Showing papers on "Renewal theory published in 2005"
Book•
02 Nov 2005
TL;DR: This book presents homogeneous and non-homogeneous semi-Markov processes, as well as Markov and semi- Markov rewards processes, which are fundamental for many applications, but are not as thoroughly presented in other books on the subject as they are here.
Abstract: Probability Tools for Stochastic Modelling.- Renewal Theory.- Markov Chains.- Markov Renewal Processes, Markov Random Walks and Semi-Markov Processes.- Functionals of (J-X) Processes.- Non-Homogeneous Markov and Semi-Markov Processes.- Markov and Semi-Markov Reward Processes.
227 citations
Book•
12 Aug 2005
TL;DR: This paper presents a meta-analysis of point processes based on Fractal Brownian Motion, which shows how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
Abstract: Preface. List of Figures. List of Tables. Authors. 1. Introduction. 2. Scaling, Fractals, and Chaos. 3. Point Processes: Definition and Measures. 4. Point Processes: Examples. 5. Fractal and Fractal-Rate Point Processes. 6. Processes Based on Fractional Brownian Motion. 7. Fractal Renewal Processes. 8. Processes Based on the Alternating Fractal Renewal Process. 9. Fractal Shot Noise. 10. Fractal-Shot-Noise-Driven Point Processes. 11. Operations. 12. Analysis and Estimation. 13. Computer Network Traffic. Appendix A: Derivations. Appendix B: Problem Solutions. Appendix C: List of Symbols. Bibliography. Author Index. Subject Index.
166 citations
TL;DR: It is shown that the Telecom process can be interpreted as the workload of a network responding to job requests with stable infinite variance intensities and durations and that fractional Brownian motion (fBM) can beinterpreted in the same way but with finite variance intensITIES.
89 citations
TL;DR: The formalism that is created here adds a new dimension to the research in perishable inventory control under uncertainty in lifetime, and some properties of optimal ordering policies and cost functions are demonstrated.
Abstract: We consider a discrete-time (s, S) inventory model in which the stored items have a random common lifetime with a discrete phase-type distribution. Demands arrive in batches following a discrete phase-type renewal process. With zero lead time and allowing backlogs, we construct a multidimensional Markov chain to model the inventory-level process. We obtain a closed-form expected cost function. Numerical results demonstrate some properties of optimal ordering policies and cost functions. When compared with the results for the constant lifetime model, the variance of the lifetime significantly affects the system behavior. Thus, the formalism that we create here adds a new dimension to the research in perishable inventory control under uncertainty in lifetime.
51 citations
TL;DR: In this paper, the authors consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process.
Abstract: We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.
43 citations
TL;DR: A strong conjecture is provided for the limiting waiting-time distribution in a general parameter setting when the load tends to 1, under proper heavy-traffic scalings.
41 citations
TL;DR: In this article, a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities is presented. But the authors do not consider whether the sample distribution is regenerative or non-regenerative.
Abstract: This thesis consists of four papers.In paper 1, we prove central limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities.In paper 2, properties of inverse subordinators are investigated, in particular similarities with renewal processes. The main tool is a theorem on processes that are both renewal and Cox processes.In paper 3, distributional properties of supercritical and especially immortal branching processes are derived. The marginal distributions of immortal branching processes are found to be compound geometric.In paper 4, a description of a dynamic population model is presented, such that samples from the population have genealogies as given by a Lambda-coalescent with mutations. Depending on whether the sample is grouped according to litters or families, the sampling distribution is either regenerative or non-regenerative.
40 citations
TL;DR: The proposed numerical method is simple, amenable to direct error analysis, performs better than previously proposed approximations in many cases of significant practical interest, and is capable of handling some cases on which other methods fail.
Abstract: The integral equation of renewal type has many applications in applied probability. However, it is rarely solvable in closed form. In this paper, we describe a numerical method for finding approximate solutions to integral equations of renewal type based on quadrature rules for Stieltjes integrals previously developed by the author. We provide error analyses for our numerical solution based on the trapezoid-like rule for Stieltjes integrals and describe computational experience with two solution algorithms based on the trapezoid-like and the Simpson-like rules for Stieltjes integrals. We discuss a link between discrete and continuous renewal theory that is illuminated by the numerical solution method. The proposed numerical method is simple, amenable to direct error analysis, performs better than previously proposed approximations in many cases of significant practical interest, and is capable of handling some cases on which other methods fail.
39 citations
TL;DR: The hit^* rate is introduced as a consistency measure, which captures the fraction of non-stale downloads from the cache, and is applied to the analysis of the fixed TTL consistency algorithm in the presence of network delays.
29 citations
Posted Content•
TL;DR: In this paper, the authors consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition and determine the precise asymptotic behavior of the partition function, from which they obtain the scaling limits of the models and an explicit construction of the infinite volume measure (thermodynamic limit) in all regimes.
Abstract: We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure (thermodynamic limit) in all regimes, including the critical one.
28 citations
TL;DR: In this paper, a compound renewal model is introduced for the probabilistic analysis of multi-year drought recurrence, which generalizes the Poisson process and incorporates the severity of droughts into the analysis by means of a threshold quantile.
Abstract: [1] Droughts influence the planning and design of water supply infrastructure. Hydrologists ascertain drought duration, severity, and pattern of recurrence from instrumental and reconstructed records (e.g., using tree rings) of streamflow and precipitation. This work introduces a compound renewal model for the probabilistic analysis of multiyear drought recurrence. The compound renewal process generalizes the Poisson process. In the former the interarrival time between two consecutive events is the duration of nondrought conditions, and the events (i.e., droughts) have a probabilistic duration of at least θ years. The sum of the interarrival time and its subsequent drought duration is called the renewal time, which regenerates over time according to probabilistic laws derived in this work. Drought severity is incorporated in the analysis by means of a threshold quantile (e.g., the median or the average), so that low-streamflow conditions become a drought whenever they last over θ years. A case study dealing with a river basin that has multiyear storage capacity, and in which droughts recurred frequently in the twentieth century, demonstrates the analytical power of the compound renewal model.
TL;DR: In this paper, an alternative for calculating the parameters of GRP models using a Genetic Algorithm (GA) approach to solve complex maximum likelihood estimation (MLE) equations is presented. And the results using this approach confirm and extend conclusions of the Kijima and Sumita, and Yanez et al. works.
Abstract: Kijima and Sumita have proposed a stochastic model called the generalized renewal process (GRP) to describe the availability characteristics of repairable systems by introducing the notion of the virtual age of the system. Yanez et al. offer maximum likelihood estimation (MLE) approach for estimating parameters of the GRP models. Due to the complexity of the equations, a close solution is not available, and numerical solutions are proposed with limited success. This paper describes an alternative for calculating the parameters of GRP models using a Genetic Algorithm (GA) approach to solve complex MLE equations. The results using this approach confirm and extend conclusions of the Kijima and Sumita, and Yanez et al. works. Examples of applications of GA have been presented. The paper also concludes that under certain conditions, application of the minimal repair assumption provide a reasonable answer for the availability of repairable units.
Journal Article•
TL;DR: In this paper, the first two moments of a compound renewal present value risk Process (CRPVR) were derived using renewal theory under the condition that the net force of interest is a Wiener process with a drift.
Abstract: Under the condition that the net force of interest is a Wiener processes with a drift, we derive the first two moments of a compound renewal present value risk Process(CRPVR) using renewal theory. An example is also given to explain the results.
TL;DR: The information gain of a point process model quantifies its predictability, relative to a reference model such as the Poisson process, and provides a bound on the utility of the model as a forecasting tool, separate from the usual “goodness of fit” assessment criteria.
Abstract: The information gain of a point process model quantifies its predictability, relative to a reference model such as the Poisson process. This is bounded above by the entropy gain, or difference between the point process entropy rates. This provides a bound on the utility of the model as a forecasting tool, separate from the usual “goodness of fit” assessment criteria. The stress release model is a point process with an underlying state variable increasing linearly with time, and decreased by events. Assuming the intensity to be an exponential function of this state, we derive an analytic expression for the entropy gain. This is illustrated, using various magnitude distributions, for earthquake data from north China, and extensions to a multivariate linked model outlined. The results measure the effectiveness of the stress release process as a predictive tool. Comparisons are made with a scale derived from the Gamma renewal process and using Molchan's ν-τ diagram.
TL;DR: In this paper, the renewal theory developed by Sarig and Gouezel in the context of non-uniformly expanding dynamical systems applies also to the study of compact group extensions of such systems.
Abstract: We show that the renewal theory developed by Sarig and Gouezel in the context of non-uniformly expanding dynamical systems applies also to the study of compact group extensions of such systems. As a consequence, we obtain results on subexponential decay of correlations for equivariant Holder observations.
Book•
01 Jan 2005TL;DR: In this article, the authors develop the theory of second-order approximations to continuous-time optimal stopping boundaries for Brownian motion and describe some new applications, including option pricing and sequential analysis.
Abstract: Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.
TL;DR: A new fluctuation technique of multivariate marked counting processes is explored that includes the time dependent analysis of queueing and busy period processes, specially developed for this process, and it yields their stationary distributions in closed analytic forms.
TL;DR: In this article, the first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process.
Abstract: The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.
TL;DR: In this article, the authors consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅, ⋅) under mixing conditions.
Abstract: Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.
TL;DR: In this article, a linear dynamical system under random trains of impulses driven by a class of non-Erlang renewal processes is considered, where the renewal events are selected from an Erlang renewal process.
Posted Content•
TL;DR: In this paper, the authors consider parameter estimation in a regression model corresponding to an i.i.d. sequence of censored observations of a finite state modulated renewal process and give conditions for consistency and asymptotic normality of estimates.
Abstract: We consider parameter estimation in a regression model corresponding to an i.i.d. sequence of censored observations of a nite state modulated renewal process. The model assumes a similar form as in Cox regression except that the baseline intensities are functions of the backwards recurrence time of the process and a time dependent covariate. As a result of this it falls outside the class of multiplicative intensity counting process models. We use kernel estimation to con- struct estimates of the regression coecien ts and baseline cumulative hazards. We give conditions for consistency and asymptotic normality of estimates. Data from a bone marrow transplant study are used to illustrate the results.
TL;DR: In this article, the authors considered an Erlang(2) risk process perturbed by diffusion, and derived the bound for the ultimate ruin probability and the ruin probability caused by claim.
Abstract: In this article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek–Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.
TL;DR: Analytical expressions for the probability mass functions and moments of the handover number distribution for a variety of combinations of the call holding time (CHT) and cell residence time (CRT) distributions are obtained.
Abstract: Knowing the distribution of the number of handovers during a call session of a given user is particularly important in cellular mobile communication networks in order to make appropriate dimensioning of virtual circuits for wireless cells. In this paper, we study the probability distributions and statistical moments for the number of handovers per call for a variety of combinations of the call holding time (CHT) and cell residence time (CRT) distributions. We assume that the first CRT in the originating cell has different statistics from the CRTs in the subsequent cells. In particular, we consider circular cells. Based on the formulation in terms of delayed renewal processes, we obtain analytical expressions for the probability mass functions and moments of the handover number distribution. We also include some numerical results for the mean number of handovers.
TL;DR: An overview of mobile license renewal issues covering the legal regime of license renewal, the renewal process, the non-renewal context and the changes in licensing conditions including spectrum implications is provided in this paper.
Abstract: This note provides an overview of mobile license renewal issues covering the legal regime of license renewal, the renewal process, the non-renewal context and the changes in licensing conditions including spectrum implications of the renewal process. It draws best practices that started to emerge in recent renewal practices, to ensure that the renewal process leads to the best outcome for all stakeholders.
07 Nov 2005
TL;DR: By employing fuzzy variables to characterize the interarrival times, the delayed renewal process is extended from random case to fuzzy case and the fuzzy form of elementary renewal theorem is established.
Abstract: By employing fuzzy variables to characterize the interarrival times, the delayed renewal process is extended from random case to fuzzy case. Furthermore, some basic properties of fuzzy delayed renewal process and relationship between the interarrival times and fuzzy renewal variable were presented. Moreover concentrated on the expected value of renewals, the fuzzy form of elementary renewal theorem is established.
TL;DR: In this article, the authors considered the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, a], defined by some stopping time a, in the case in which the randomly stopped sum is negative.
Abstract: Let (Yn, Nn)n>l be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean v and the Y, have a common heavy-tailed distribution F. We consider the process (Zn)n>1 defined by Zn = Yn - En-1, where En-1 = Ek-I Nk. It is shown that the probability that the maximum M = maxn>l Zn exceeds x is approximately v-1 fL F(u) du, as x -> o , where F := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, a], defined by some stopping time a, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.
Journal Article•
TL;DR: In this paper, a distribution of claim numbers in insurance has been introduced, which include two parameters and is a generalization of Poisson's It can be used to describe an extra zero claim numbers, so it is significant to apply the distribution in insurance practice based on this distribution, a stochastic counting process with independent and stationary increments can be generated in our paper.
Abstract: A distribution of claim numbers in insurance has been introduced, which include two parameters and is a generalization of Poisson's It can be used to describe an extra zero claim numbers, so it is significant to apply the distribution in insurance practice Based on this distribution, a stochastic counting process with independent and stationary increments can be generated in our paper A risk model with the counting process has been studied The ruin probability formula and the renewal equation have been given Some important results on classical risk model with poisson process are the special cases in this paper
Dissertation•
02 Dec 2005
TL;DR: Jacopino et al. as mentioned in this paper proposed a Bayesian solution to the problem of estimating the reliability of a repairable component based on incomplete inspection and maintenance data, which can provide insights into the relative maintenance effectiveness and the merit of optimising imperfect maintenance or inspection.
Abstract: Title: GENERALISATION AND BAYESIAN SOLUTION OF THE GENERAL RENEWAL PROCESS FOR MODELLING THE RELIABILITY EFFECTS OF IMPERFECT INSPECTION AND MAINTENANCE BASED ON IMPRECISE DATA Andrew Guiseppe Jacopino, Doctor of Philosophy (Ph.D.) in Reliability Engineering, 2005 Directed By: Professor Ali Mosleh, Department of Mechanical Engineering Common Stochastic Point Processes used in the analysis of Repairable Systems do not accurately represent the true life of a repairable component mainly due to the underlying repair assumption. Specifically, the Ordinary Renewal Process uses an asgood-as-new repair assumption while the Non-Homogenous Poisson Process uses an as-bad-as-old repair assumption. However, it is highly unlikely that any repairable system will readily fit into either repair assumption. Additionally, there is the possibility that any inspection or maintenance activity may actually worsen the system; worse-than-old. Regardless of the underlying repair assumptions and the limitation they impose on any solution, these point processes continue to be used to assist engineering and logistic decision making. While other solutions, mainly GRP based, have offered some resolution, no solution has sufficiently resolved the combined complexities of imperfect maintenance of multiple dependent failure modes, imperfect inspections and data uncertainty, specifically unknown times to failure. Accordingly, the solution offered here offers a model that can contend with all these factors through a Bayesian solution thereby allowing additional “soft-data” to be utilised during the analysis. The modelling scheme consisted of 10 cases divided into 2 main types; Type I with known failure times, and Type II with unknown failure times (data uncertainty). Each of the cases are incrementally modified through the addition of factors including imperfect maintenance of a single failure mode through to multiple dependent failure modes, and finally imperfect inspection. Generalisation of the GRP equations and Bayesian estimation models were developed for these cases. As a closed form solution to each of these cases is unavailable, numerical procedures were formulated. Specifically, an alternative Markov Chain sampling methods, Slice Sampling, was utilised to solve the Bayesian implementation of the needed extensions to the KIJIMA Type I GRP model with an underlying 2-parameter Weibull Time-To-Failure distribution. Based on a number of examples the resulting models have shown the ability to accurately predict future failure trends. Furthermore, the model provides a number of insights into the results including relative maintenance effectiveness and the merit of optimising imperfect maintenance or inspection to maximise availability. GENERALISATION AND BAYESIAN SOLUTION OF THE GENERAL RENEWAL PROCESS FOR MODELLING THE RELIABILITY EFFECTS OF IMPERFECT INSPECTION AND MAINTENANCE BASED ON IMPRECISE DATA
TL;DR: A production-inventory system in which an unreliable machine is susceptible to failure following which it must be repaired to make it operative again, and its steady-state distribution is obtained which is used to calculate some performance measures of the system.