Showing papers on "Renewal theory published in 1999"
Book•
01 Jan 1999
TL;DR: In this paper, the authors introduce Markov Chains, Martingales, Poisson Processes, and Renewal Theory, as well as Brownian Motion, and conclude that renewal theory can be viewed as a form of Markov chains.
Abstract: 1. Markov Chains 2. Martingales 3. Poisson Processes 4. Markov Chains 5. Renewal Theory 6. Brownian Motion
286 citations
TL;DR: In this article, the authors derived a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of Kalikow.
Abstract: We derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of Kalikow. The approach is based on the existence of a renewal structure under an assumption of “transience in the direction $l$ .” This extends, to a multidimensional context, previous work of Kesten. Our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution.
252 citations
TL;DR: In this paper, an estimator for the spectral measure associated with solutions of stochastic recurrence equations (SREs) was proposed and proved its consistency, which is the tail empirical measure of multivariate time series.
152 citations
TL;DR: This work analyzes an (s, S) continuous review perishable inventory system with a general renewal demand process and instantaneous replenishments, and obtains closed-form solutions for the steady state probability distribution of the inventory level and system performance measures.
Abstract: We analyze an (s, S) continuous review perishable inventory system with a general renewal demand process and instantaneous replenishments. Though continuous review systems seem more amenable to optimization analysis than do periodic review systems, the existing literature addressing this type of model is rather limited. This limitation motivated us to seek greater understanding of this important class of inventory models. Using a Markov renewal approach, we obtain closed-form solutions for the steady state probability distribution of the inventory level and system performance measures. We then construct a closed-form expected cost function. Useful analytical properties for the cost function are identified and extensive computations are conducted to examine the impact of different parameters. The numerical results illustrate the system behavior and lead to managerial insights into controlling such inventory systems.
113 citations
TL;DR: In this article, a renewal process is fitted to paleoearthquake occurrence data, and the posterior of the selected model provides a reasonable estimate of distribution of uncertain occurrence times associated with the fitted renewal process model.
Abstract: When a renewal process is fitted to paleoearthquake occurrence data, it is more informative to use the data including the two interval ends before the first event to the beginning of observation period and after the last event till the present time because the sample size is usually small. Our main concern is analysis of the data in which some occurrence times of events themselves are uncertain and given by intervals or some distributions. Specifically, we consider a Bayesian inference where each uncertainty is interpreted as a prior distribution associated with likelihood of a renewal process model. Integration of the posterior with respect to the occurrence times and its maximization with respect to the parameters of the renewal process model are implemented in order to estimate the parameters and further to compare the goodness of fit of other competing renewal process models. Thus the posterior of the selected model provides a reasonable estimate of distribution of uncertain occurrence times associated with the fitted renewal process model. Finally, in a case where occurrence time of the last event is uncertain, a practical method for the assessment of current and future hazard of the forthcoming rupture is provided.
84 citations
TL;DR: A theorem which presents the long-run average fuzzy reward per unit time is stated and a procedure to obtain the best T-age replacement policy with fuzzy cost structure is developed.
62 citations
TL;DR: A generalized replacement model where a deteriorating system has two types of failures and is replaced at the Nth type I failure or first type II failure or at the working age T, whichever occurs first, is proposed.
61 citations
TL;DR: In this paper, a reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights.
Abstract: A reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights
53 citations
TL;DR: In this paper, the authors give a probabilistic analogue of the mean value theorem, which is shown to be useful in various contexts of reliability theory, and provide various applications to evaluate the mean total profits of devices having random lifetimes.
Abstract: In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whitt [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.
44 citations
TL;DR: In this article, the authors conjecture the limit distributions of the "understandingshoot" and "overshoot" at the passage of a high level by subordinators, and then prove these conjectures by Levy-process methods.
44 citations
01 Jan 1999
TL;DR: In this article, the authors consider the class of so-called trend-renewal processes (TRP), which have both the NHPP and the renewal process as special cases.
Abstract: A repairable system can briefly be characterized as a system which is repaired rather than replaced after a failure. The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes (NHPP), corresponding to minimal repairs, and renewal processes (RP), corresponding to perfect repairs. The paper reviews models for more general repair actions, often called “better-than-minimal repair” models. In particular we study the class of so called trend-renewal processes (TRP), which has both the NHPP and the RP as special cases. Parametric inference in TRP models is considered, including cases with several systems involving unobserved heterogeneity. Trend testing is discussed when the null hypothesis is that the failure process is an RP. It is shown how Monte Carlo trend tests for this case can be made from the commonly used trend tests for the null hypothesis of a homogeneous Poisson process (e.g. the Laplace test and the Military Handbook test). Simulations show that the Monte Carlo tests have favorable properties when the sample sizes are not too small.
TL;DR: This work measures the effectiveness of a repairable system by the proportion of time the system is on, where on-time and off-times are assumed independent and both gamma-distributed.
Abstract: We measure the effectiveness of a repairable system by the proportion of time the system is on, where on-time and off-times are assumed independent and both gamma-distributed. This measure is helpful for system planning and control in the short term, before the steady-state is reached, and its mean value is intermediary between instantaneous and steady-state availabilities. We also present other significant results concerning the Gamma Alternating Renewal Process. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 822–844, 1999
TL;DR: A model based on a two-state mixed renewal process and its use for modelling disease activity in studies of chronic conditions and stratified analyses in which strata are defined by the number of past exacerbations is proposed.
Abstract: We develop a model based on a two-state mixed renewal process and propose its use for modelling disease activity in studies of chronic conditions. We specify Weibull forms for the conditional transition intensities to allow for time trends, and bivariate frailties to accommodate subject-to-subject variability in the disease process. Extensions of this model are considered for stratified analyses in which strata are defined by the number of past exacerbations. Data from a motivating study of chronic bronchitis are analysed to illustrate the methodology.
TL;DR: This work considers compound processes that are linear with constant slope between i.i.d. jumps at time points forming a renewal process in queueing, dam and risk theory and derives the distribution of the first crossing time of a prespecified level.
Abstract: We consider compound processes that are linear with constant slope between i.i.d. jumps at time points forming a renewal process. These processes are basic in queueing, dam and risk theory. For positive and for negative slope we derive the distribution of the first crossing time of a prespecified level. The related problem of busy periods of single-server queueing systems is also studied.
TL;DR: In this paper, the authors consider a class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewing process possesses some block-structured property, including repeating rows.
Abstract: In this paper, we consider a certain class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewal process possesses some block-structured property, including repeating rows. Duality conditions and properties are obtained on two probabilistic measures which often play a key role in the analysis and computations of such a block-structured process. The method used here unifies two different concepts of duality. Applications of duality are also provided, including a characteristic theorem concerning recurrence and transience of a transition matrix with repeating rows and a batch arrival queueing model.
TL;DR: In this paper, the distribution of the first time a prespecified level is reached or exceeded, and of the position at this time is derived for a process that increases linearly, with unit slope, between jumps of i.i.d. positive sizes occurring at renewal times.
Abstract: For a process that increases linearly, with unit slope, between jumps of i.i.d. positive sizes occurring at renewal times, we present methods to compute the distribution of the first time a prespecified level is reached or exceeded, and of the position at this time. In the exponential case the Laplace transform of this first-exit time is derived in closed form. A general formula for the distribution of the stopping time is given, and shown to yield explicit results in certain cases. An effective method of successive approximation is also derived. The problem is equivalent to that of determining the distribution of the total ON time in [0, t] of a system changing between the states ON and OFF at the points of an alternating renewal process.
TL;DR: In this article, generalized bounds for the ruin probability under a Markovian modulated risk model were developed for the ladder height distribution and a numerical iteration method was proposed for obtaining tightened monotone bounds by using the developed lower and upper bounds.
Abstract: In this paper, generalized bounds are developed for the ruin probability under a Markovian modulated risk model. First, lower and upper bounds in terms of NBU and NWU type distributions or functions are obtained which includes the exponential bound as a special case when the Cramer Condition is satisfied. The technique used here is a combination of the conjugation idea, the Markov renewal theorem and the induction method. Then, a numerical iteration method is proposed for obtaining tightened monotone bounds by using the developed lower and upper bounds. Finally, improved bounds are obtained in terms of the ladder height distribution and an example is used for illustration
TL;DR: In this article, the integral equation for the multiple availability for arbitrary Cdf's of periods of operation and repair is developed for a nonhomogeneous Poisson point process of demand.
Abstract: Stochastic models for multiple availability are analyzed for a system with periods of operation and repair that form an alternating process. The system is defined as available in time interval (0, T] if it is available at each moment of demand. System unavailability at the moment of demand is called a breakdown. The approximate probability of functioning without breakdowns is derived and analyzed for the nonhomogeneous Poisson point process of demand. Specific cases, which can be of interest in practical applications, are investigated. The integral equation for the multiple availability for arbitrary Cdf's of periods of operation and repair is developed.
TL;DR: In this paper, the authors derive convergence rates for Markov processes associated with a renewal process with a common inter-renewal time distribution, based on recent results on the exploitation of "drift criteria" for general state-space Markov process.
TL;DR: In this paper, an appropriate stopping rule is used to determine the sample size when estimating the parameters in a stationary and ergodic threshold AR(1) model, and the sequential least-squares estimator is asymptotically risk efficient.
TL;DR: In this article, an exact expression for the mean rate of coincidence registration is developed using techniques from renewal theory, and it is shown that the traditional approximate rate, in certain situations, leads to the overestimation of the actual rate.
TL;DR: It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.
Abstract: In this paper we consider the determination of the reorder point s in an (R, s, Q) inventory model subject to a fill rate service level constraint. We assume that the underlying demand process is a compound renewal process. We then derive an approximation method to compute the reorder level such that a target service level is achieved. Restrictions on the input parameters are given, within which this method is applicable. Moreover, we will investigate the effects on the fill rate performance in case the underlying demand process is indeed a compound renewal process, while the demand process is modelled as a discrete-time demand process. That is, the time axis is divided in time units (for example, days) and demands per time unit are independent and identically distributed random variables. It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.
TL;DR: In this article, a simple algorithm for computing the distribution of the residual life when the renewal process is discrete was developed, which does not require convolutions and can be applied to an inventory problem.
Abstract: We develop a simple algorithm, which does not require convolutions, for computing the distribution of the residual life when the renewal process is discrete. We also analyze the algorithm for the particular case of lattice distributions, and we show how it can apply to an inventory problem. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 435–443, 1999
TL;DR: The joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is only slightly less than the service rates, is analyzed.
Abstract: We consider two tandem queues with exponential servers. Arrivals to the first queue are governed by a general renewal process. If the arrivals were also exponentially distributed, this would be a simple example of a Jackson network. However, the structure of the model is much more complicated for general arrivals. We analyze the joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is only slightly less than the service rates. We formulate and solve the boundary value problem for the diffusion approximation to this model. We obtain simple integral representations for the (asymptotic) steady-state queue length distribution.
TL;DR: In this paper, the higher-order version of the mean remaining lifetime in an equilibrium renewal process is defined and connections are made with the decreasing variance residual lifetime class and stochastic ordering.
Abstract: Bondesson's functions in reliability theory are shown to be related to a recursive sequence of probability distributions. These are the 'higher-order' versions of the mean remaining lifetime in an equilibrium renewal process. Based on these functions, classes of distribution functions can be defined. This paper will investigate these classes and place Bondesson's work in the content of the other work done in reliability theory. Connections are made with the decreasing variance residual lifetime class and stochastic ordering.
TL;DR: In this article, a generalization of the alternating renewal process is presented, where the random walk is driven by a Harris recurrent Markov chain, and some interesting applications are given.
Abstract: Some results for stopped random walks are extended to the Markov renewal setup where the random walk is driven by a Harris recurrent Markov chain. Some interesting applications are given; for example, a generalization of the alternating renewal process.
TL;DR: In this paper, the authors present an analytic method which gives upper and lower bounds for the reliability of systems subject to inspections at Poisson random times, and derive the asymptotic failure rate.
Abstract: For systems subject to inspections at Poisson random times, we present an analytic method which gives upper and lower bounds for the reliability. We also study its asymptotic behaviour and derive the asymptotic failure rate.
01 Jan 1999
TL;DR: In this article, an exact expression for the mean rate of coincidence registration is developed using techniques from renewal theory, and it is shown that the traditional approximate rate, in certain situations, leads to the overestimation of the actual rate.
Abstract: The statistics of photon coincidence counting in photon-correlated beams is thoroughly investigated considering the effect of the finite coincidence resolving time. The correlated beams are assumed to be generated using parametric downconver- sion, and the photon streams in the correlated beams are modeled by two partially correlated Poisson point processes. An exact expression for the mean rate of coincidence registration is developed using techniques from renewal theory. It is shown that the use of the traditional approximate rate, in certain situations, leads to the overestimation of the actual rate. The error between the exact and approximate coincidence rates increases as the coincidence-noise parameter, defined as the mean number of uncorrelated photons detected per coincidence resolving time, increases. The use of the exact statistics of the coincidence becomes crucial when the background noise is high or in cases when high precision measurement of coincidence is required. Such cases arise whenever the coincidence-noise parameter is even slightly in excess of zero. It is also shown that the probability distribution function of the time between consecutive coincidence registration can be well approximated by an exponential distribution function. The well-known and experimentally verified Poissonian model of the coincidence registration process is therefore theoretically justified. The theory is applied to an on-off keying communication system proposed by Mandel which has been shown to perform well in extremely noisy conditions. It is shown that the . bit-error rate BER predicted by the approximate coincidence-rate theory can be significantly lower than the actual BER obtained using the exact theory. q 1999 Elsevier Science B.V. All rights reserved.
TL;DR: In the population coding framework, how the response distributions affect output distribution is considered and a general theory for the output of neuronal population code is presented when the spike train is a renewal process.
Abstract: In the population coding framework, we consider how the response distributions affect output distribution. A general theory for the output of neuronal population code is presented when the spike train is a renewal process. Under a given condition on the response distribution, the most probable value of the output distribution is the center of input-preferred values, whereas in the other cases the most improbable value of the output distribution is the center of input-preferred values or there are no most probable states. Depending on the exact form of the response distributions, the variance of the output distributions can either enlarge or reduce the tuning width of the tuning curves.
TL;DR: In this article, the authors considered two tandem queues with exponential servers and formulated and solved the boundary value problem for the diffusion approximation to this model, and obtained simple integral representations for the (asymptotic) steady-state queue length distribution.
Abstract: We consider two tandem queues with exponential servers. Arrivals to the first queue are governed by a general renewal process. If the arrivals were also exponentially distributed, this would be a simple example of a Jackson network. However, the structure of the model is much more complicated for general arrivals. We analyze the joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is only slightly less than the service rates. We formulate and solve the boundary value problem for the diffusion approximation to this model. We obtain simple integral representations for the (asymptotic) steady-state queue length distribution. We also do a detailed study of the tail behavior of the diffusion approximation