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Showing papers on "Renewal theory published in 2022"


Journal ArticleDOI
01 Apr 2022-Test
TL;DR: In this article , the authors introduced a general $$\delta $$ -shock model when the recovery time depends on both the arrival times and the magnitudes of shocks, and also considered a more general and flexible shock process, namely, the Poisson generalized gamma process.
Abstract: The $$\delta $$ -shock model is one of the basic shock models which has a wide range of applications in reliability, finance and related fields. In existing literature, it is assumed that the recovery time of a system from the damage induced by a shock is constant as well as the shocks magnitude. However, as technical systems gradually deteriorate with time, it takes more time to recover from this damage, whereas the larger magnitude of a shock also results in the same effect. Therefore, in this paper, we introduce a general $$\delta $$ -shock model when the recovery time depends on both the arrival times and the magnitudes of shocks. Moreover, we also consider a more general and flexible shock process, namely, the Poisson generalized gamma process. It includes the homogeneous Poisson process, the non-homogeneous Poisson process, the Pólya process and the generalized Pólya process as the particular cases. For the defined survival model, we derive the relationships for the survival function and the mean lifetime and study some relevant stochastic properties. As an application, an example of the corresponding optimal replacement policy is discussed.

8 citations


Journal ArticleDOI
TL;DR: In this article , the authors generalize several calendar time-based minimal repair policies, including periodic replacement policy with minimal repair, reference time policy, and bivariate T−N maintenance policy.

7 citations


Journal ArticleDOI
TL;DR: Taking random repair time into account, several calendar time-based minimal repair policies, including periodic replacement policy with minimal repair, reference time policy, and bivariate T − N maintenance policy are generalized and the system long-run average cost rate under each minimal repair policy is gained explicitly.

7 citations


Journal ArticleDOI
TL;DR: In this paper , a renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time has been investigated, in order to provide some insight into the performance measures in renewal process and sequence.
Abstract: A renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time has been investigated, in order to provide some insight into the performance measures in renewal process and sequence such as, the mean time between successive renewals, ; Laplace-Stiltjes transform (LST) of the mean time, ; the Laplace-Stieltjes transform (LST) of the mean time distribution function, ; Laplace-Stiltjes transform (LST) of fold convolution of distribution function, ; the time at which the renewal occurs, the average number of renewals per unit time over the interval (0, t], and expected reward, . Our quest is to analyse the distribution function of the renewal process and sequence using the concept of discrete time Markov chain to obtain the aforementioned performance measures. Some properties of the Erlang , exponential and geometric distributions are used with the help of some existing laws, theorems and formulas of Markov chain. We concluded our study through illustrative examples that, it is not possible for an infinite number of renewals to occur in a finite period of time; Also, the expected number of renewals increases linearly with time; and from the uniqueness property, we affirmed that, the Poisson process is the only renewal process with a linear mean-value function; and lastly, we obtained the optimal replacement policy for a manufacturing machine which showed that, the exponential property of the lifetime

4 citations


Journal ArticleDOI
TL;DR: In this paper , the authors proved counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell's theorem, and the key renewal theorem) for the number of j th-generation individuals with birth times, when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}big)$ .
Abstract: Abstract An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of j th-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations.

4 citations


Journal ArticleDOI
TL;DR: In this article, a Sparre Andersen model was studied in which the business activity of a company is described by a compound renewal process with drift assuming that the capital reserves are invested in a risky asset.

2 citations


Journal ArticleDOI
TL;DR: In this article , a characterisation of mixed renewal processes in terms of exchangeability and of different types of regular conditional probabilities is given, and an existence result for mixed renewal process, providing also a new construction for them, is obtained.
Abstract: Abstract Some characterizations of mixed renewal processes in terms of exchangeability and of different types of regular conditional probabilities are given. As a consequence, an existence result for mixed renewal processes, providing also a new construction for them, is obtained. As an application, some concrete examples of constructing such processes are presented and the corresponding regular conditional probabilities are explicitly computed.

2 citations


Journal ArticleDOI
TL;DR: In this article , a new class of asymmetric random walks on the one-dimensional infinite lattice is introduced, where the direction of the jumps (positive or negative) is determined by a discrete-time renewal process.

2 citations




Journal ArticleDOI
TL;DR: In this article , it was shown that the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process, and that the removal of interval correlations does not change the long-term variability and its effect on information transmission.
Abstract: Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer-Schwabedal-Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin-Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.

Journal ArticleDOI
TL;DR: The generalized Sibuya random walk (GSD) as mentioned in this paper is a semi-Markovian discrete-time generalization of the telegraph process, where the step direction is reversed at arrival times of a discrete time renewal process and remains unchanged at uneventful time instants, and the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m.
Abstract: In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m=1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.

Journal ArticleDOI
TL;DR: In this paper , the authors describe the limiting distribution of the extremes of observations that arrive in clusters and apply the developed theory to determine the limiting distributions of observations arriving in a marked renewal cluster.
Abstract: Abstract This article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster, and then we apply the developed theory to determine the limiting distribution of $\max\{X_j\,:\, j=0,\ldots, K(t)\}$ , where K(t) is the number of independent and identically distributed observations $(X_j)$ arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.

Journal ArticleDOI
TL;DR: In this paper , the authors study renewal in a system with two components connected in series and propose a "coupled" lifetime that combines the lifetimes of the two serially connected components to represent the joint lifetime of the system.
Abstract: In this paper, we study renewals in a system with two components connected in series. Both components can undergo corrective maintenance (i.e., either full replacement/perfect repair or minimal repair) when a failure occurs. When one of the units fails and is correctively maintained, the other one is either preventively replaced (if cost-feasible) or simply left working as is. When a component is minimally repaired or left working as is, its remaining lifetime is reevaluated (“memory effect”) and is taken into consideration in calculating the next renewal. A “coupled” lifetime that combines the lifetimes of the two serially connected components is proposed to represent the joint lifetime of the system. We develop renewal functions based on the coupled lifetimes and show that they follow the classical or generalized renewal theory depending on whether the components work without memory or not. Approximation formulas for the new renewal functions are also obtained and validated by Monte Carlo simulations for various combinations of distributions, and a comparative cost analysis is conducted.

Journal ArticleDOI
TL;DR: In this article , the renewal density of a beam to a train of moving forces driven by a Poisson and by an Erlang renewal process is determined by numerical evaluation of integrals and verified against direct Monte Carlo simulations.

Journal ArticleDOI
TL;DR: In this article , the authors considered a class of discrete-time random walks with directed unit steps on the integer line and showed that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process.

Journal ArticleDOI
TL;DR: In this article , a non-standard renewal risk model was considered, in which each main claim may produce a random number of delayed claims, and the claim size distributions were restricted to the smaller class of extended regular variation.
Abstract: In this paper, we consider a non-standard renewal risk model, in which each main claim may produce a random number of delayed claims. Such a model takes full account of lasting impacts of severe claims and hence may avoid the underestimation of an insurer's operation risk within a relatively long period. We obtain asymptotic expansions for the finite-time ruin probability, given that the claim size distributions belong to the general subexponential class. When the claim size distributions are restricted to the smaller class of extended regular variation, we push our study forward to the infinite-time ruin probability.

Journal ArticleDOI
TL;DR: In this paper , an N warm standby system under shocks and inspections governed by Markovian arrival processes is presented, where inspections detect the number of down units, and their replacement is carried out if there are a minimum K of failed units.
Abstract: This paper presents an N warm standby system under shocks and inspections governed by Markovian arrival processes. The inspections detect the number of down units, and their replacement is carried out if there are a minimum K of failed units. This is a policy of the type (K,N) used in inventory theory. The study is performed via the up and down periods of the system (cycle); the distribution of these random times and the expected costs for each period comprising the cycle are determined on the basis of individual costs due to maintenance actions (per inspection and replacement of every unit) and others due to operation or inactivity of the system, per time unit. Intermediate addressed calculus are the distributions of the number of inspections by cycle and the expected cost involving every inspection, depending on the number of replaced units. The system is studied in transient and stationary regimes, and some reliability measures of interest and the cost rate are calculated. An optimization of these quantities is performed in terms of the number K in a numerical example. This general model extends to many others in the literature, and, by using the matrix-analytic method, compact and algorithmic expressions are achieved, facilitating its potential application.

Journal ArticleDOI
TL;DR: In this paper , the impact of an inhibitory autapse on neuronal activity is analyzed analytically. But, the authors focus on a set of non-adaptive spiking neuron models with delayed feedback inhibition, instead of considering a particular neuronal model.
Abstract: In this paper, we study analytically the impact of an inhibitory autapse on neuronal activity. In order to do this, we formulate conditions on a set of non-adaptive spiking neuron models with delayed feedback inhibition, instead of considering a particular neuronal model. The neuron is stimulated with a stochastic point renewal process of excitatory impulses. Probability density function (PDF) [Formula: see text] of output interspike intervals (ISIs) of such a neuron is found exactly without any approximations made. It is expressed in terms of ISIs PDF for the input renewal stream and ISIs PDF for that same neuron without any feedback. Obtained results are applied to a subset of neuronal models with threshold 2 when the time intervals between input impulses are distributed according to the Erlang-2 distribution. In that case, we have found explicitly the model-independent initial part of ISIs PDF [Formula: see text] defined at some initial interval [Formula: see text] of ISI values.

Journal ArticleDOI
TL;DR: In this article , an alternating renewal process is proposed to mathematically model a multi-failure complex system in order to develop its availability and maintenance measures, where inspections are performed at regular intervals to detect breakdowns.
Abstract: In this paper, it is aimed to propose a notion of the alternating renewal process to mathematically model a multi-failure complex system in order to develop its availability and maintenance measures. In the study, M/E2/1 queueing model with infinite waiting space during service times of components has been worked out under the FCFS discipline. Inspections are performed at regular intervals to detect breakdowns. The primary objective of the paper is about obtaining the system’s reliability, availability, and the optimal interval period with minimum maintenance cost. To validate the mentioned model a case study has been done on the Power Distribution System. Finally, a comparative study between M/E2/1 and M/M/1 queuing models is done and the various aspects based on availability and long-run cost are focused in the analytical result. The results were direct toward enhancing the availability of the system and making the decision about an appropriate inspection period.

Journal ArticleDOI
TL;DR: For a random inspection time, which includes the deterministic case, and a delayed renewal process, representations of the expected length of an inspection interval and related inequalities in terms of covariances are shown in this paper .
Abstract: The well-known inspection paradox of renewal theory states that, in expectation, the inspection interval is larger than a common renewal interval, in general. For a random inspection time, which includes the deterministic case, and a delayed renewal process, representations of the expected length of an inspection interval and related inequalities in terms of covariances are shown. Datasets of eruption times of Beehive Geyser and Riverside Geyser in Yellowstone National Park, as well as several distributional examples, illustrate the findings.

Posted ContentDOI
05 Oct 2022
TL;DR: In this paper , a Markov renewal process is used to obtain approximate queue system parameters values, obtained through the consideration of an adequate renewal process, and the results of the renewal process are studied.
Abstract: Some $M|G|\infty$ queue systems parameters values approximations, obtained through the consideration of an adequate Markov renewal process, are presented, and studied.

Journal ArticleDOI
TL;DR: In this paper , the authors show general conditions under which the sampled process is strongly mixing or weakly dependent and explicitly compute the strong-mixing or weak-dependent coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of X is preserved (at least asymptotically).
Abstract: Abstract Let X be a continuous-time strongly mixing or weakly dependent process and let T be a renewal process independent of X . We show general conditions under which the sampled process $(X_{T_i},T_i-T_{i-1})^{\top}$ is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of X is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process X are at our disposal.

Journal ArticleDOI
05 Aug 2022
TL;DR: In this paper , a renewal risk model with by-claims was considered, where the price process of the investment portfolio followed an exponential Lévy process and the main claim was a one-sided linear process.
Abstract: In this article, we consider a renewal risk model with by-claims, where the price process of the investment portfolio follows an exponential Lévy process. We further assume that the main claim is a one-sided linear process and there exists a certain dependence structure between the innovations and by-claims. In the presence of heavy tails, we obtain a series of uniform formulas in finite and infinite intervals. In order to better describe the obtained results, we carry on the numerical simulations.




Posted ContentDOI
12 Dec 2022
TL;DR: In this paper , the authors considered coherent systems subject to random shocks that can damage a random number of components of a system and proposed an optimal replacement policy for repairable systems. But the authors focused on the distribution of the number of failed components and not the reliability properties of coherent systems.
Abstract: Abstract We consider coherent systems subject to random shocks that can damage a random number of components of a system. Based on the distribution of the number of failed components, we discuss three models, namely, (i) a shock can damage any number of components (including zero) with the same probability, (ii) each shock damages, at least, one component, and (iii) a shock can damage, at most, one component. Shocks arrival times are modeled using three important counting processes, namely, the Poisson generalized gamma process, the Poisson phase-type process and the renewal process with matrix Mittag-Leffler distributed inter-arrival times. For the defined shock models, we discuss relevant reliability properties of coherent systems. An optimal replacement policy for repairable systems is considered as an application of the proposed modeling.

Journal ArticleDOI
TL;DR: In this article , it was shown that convergence of the iterative renewal transform to quasi-stationary distributions is equivalent to a condition on the moment growth rate of the lifetime, which is at the same time a necessary condition for the existence of Yaglom limits.
Abstract: We consider quasi-stationary distributions for one-dimensional diffusions via the renewal dynamical approach. We show that convergence of the iterative renewal transform to quasi-stationary distributions is equivalent to a condition on the moment growth rate of the lifetime, which is at the same time a necessary condition for the existence of Yaglom limits.

Posted ContentDOI
29 Jun 2022
TL;DR: In this article , the authors considered a class of discrete-time random walks with directed unit steps on the integer line and showed that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process.
Abstract: We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the `aging effect' as a hallmark of non-Markovianity where the discrete-time version of the `aging renewal process' comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a $t^2$-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (`narrow' with finite mean and variance) the walk exhibits normal diffusion and for `broad' waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.