abstract: Partly motivated by entropy-estimation problems in neuroscience, we present adetailed and extensive comparison between some of the most popular and effective entropyestimation methods used in practice: The plug-in method, four different estimators basedon the Lempel-Ziv (LZ) family of data compression algorithms, an estimator based on theContext-Tree Weighting (CTW) method, and the renewal entropy estimator.M ETHODOLOGY : Three new entropy estimators are introduced; two new LZ-basedestimators, and the “renewal entropy estimator,” which is tailored to data generated by abinary renewal process. For two of the four LZ-based estimators, a bootstrap procedure isdescribed for evaluating their standard error, and a practical rule of thumb is heuristicallyderived for selecting the values of their parameters in practice. T HEORY : We prove that,unlike their earlier versions, the two new LZ-based estimators are universally consistent,that is, they converge to the entropy rate for every finite-valued, stationary and ergodicprocess. An effective method is derived for the accurate approximation of the entropy rateof a finite-state hidden Markov model (HMM) with known distribution. Heuristiccalculations are presented and approximate formulas are derived for evaluating the bias andthe standard error of each estimator. S
109 citations
TL;DR: In this paper, the authors studied the tail probability of discounted aggregate claims in a continuous-time renewal model and obtained an asymptotic formula, which holds uniformly for all time horizons within a finite interval.
Abstract: In this paper we study the tail probability of discounted aggregate claims in a continuous-time renewal model. For the case that the common claim-size distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and inter-arrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Probab. 44 (2), 285–294].
71 citations
TL;DR: In this paper, the authors consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions, and they model the occurrence of claims according to a renewal process.
Abstract: Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramer light-tail assumption on the claim size distribution.
64 citations
TL;DR: In this article, the authors consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions, and they model the occurrence of claims according to a renewal process.
Abstract: Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cram\'{e}r light-tail assumption on the claim size distribution.
60 citations
TL;DR: It is proved that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated.
55 citations
TL;DR: This work considers a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered, and develops exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process.
Abstract: The order fill rate (OFR) is sometimes suggested as an alternative to the volume fill rate (VFR) (most often just denoted fill rate) as a performance measure for inventory control systems. We consider a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered. For this policy, we develop exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process. We also elaborate on when the OFR can be interpreted as the (extended) ready rate. For the case when customer orders are generated by a negative binomial distribution, we show that it is the size of the shape parameter of this distribution that determines the relative magnitude of the two fill rates. In particular, we show that when customer orders are generated by a geometric distribution, the OFR and the VFR are equal.
53 citations
30 Sep 2008
TL;DR: In this paper, the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting timedistribution was shown.
Abstract: We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting timedistribution, characteristic for a time fractional continuous time random walk. This asymptotic equivalence is effected by a combination of “rescaling” time and “respeeding” the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. As far as we know such procedure has been first applied in the 1960s by Gnedenko and Kovalenko in their theory of “thinning” a renewal process. Turning our attention to spatially one-dimensional continuous time random walks with a generic power law jump distribution, “rescaling” space can be interpreted as a second kind of “respeeding” which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the ‘time fractional drift” process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.
52 citations
Book•
09 May 2008
TL;DR: In this article, the reliability of repairable systems with three or more repairable units is investigated. But the results for the repairable system with three repair units follow a general distribution.
Abstract: Preliminaries The Poisson process and distribution Waiting time distributions for a Poisson process Statistical estimation theory Generating a Poisson process Nonhomogeneous Poisson process Binomial, geometric, and negative binomial distributions Statistical Life Length Distributions Stochastic life length models Models based on the hazard rate General remarks on large systems Reliability of Various Arrangements of Units Series and parallel arrangements Series-parallel and parallel-series systems Various arrangements of switches Standby redundancy Reliability of a One-Unit Repairable System Exponential times to failure and repair Generalizations Reliability of a Two-Unit Repairable System Steady-state analysis Time-dependent analysis via Laplace transform On model 2(c) Continuous-Time Markov Chains The general case Reliability of three-unit repairable systems Steady-state results for the n-unit repairable system Pure birth and death processes Some statistical considerations First Passage Time for Systems Reliability Two-unit repairable systems Repairable systems with three (or more) units Repair time follows a general distribution Embedded Markov Chains and Systems Reliability Computations of steady-state probabilities Mean first passage times Integral Equations in Reliability Theory Introduction Example 1: Renewal process with a general distribution Example 2: One-unit repairable system Example 3: Effect of preventive replacements or maintenance Example 4: Two-unit repairable system Example 5: One out of n repairable systems Example 6: Section 7.3 revisited Example 7: First passage time distribution References Index A Problems and Comments section appears at the end of each chapter.
45 citations
TL;DR: The economic and economic-statistical design of a χ 2 chart for a maintenance application is considered, and an additional constraint guaranteeing the occurrence of the true alarm signal on the chart before failure with given probability is considered.
TL;DR: In this article, a consistency test between time dependent and time independent recurrence distributions is made using a Monte Carlo method to replicate the paleoseismic series on the south Hayward fault.
Abstract: [1] Elastic rebound and stress renewal are important components of earthquake forecasting because if large earthquakes can be shown to be periodic, then rupture probability is time dependent. While renewal models are used in formal forecasts, it has not been possible to exclude the alternate view that repeated large earthquakes can happen in rapid succession without requiring time for stress regeneration. Here a consistency test between time dependent and time independent recurrence distributions is made using a Monte Carlo method to replicate the paleoseismic series on the south Hayward fault. Time dependent distributions with recurrence interval of 210 years and coefficient of variation of 0.6 reproduce the event series on the south Hayward 5 times more often than any exponential distribution: a highly significant difference as determined using a two-tailed Z-test for relative proportions. Therefore large Hayward fault earthquakes are quasi-periodic, and are most consistent with a stress renewal process.
TL;DR: A deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections, and the distribution of the number of minimal and perfect repairs between two inspections is determined.
Posted Content•
TL;DR: In this article, it was shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline.
Abstract: Sampling from a random discrete distribution induced by a `stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline.
TL;DR: In this paper, it was shown that close to criticality, under general assumptions, the correlation decay rate or the renewal convergence rate, coincides with the inter-arrival decay rate.
Abstract: A class of discrete renewal processes with exponentially decaying inter-arrival distributions coincides with the infinite volume limit of general homogeneous pinning models in their localized phase. Pinning models are statistical mechanics systems to which a lot of attention has been devoted both for their relevance for applications and because they are solvable models exhibiting a non-trivial phase transition. The spatial decay of correlations in these systems is directly mapped to the speed of convergence to equilibrium for the associated renewal processes. We show that close to criticality, under general assumptions, the correlation decay rate, or the renewal convergence rate, coincides with the inter-arrival decay rate. We also show that, in general, this is false away from criticality. Under a stronger assumption on the inter-arrival distribution we establish a local limit theorem, capturing thus the sharp asymptotic behavior of correlations.
TL;DR: A repair model with or without preceding inspections is introduced, with finite renewal (repair) times independent of failure times, for optimizing design and maintenance strategies of aging structural components.
Posted Content•
TL;DR: In this article, a mathematical treatment of the finite size scaling limit of pinning models is given, namely studying the limit (in law) of the process close to criticality when the system size is proportional to the correlation length.
Abstract: Pinning models are built from discrete renewal sequences by rewarding (or penalizing) the trajectories according to their number of renewal epochs up to time $N$, and $N$ is then sent to infinity. They are statistical mechanics models to which a lot of attention has been paid both because they are very relevant for applications and because of their {\sl exactly solvable character}, while displaying a non-trivial phase transition (in fact, a localization transition). The order of the transition depends on the tail of the inter-arrival law of the underlying renewal and the transition is continuous when such a tail is sufficiently heavy: this is the case on which we will focus. The main purpose of this work is to give a mathematical treatment of the {\sl finite size scaling limit} of pinning models, namely studying the limit (in law) of the process close to criticality when the system size is proportional to the correlation length.
TL;DR: In this article, it was shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the half line.
Abstract: Sampling from a random discrete distribution induced by a 'stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the half-line.
TL;DR: In this article, the arrival times of a renewal process are defined so that the number of events occurring before a randomly distributed time, independent of the process, preserves the aging properties of the renewal process.
Abstract: In this work we provide sufficient conditions for the arrival times of a renewal process so that the number of its events occurring before a randomly distributed time, T, independent of the process preserves the aging properties of T.
TL;DR: In this paper, it was shown that the increasing geometric process grows at most logarithmically in time while the decreasing geometric process is almost certain to have a time of explosion.
Abstract: The geometric process has been proposed as a simple model for use in reliability. Recently, the α-series process was proposed as a complementary model which can be used in situations where the geometric process is inappropriate. In this article, we show that the increasing geometric process grows at most logarithmically in time while the decreasing geometric process is almost certain to have a time of explosion. The α-series process grows either as a polynomial in time or exponentially in time. We also show that, unlike most renewal processes, the geometric process does not satisfy a central limit theorem, while the α-series process does.
TL;DR: This approach combines the elements of renewal theory to estimate the essential features of the resulting stochastic process as functions of the parameters of the controlling term.
Abstract: We present an approach for the analytical treatment of excitable systems with noise-induced dynamics in the presence of time delay. An excitable system is modeled as a bistable system with a time delay, while another delay enters as a control term taken after Pyragas [K. Pyragas, Phys. Lett. A 170, 421 (1992)] as a difference between the current system state and its state tau time units before. This approach combines the elements of renewal theory to estimate the essential features of the resulting stochastic process as functions of the parameters of the controlling term.
TL;DR: This work derives closed-form expressions for the expected delay in heavy-traffic (HT) limits from gated polling systems with general service and switch-over times and with renewal arrival processes.
TL;DR: In this paper, the authors considered the conditional LDP problem for the conditional empirical process of words, where one conditions on a typical underlying (i.i.d.) sequence of words and showed that the level 3 large deviation behavior of this sequence is governed by specific relative entropy.
TL;DR: In this paper, it was shown that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale and that its rate function is not convex.
Abstract: This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.
TL;DR: In this article, the authors present new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbola, and no discounting).
Abstract: To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.
IBM1
TL;DR: In this article, the authors study multi-product and multi-item assemble-to-order systems under general assumptions on demand patterns and replenish leadtime distributions, and develop procedures for approximating key performance measures of these inventory systems, such as average inventory and immediate order fill rate.
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.
Journal Article•
TL;DR: In this article, the arrival times of a renewal process are defined so that the number of events occurring before a randomly distributed time, independent of the process, preserves the aging properties of the renewal process.
Abstract: In this work we provide sufficient conditions for the arrival times of a renewal process so that the number of its events occurring before a randomly distributed time, T, independent of the process preserves the aging properties of T.
TL;DR: In this paper, the asymptotic behavior of small deviation probabilities for compound renewal processes is investigated, and the authors show that the small deviation probability of small deviations is a function of the compound renewal process.
Abstract: The asymptotic behavior of small deviation probabilities for compound renewal processes is investigated.
TL;DR: In this article, the authors prove universal estimates for the expected time to renewal as well as the conditional distribution of the time-to-renewal distribution of a binary renewal process.
Abstract: A binary renewal process is a stochastic process $\{X_n\}$ taking values in $\{0,1\}$ where the lengths of the runs of 1's between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.