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Showing papers on "Renewal theory published in 1988"


Book
01 Jan 1988
TL;DR: In this paper, limit theorems for stopping random walks with positive drift have been studied in Probability Theory and Regularly Varying Functions (RVF) theory.
Abstract: Preface- Notations and Symbols- Introduction- Limit Theorems for Stopped Random Walks- Renewal Processes and Random Walks- Renewal Theory for Random Walks with Positive Drift- Generalizations and Extensions- Functional Limit Theorems- Perturbed Random Walks- Appendix A: Some Facts from Probability Theory- Appendix B: Some Facts about Regularly Varying Functions- Bibliography- Index

351 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduced the geometric process which is a sequence of independent nonnegative random variables, such that the distribution function of a random variable X n is F (a n−1 −1 normalized x), wherea is a positive constant, and the explicit expressions of the long-run average costs per unit time under each replacement policy are calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.
Abstract: In this paper, we introduce and study the geometric process which is a sequence of independent non-negative random variablesX 1,X 2,... such that the distribution function ofX n isF (a n−1 x), wherea is a positive constant. Ifa>1, then it is a decreasing geometric process, ifa<1, it is an increasing geometric process. Then, we consider a replacement model as follows: the successive survival times of the system after repair form a decreasing geometric process or a renewal process while the consecutive repair times of the system constitute an increasing geometric process or a renewal process. Besides the replacement policy based on the working age of the system, a new kind of replacement policy which is determined by the number of failures is considered. The explicit expressions of the long-run average costs per unit time under each replacement policy are then calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.

266 citations


Book
25 Aug 1988
TL;DR: A Poisson model of equipment wearout of reactor safety studies, and the application of point processes to a theory of safety assessment.
Abstract: 1 Introduction.- 1.1 Arrivals in time.- 1.2 Reliability.- 1.3 Safety assessment.- 1.4 Random stress and strength.- Notes on the literature.- Problems.- 2 Point processes.- 2.1 The probabilistic context.- 2.2 Two methods of representation.- 2.3 Parameters of point processes.- 2.4 Transformation to a process with constant arrival rate.- 2.5 Time between arrivals.- Notes on the literature.- Problems.- 3 Homogeneous Poisson processes.- 3.1 Definition.- 3.2 Characterization.- 3.3 Time between arrivals for the hP process.- 3.4 Relations to the uniform distribution.- 3.5 A process with simultaneous arrivals.- Notes on the literature.- Problems.- 4 Application of point processes to a theory of safety assessment.- 4.1 The Reactor Safety Study.- 4.2 The annual probability of a reactor accident.- 4.3 A stochastic consequence model.- 4.4 A concept of rare events.- 4.5 Common mode failures.- 4.6 Conclusion.- Notes on the literature.- Problems.- 5 Renewal processes.- 5.1 Probabilistic theory.- 5.2 The renewal process cannot model equipment wearout.- Notes on the literature.- Problems.- 6 Poisson processes.- 6.1 The Poisson model.- 6.2 Characterization of regular Poisson processes.- 6.3 Time between arrivals for Poisson processes.- 6.4 Further observations on software error detection.- Notes on the literature.- Problems.- 7 Superimposed processes.- Notes on the literature.- Problems.- 8 Markov point processes.- 8.1 Theory.- 8.2 The Poisson process.- 8.3 Facilitation and hindrance.- Notes on the literature.- Problems.- 9 Applications of Markov point processes.- 9.1 Egg-laying dispersal of the bean weevil.- 9.2 Application of facilitation - hindrance to the spatial distribution of benthic invertebrates.- 9.3 The Luria-Delbruck model.- 9.4 Chance placement of balls in cells.- 9.5 A model for multiple vehicle automobile accidents.- 9.6 Engels' model.- Notes on the literature.- Problems.- 10 The order statistics process.- 10.1 The sampling of lifetimes.- 10.2 Derivation from the Poisson process.- 10.3 A Poisson model of equipment wearout.- Notes on the literature.- Problems.- 11 Competing risk theory.- 11.1 Markov chain model.- 11.2 Classical competing risks.- 11.3 Competing risk presentation of reactor safety studies.- 11.4 Delayed fatalities.- 11.5 Proportional hazard rates.- Notes on the literature.- Problems.- Further reading.- Appendix 1 Probability background.- A1.1 Probability distributions.- A1.2 Expectation.- A1.3 Transformation of variables.- A1.4 The distribution of order statistics.- A1.5 Conditional probability.- A1.6 Operational methods in probability.- A1.7 Convergence concepts and results in the theory of probability.- Notes on the literature.- Appendix 2 Technical topics.- A2.1 Existence of point process parameters.- A2.2 No simultaneous arrivals.- Solutions to a few of the problems.- References.- Author index.

106 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used renewal theory to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non-Markovian) fluid chord length distributions.
Abstract: Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model, as well as the simpler model involving the effective cross sections and source.

94 citations


01 Jan 1988
TL;DR: In this paper, the state of the art Laplace transforms are used for explaining probability, statistics, and special math needed to use results, in the context of renewal theory and reliability analysis.
Abstract: Purpose: Widen state of art Special math needed for explanations: Probability, statistics, Special math needed to use results: Probability and statistics Results useful to: Reliability analysts and theoreticians Laplace transforms, renewal theory

84 citations


Journal ArticleDOI
TL;DR: A framework which leads to expressing the expected costs in terms of complex functions arising in renewal theory is reviewed and several approximations are discussed, finding that a straight line approximation performs well.
Abstract: In this paper, we survey a portion of the literature in the analysis of warranty costs. In order to compare the price of different warranty policies, we review a framework which leads to expressing the expected costs in terms of complex functions arising in renewal theory. These functions and several approximations are discussed. In particular, a straight line approximation performs well.

43 citations


Journal ArticleDOI
TL;DR: In this paper, renewal processes are proposed to describe any statistically homogeneous and isotropic, two-component, random medium with structure, and an effective opacity is calculated in the diffusion limit as a function of the chord-length distributions through the grains of matter.
Abstract: As a generalization of Markov processes, renewal processes are proposed to describe any statistically homogeneous and isotropic, two-component, random medium with structure. An effective opacity is calculated in the diffusion limit as a function of the chord-length distributions through the grains of matter.

36 citations


Journal ArticleDOI
TL;DR: In this article, a procedure for computing the renewal function, the renewal density, the integral of the renewal functions, and the variance function of phase-type renewal processes is presented, based on the computation of the state probability vector of a continuous-time Markov chain.
Abstract: This article presents a procedure for computing the renewal function, the renewal density, the integral of the renewal function, and the variance function of phase-type renewal processes. The procedure hinges on the computation of the state probability vector of a continuous-time Markov chain. This is accomplished by using a randomization approach that is simple, efficient, and numerically stable and does not require numerical integration. I discuss approximating arbitrary interrenewal distributions by phase-type distributions so that the procedure can be used to approximate renewal and related functions.

35 citations


Journal ArticleDOI
TL;DR: In this paper, the distribution of the excess over the boundary, the expected stopping time $ET$ and the variance of the stopping time $\operatorname{Var}(T)$ were obtained by using linear renewal theorems with varying drift.
Abstract: Let $T$ be the first time that a perturbed random walk crosses a nonlinear boundary. This paper concerns the approximations of the distribution of the excess over the boundary, the expected stopping time $ET$ and the variance of the stopping time $\operatorname{Var}(T)$. Expansions are obtained by using linear renewal theorems with varying drift.

32 citations



Journal ArticleDOI
TL;DR: Some illustrative applications of semi-Markov processes in biostatistics, demography, and queuing theory are discussed, with suggestions for statistical inference based on models constructed from renewal theory and semi- Markov processes.
Abstract: Some illustrative applications of semi-Markov processes in biostatistics, demography, and queuing theory are discussed. Algorithms for implementing such processes on a computer are also described, with suggestions for statistical inference based on models constructed from renewal theory and semi-Markov processes. An illustrative numerical example, based on a simple illness-death process of Fix and Neyman, is also provided.

Journal ArticleDOI
TL;DR: In this article, it was shown that the gamma distribution with shape parameter a can be obtained through a p-thinning for every 0 1, while for any ε > 0, the distribution cannot be obtained by thinning.
Abstract: It is shown that the gamma distribution with shape parameter a can be obtained through a p-thinning for every 0 1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0 < a < 1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.

Journal ArticleDOI
TL;DR: In this paper, the renewal function is used to estimate the expected number of renewals for a random sample of size n. The problem of estimating the renewal cost can be reduced to estimating these functions.
Abstract: The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.

Journal ArticleDOI
TL;DR: In this paper, the class of inverses of a p-thinned renewal process is considered, and it is shown that this class consists of Cox and renewal processes if and only if the given thinned process is Cox and renewing.
Abstract: The class of inverses of a p-thinned renewal process is considered. It is shown that this class consists of renewal processes. It consists of Cox and renewal processes if and only if the given thinned process is Cox and renewal. In the non-Cox case, there exists a unique top renewal process, which by thinning generates all the possible inverses. Conditions for a renewal process to be a top process are given. Finally, a gamma renewal process is shown to be a top process when a > 1, where a is the shape parameter of the gamma distribution. POINT PROCESS; THINNED POINT PROCESS; INVERSE THINNING; COX PROCESS; GAMMA RENEWAL PROCESS

Journal ArticleDOI
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.

Book ChapterDOI
Allan Gut1
01 Jan 1988
TL;DR: In this paper, the authors consider the first passage times across horizontal levels, where the zero level corresponds to the first ascending ladder epoch, and the renewal counting process generates the time points generated by renewal counting processes.
Abstract: Classical limit theorems such as the law of large numbers, the central limit theorem and the law of the iterated logarithm are statements concerning sums of independent and identically distributed random variables, and thus, statements concerning random walks. Frequently, however, one considers random walks evaluated after a random number of steps. In sequential analysis, for example, one considers the time points when the random walk leaves some given finite interval. In renewal theory one considers the time points generated by the so called renewal counting process. For random walks on the whole real line one studies the first passage times across horizontal levels, where, in particular, the zero level corresponds to the first ascending ladder epoch. In reliability theory one may, for example, be interested in the total cost for the replacements made during a fixed time interval and so on.

Journal ArticleDOI
TL;DR: Renewal theory is used to develop the probability of successfully transmitting variable-length packets in a finite-user unslotted ALOHA channel operating in an satellite network of very small aperture terminals (VSAT).
Abstract: Renewal theory is used to develop the probability of successfully transmitting variable-length packets in a finite-user unslotted ALOHA channel operating in an satellite network of very small aperture terminals (VSAT). This result, which does not depend on the arrival distribution, is then used to determine the equilibrium or operating point from which the normalized throughput, traffic and delay are estimated. >

Journal ArticleDOI
TL;DR: In this paper, the Alternating Renewal Process is employed for the evaluation of probability distribution functions for total wet and dry periods over a homogeneous time interval (0, t).
Abstract: The paper is concerned with the modelling of rainfall occurrence in continuous time. The Alternating Renewal Process is employed for the evaluation of probability distribution functions for total wet and dry periods over a homogeneous time interval (0, t). The derived general solution is simplified by assuming that the individual wet and dry intervals are random variables following an Erlang distribution, in particular an exponential distribution. Data on a continuous time scale from the Mikra Station in Greece are used to illustrate the proposed methodology.

Journal ArticleDOI
01 May 1988
TL;DR: An approximation method is derived to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption and it is shown that the renewal property is also given for this general class of Interrupted Poison processes having generally distributed off-phase.
Abstract: Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.

Journal ArticleDOI
TL;DR: In this paper, Einmahl et al. developed corresponding invariance principles for associated renewal processes and random sums and proved the optimality of the approximation in the case when only two moments exist.

Journal ArticleDOI
Jinhua Cao1
TL;DR: In this paper, the authors considered a repairable system in a changing environment subject to a general alternating renewal process and used Markov renewal theory to obtain the system availability, failure frequency and reliability function.

Journal ArticleDOI
TL;DR: In this paper, the authors generalize the inspection paradox, show a stochastic ordering of X1, XN, and then interpret and consider the ramifications of this stochastically ordering.

Journal ArticleDOI
TL;DR: An approximate evaluation is proposed for the individual mean waiting time, and loss probability for mixed delay and loss (nondelay) systems with renewal and Poisson inputs handled by servers with exponential service time.
Abstract: An approximate evaluation is proposed for the individual mean waiting time, and loss probability for mixed delay and loss (nondelay) systems with renewal and Poisson inputs handled by servers with exponential service time. The approximation is based on the GI approximation previously proposed by H. Akimaru, et al. (1983, 1985), in which the mixed input process is regarded as renewal. The systems with mixed delay renewal and nondelay Poisson inputs, and ones with mixed nondelay renewal and delay Poisson inputs are analyzed. Approximate formulas for the mean waiting time and loss probability for the respective inputs are presented in simple closed form, and comparisons to simulations show good accuracy. The formulas are expected to be useful for analysis and optimum design of the mixed delay and loss systems. >

Journal ArticleDOI
TL;DR: In this paper, three term asymptotic expansions for R far from the origin were obtained for distributions in linear boundary crossing problems, where the renewal measure R = ΣP*n where n denotes convolution.
Abstract: Let P be a distribution in the plane and define the renewal measure R=ΣP*nwhere * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.

Journal ArticleDOI
TL;DR: In this paper, the inverse problem for thinned renewal processes was shown to be equivalent to inverse problems for thinning renewal processes, and applications to gamma renewal processes were also discussed.
Abstract: The following problem is considered: for which p ∈ (0, 1) and completely monotone functions g is g/[p+(1-p)g] completely monotone? This problem is shown to be equivalent to the inverse problem for thinned renewal processes. Some applications to gamma renewal processes are also discussed.

Journal ArticleDOI
TL;DR: On applique certains concepts de la theorie du renouvellement a la recherche par enquete avec une reference particuliere aux questions retrospectives as discussed by the authors.
Abstract: On applique certains concepts de la theorie du renouvellement a la recherche par enquete avec une reference particuliere aux questions retrospectives

Book ChapterDOI
TL;DR: In this paper, the authors considered the behavior of the renewal process up to a given time t > 0 or up to S n = s, where s is the number of interarrivals.

Journal ArticleDOI
TL;DR: In this paper, the optimality of certain approximation rates appearing in strong invariance principles for partial sums indexed by a renewal process is discussed, and the results extend and unify earlier work on the best rates in the invariance principle for renewal counting processes.

Book ChapterDOI
Allan Gut1
01 Jan 1988
TL;DR: In this paper, the authors present a survey of the basic facts about random walks and prove various limit theorems for stopping random walks, which can be used in order to obtain results for random walks stopped according to specific stopping procedures as well as for the families of stopping times (random indices) themselves.
Abstract: In the first chapter we stated and proved various limit theorems for stopped random walks. These limit theorems shall, in subsequent chapters, be used in order to obtain results for random walks stopped according to specific stopping procedures as well as for the families of stopping times (random indices) themselves. However, before doing so we shall, in this chapter, survey some of the basic facts about random walks.

Proceedings ArticleDOI
07 Dec 1988
TL;DR: In this paper, a class of cyclic hybrid-state stochastic processes arising in the modeling of thermostat-controlled electric power system loads, and particularly important in load management applications, are discussed.
Abstract: Summary form only given. The author reviews a class of cyclic hybrid-state stochastic processes arising in the modeling of thermostat-controlled electric power system loads, and particularly important in load management applications. He shows that there are two distinct viewpoints for analyzing the statistical properties of these processes: an internal viewpoint and an external one. The internal viewpoint, where the full state is considered, yields the complete statistical picture. It is possible to establish an associated system of coupled forward Kolmogorov equations, together with appropriate boundary conditions. In general, this system of equations is difficult to analyze. By contrast, in the external viewpoint the focus is on the discrete part of the process only. It is then possible to uncover an underlying renewal structure. This renewal structure is useful in a rapid establishment of the existence of a statistical steady state as well as its precise determination. Partial transient results can also be obtained in this manner. The two analytical viewpoints usefully complement one another. >