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Showing papers on "Renewal theory published in 2016"


Journal ArticleDOI
TL;DR: In this article, a new class of time inhomogeneous Polya-type urn schemes and optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters were studied, a class which includes the Rayleigh, half-normal and gamma distributions.
Abstract: We study a new class of time inhomogeneous Polya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.

42 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sublinear expectations initiated by Peng.
Abstract: We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.

37 citations


Journal ArticleDOI
TL;DR: Two EM algorithms are introduced: the first extends the existing EM algorithm for the Hawkes process to consider renewal immigration, and the second reduces the amount of missing data, considering only if a point is an immigrant or not as missing data.

36 citations


Journal ArticleDOI
TL;DR: A new TBE control chart is introduced, based on the renewal process, where the distribution of the TBE belongs to a parametric class of absolutely continuous distributions, which includes some well-known and commonly used lifetime distributions, i.e., exponential, Rayleigh, Weibull, Burr type XII, Pareto and Gompertz.

29 citations


Journal ArticleDOI
TL;DR: In this article, the authors present an abstract framework that allows to obtain mixing (and in some cases sharp mixing) rates for a reasonable large class of invertible systems preserving an infinite measure.
Abstract: In this work, we present an abstract framework that allows to obtain mixing (and in some cases sharp mixing) rates for a reasonable large class of invertible systems preserving an infinite measure. The examples explicitly considered are the invertible analogue of both Markov and non-Markov unit interval maps. For these examples, in addition to optimal results on mixing and rates of mixing in the infinite case, we obtain results on the decay of correlation in the finite case of invertible non-Markov maps, which, to our knowledge, were not previously addressed. The proposed method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig (Invent Math 150:629–653, 2002), with the framework of function spaces of distributions developed in the recent years along the lines of Blank et al. (Nonlinearity 15:1905–1973, 2001).

27 citations


Journal ArticleDOI
TL;DR: An uncertain random block replacement problem is proposed and an unconstrained optimization model is formulates by using the uncertain random renewal reward process, in which the interarrival times and the rewards are assumed to be random variables and uncertain variables, respectively.
Abstract: As a mixture of uncertain variable and random variable, uncertain random variable is an important tool to describe indeterminacy phenomena. In order to model the evolution of uncertain random phenomena, a concept of uncertain random process has been proposed, and an uncertain random renewal process has been designed as an example. This paper aims to propose a new type of uncertain random process, called uncertain random renewal reward process, in which the interarrival times and the rewards are assumed to be random variables and uncertain variables, respectively. The chance distribution of the renewal reward process is obtained, and the reward rate is derived. A renewal reward theorem is verified, which shows that the reward rate converges in distribution to an uncertain variable derived from the random interarrival times and the uncertain rewards. As an application, this paper also proposes an uncertain random block replacement problem and formulates an unconstrained optimization model by using the uncertain random renewal reward process.

22 citations


Journal ArticleDOI
TL;DR: In this article, an optimization model for scheduling railway ballast, rail, and sleeper renewal operations at a network level is presented to minimize the expected railway track life-cycle cost (LCC) and track unavailability costs derived from user impacts caused by traffic disruption during track renewal operations.
Abstract: This paper presents an optimization model for scheduling railway ballast, rail, and sleeper renewal operations at a network level. The objective of the model is to minimize the expected railway track life-cycle cost (LCC) and track unavailability costs derived from user impacts caused by traffic disruption during track renewal operations. To minimize costs, the model assesses the opportunistic renewal of railway track components and takes advantage of planning from a network perspective to study the possibility of reusing track components on different lines. The practical utility of the model is illustrated with a case study involving the Portuguese railway network. The results indicate that user costs have an important influence on decision making in the track renewal process and that the network perspective of renewal planning can reduce the direct costs of these operations.

18 citations


Journal ArticleDOI
TL;DR: Borovkov and Mogul'skii as mentioned in this paper obtained the first partial local large deviation principle for the trajectories of a compound renewal process under some condition on the distribution of the process.
Abstract: The present paper continues studies of large deviation principles for compound renewal processes that were started in [A. A. Borovkov, Asymptotic Analysis of Random Walking. Fast Decreasing Increment Distributions, Fizmatlit, Moscow, 2013 (in Russian)], [A. A. Borovkov and A. A. Mogul'skii, Siberian Math. J., 56 (2015), pp. 28--53]. The main subject of this research is probabilities of large deviations of the trajectories of compound renewal processes. The paper consists of two parts. In part I, under some condition on the distribution of the process, we obtain the so-called first partial local large deviation principle for the trajectories of a compound renewal process. In part II, under an alternative condition, we obtain the second partial local large deviation principle. Under additional conditions, we also obtain the “total local” and “total integral” large deviation principles for compound renewal processes.

17 citations


Journal ArticleDOI
TL;DR: In this paper, the authors track the sign of fluctuations governed by the Kardar-Parisi-Zhang (KPZ) universality class and find an unexpected link to a simple stochastic model called the renewal process studied in the context of aging and ergodicity breaking.
Abstract: Tracking the sign of fluctuations governed by the $$(1+1)$$ -dimensional Kardar–Parisi–Zhang (KPZ) universality class, we show, both experimentally and numerically, that its evolution has an unexpected link to a simple stochastic model called the renewal process, studied in the context of aging and ergodicity breaking. Although KPZ and the renewal process are fundamentally different in many aspects, we find remarkable agreement in some of the time correlation properties, such as the recurrence time distributions and the persistence probability, while the two systems can be different in other properties. Moreover, we find inequivalence between long-time and ensemble averages in the fraction of time occupied by a specific sign of the KPZ-class fluctuations. The distribution of its long-time average converges to nontrivial broad functions, which are found to differ significantly from that of the renewal process, but instead be characteristic of KPZ. Thus, we obtain a new type of ergodicity breaking for such systems with many-body interactions. Our analysis also detects qualitative differences in time-correlation properties of circular and flat KPZ-class interfaces, which were suggested from previous experiments and simulations but still remain theoretically unexplained.

16 citations


Journal ArticleDOI
TL;DR: Applying a matrix analytic approach, fluid flow techniques and martingales, methods are developed to obtain explicit formulas for the cost functionals (setup, holding, production and lost demand costs) in the discounted case and under the long-run average criterion.
Abstract: We study the performance of a reflected fluid production/inventory model operating in a stochastic environment that is modulated by a finite state continuous time Markov chain. The process alternates between ON and OFF periods. The ON period is switched to OFF when the content level reaches a predetermined level q and returns to ON when it drops to 0. The ON/OFF periods generate an alternative renewal process. Applying a matrix analytic approach, fluid flow techniques and martingales, we develop methods to obtain explicit formulas for the cost functionals (setup, holding, production and lost demand costs) in the discounted case and under the long-run average criterion. Numerical examples present the trade-off between the holding cost and the loss cost and show that the total cost appears to be a convex function of q.

15 citations


Journal ArticleDOI
TL;DR: An original algorithm is developed to estimate the conditional intensity function by preserving its structure in terms of the trend function and the underlying renewal process by using kernel smoothing techniques.

Journal ArticleDOI
TL;DR: In this paper, a stochastic signal described by a renewal process was investigated for a system with N states, where each state has an associated joint distribution for the signal's intensity and its holding time.
Abstract: We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal’s intensity and its holding time. We calculate multi-point distributions, correlation functions, and the power-spectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.

Journal ArticleDOI
TL;DR: In this article, the probability of events related to the intersection (or nonintersection) of arbitrary remote boundaries by the trajectory of a compound renewal process is investigated, and explicit logarithmic asymptotics for the probability are derived.
Abstract: We find explicit logarithmic asymptotics for the probability of events related to the intersection (or nonintersection) of arbitrary remote boundaries by the trajectory of a compound renewal process.

Journal ArticleDOI
TL;DR: This work develops more a realistic analytical model which investigates the average IPS based on the number of renewal cycles that a piece of information needs to be delivered and provides helpful insights towards designing new applications on VANETs.
Abstract: The Information propagation process is one the main challenges in delay tolerant networks especially in vehicular ad hoc networks (VANETs). A cycle of information propagation in a time-varying vehicular speed situation starts with physical movement of the vehicles as a catch-up process and ends with multihop transmission through connected vehicles as a forwarding process. Based on these two alternating processes information propagation cyclically renews. In the literature of VANET information propagation speed (IPS) is formulated based on one propagation cycle. This motivated us to develop more a realistic analytical model which investigates the average IPS based on the number of renewal cycles that a piece of information needs to be delivered. Using this renewal process, unlike traditional models, the expected length and expected duration of renewal cycles are formulated mathematically and subsequent closed-form equations are proposed for average IPS. The accuracy of the proposed model is confirmed using simulation. The concluded results provide helpful insights towards designing new applications on VANETs.

Journal ArticleDOI
TL;DR: A system with n components, one online and the rest in standby subject to repair, and the source of failures of the online unit is different from the one of the standby units, which means the interarrival times between failures are dependent and the same for the consecutive repair times.

Journal ArticleDOI
TL;DR: This paper proves that the spectrum access of SUs with respect to the primary user (PU) traffic behavior forms a renewal process and the corresponding renewal cycle is derived and metrics such as collision probability and interference time due to both sensing error and PU re-occupancy are formulated in the renewal cycle.

Book ChapterDOI
01 Jan 2016
TL;DR: In this article, the reliability of motor subsystem of a dump truck in Miduk Copper Mine in Iran has been analyzed and the failure data were collected during 20 months of dump truck operation.
Abstract: Dump truck is one of the main machinery in open pit mines. From an economic point of view, more than 50–60 % of production costs in open pit mines are allocated to hauling and loading costs, so it is important to keep equipment in good condition. Reliability is a useful tool for evaluating the performance of this machine. In this research, the reliability of motor subsystem of a dump truck in Miduk Copper Mine in Iran has been analyzed. The failure data were collected during 20 months of dump truck operation. Trend and serial correlation tests were used to validate the assumption of independent and identically distribution (IID). According to tests, the data are independent and identically distributed therefore the renewal process technique is used for modelling. For finding the best-fit distribution, different types of statistical distributions were tested using the Easyfit software. The analysis results indicated the time between failures (TBF) data obey the Weibull (3p) distribution. The developed model based on these data showed that the reliability of the motor subsystem decreases to a zero value after approximately 430 h of operation. Regarding to the obtained reliability plot, preventive reliability-based maintenance time interval for 90 % reliability levels for machine in the motor subsystem is 21 h.

Journal ArticleDOI
TL;DR: In this paper, distributional properties pertaining to the homogeneous Poisson process (HPP) when observed over a possibly random window are presented, and properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained.
Abstract: In this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. We envision that through the results in this article, a better appreciation of the subtleties involved in the modeling and analysis of recurrent events data will ensue, since the HPP is arguably one of the simplest among recurrent event models. In addition, the use of the theorem of total probability, Bayes’ theorem, the iterated rules of expectation, variance and covariance, and the renewal equation could be illustrative when teaching distribution theory, mathematical statistics, and stochastic processes at both the undergraduate and graduat...

Journal ArticleDOI
TL;DR: In this paper, the large and moderate deviations for a renewal randomly indexed branching process (ZNt) were derived, where Zn is a Galton-Watson process and Nt is a renewal process which is independent of Zn.

Journal ArticleDOI
TL;DR: In this paper, the mean probabilities of particle passage through a stochastic medium are constructed based on Monte Carlo simulation and probabilistic analysis, and the parameters of the averaged models are estimated based on the properties of the exponential distribution and the renewal theory.
Abstract: Based on the Monte Carlo simulation and probabilistic analysis, stochastic radiative models are effectively averaged; that is, deterministic models that reproduce the mean probabilities of particle passage through a stochastic medium are constructed. For this purpose, special algorithms for the double randomization and conjugate walk methods are developed. For the numerical simulation of stochastic media, homogeneous isotropic Voronoi and Poisson mosaic models are used. The parameters of the averaged models are estimated based on the properties of the exponential distribution and the renewal theory.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the limit law of the supremum of the local time, as well as the position of the favorite sites of a one-dimensional diffusion in a drifted Brownian potential.
Abstract: We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W_\kappa$, with $ 0 0$. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Levy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.

Journal ArticleDOI
TL;DR: The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Rosen and Wendel, then extended by the latter to stable processes.
Abstract: The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Rosen and Wendel, then extended by the latter to stable processes. We recover and extend these results using simple concepts of renewal theory, which allows to revisit past or recent works of the physics literature.

Journal ArticleDOI
TL;DR: A novel maintenance model is first presented based on a new defined renewal-geometric process, which splits the operation process into an early renewal process and a late geometric process to characterize such a special deterioration delay.
Abstract: The optimal replacement policy is proposed for a new maintenance model of a repairable deteriorating system to minimize the average cost rate throughout the system life cycle. It is assumed that the system undergoes deterioration with an increasing trend of deterioration probability after each repair. More specifically, a novel maintenance model is first presented based on a new defined renewal-geometric process, which splits the operation process into an early renewal process and a late geometric process to characterize such a special deterioration delay. Then, the average cost rate for the new model is formulated according to the renewal-reward theorem. Next, a theorem is presented to derive the theoretical relationships of optimal replacement policies for the geometric-process maintenance model and the new proposed model, respectively. Finally, numerical examples suggest that the optimum values can be determined to minimize the average cost rates.

Posted Content
TL;DR: In this article, the authors considered the GI/GI/N queue and established convergence of the corresponding sequence of centered and scaled stationary distributions in the Halfin-Whitt asymptotic regime.
Abstract: We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by $N$ identical parallel servers in a first-come-first-serve manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin-Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite-dimensional. This approach is more broadly applicable to a larger class of networks.

Proceedings ArticleDOI
02 Jun 2016
TL;DR: In this article, a count distribution is presented by considering a renewal process where the distribution of the duration is a finite mixture of exponential distributions, and the computation of the probabilities and renewal function (expected number of renewals) are examined.
Abstract: A count distribution is presented by considering a renewal process where the distribution of the duration is a finite mixture of exponential distributions. This distribution is able to model over dispersion, a feature often found in observed count data. The computation of the probabilities and renewal function (expected number of renewals) are examined. Parameter estimation by the method of maximum likelihood is considered with applications of the count distribution to real frequency count data exhibiting over dispersion. It is shown that the mixture of exponentials count distribution fits over dispersed data better than the Poisson process and serves as an alternative to the gamma count distribution.

Journal ArticleDOI
12 Sep 2016
TL;DR: In this paper, a generalization of the alternating Poisson process from the point of view of offractional calculus is proposed, which produces a fractional 2-state point process.
Abstract: We propose a generalization of the alternating Poisson process from the point of view offractional calculus. We consider the system of differential equations governing the state probabilitiesof the alternating Poisson process and replace the ordinary derivative with the fractional derivative (inthe Caputo sense). This produces a fractional 2-state point process. We obtain the probability massfunction of this process in terms of the (two-parameter) Mittag-Leffler function. Then we show thatit can be recovered also by means of renewal theory. We study the limit state probability, and certainproportions involving the fractional moments of the sub-renewal periods of the process. In conclusion,in order to derive new Mittag-Leffler-like distributions related to the considered process, we exploit atransformation acting on pairs of stochastically ordered random variables, which is an extension of theequilibrium operator and deserves interest in the analysis of alternating stochastic processes.

Journal ArticleDOI
TL;DR: In this paper, a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet was proved, extending results by M. Pollicott from the finite to the infinite alphabet setting.
Abstract: We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.

Journal ArticleDOI
TL;DR: In this article, the aging effects of the renewal process with the tempered power-law waiting time distribution were discussed, and the p-th moment of the number of renewal events was derived for both the weakly and strongly aged systems.
Abstract: In the renewal processes, if the waiting time probability density function is a tempered power-law distribution, then the process displays a transition dynamics; and the transition time depends on the parameter $$\lambda $$ of the exponential cutoff. In this paper, we discuss the aging effects of the renewal process with the tempered power-law waiting time distribution. By using the aging renewal theory, the p-th moment of the number of renewal events $$n_a(t_a, t)$$ in the interval $$(t_a, t_a+t)$$ is obtained for both the weakly and strongly aged systems; and the corresponding surviving probabilities are also investigated. We then further analyze the tempered aging continuous time random walk and its Einstein relation, and the mean square displacement is attained. Moreover, the tempered aging diffusion equation is derived.

Journal ArticleDOI
TL;DR: This paper investigates renewal theories in the T -independent random fuzzy environment based on the concept of fuzzy variable and random fuzzy variable with special cases for T = min and T =Archimedean t-norm.
Abstract: In this paper, we investigate renewal theories in the T -independent random fuzzy environment based on the concept of fuzzy variable and random fuzzy variable. For special cases, we consider the case for T = min and T =Archimedean t-norm. Mathematics Subject Classification: 60A86

Journal ArticleDOI
23 Dec 2016
TL;DR: In this paper, the authors considered two time-inhomogeneous birth-death Markov chains with discrete time on a general state space and derived an upper bound for the expectation of the renewal time.
Abstract: In this paper, we consider two time-inhomogeneous Markov chains $X^{(l)}_t$, $l\in\{1,2\}$, with discrete time on a general state space. We assume the existence of some renewal set $C$ and investigate the time of simultaneous renewal, that is, the first positive time when the chains hit the set $C$ simultaneously. The initial distributions for both chains may be arbitrary. Under the condition of stochastic domination and nonlattice condition for both renewal processes, we derive an upper bound for the expectation of the simultaneous renewal time. Such a bound was calculated for two time-inhomogeneous birth–death Markov chains.