Ordinary Renewal Process Research Articles

The Robust Classical MTBF Test

For the reliability demonstration of a repairable system, testers expediently assume that the rate of occurrence of failures (ROCOF) can be modeled by a homogenous Poisson process (HPP), also called an ordinary renewal process (ORP), and therefore the times between failure, or interarrival times, are exponentially distributed with a constant mean time between failures (MTBF). This often-used test is colloquially called the “classical MTBF test” because it has been widely used by both government and industry for the past 50 years. The test is simple; operate a system for a period and the sample estimate for MTBF is the operating time divided by the number of failures. A confidence interval is applied to the sample estimate to determine the test result. This paper considers the validity of the test result if the repairable system has an increasing or decreasing MTBF, that is, a nonhomogeneous Poisson process (NHPP). Specifically, the change in consumer’s risk is analyzed if the repairable system has a decreasing MTBF (increasing ROCOF). A simple risk mitigation strategy consisting of a burn-in period equal to the threshold MTBF is also considered and found to be effective. The classical MTBF test is shown to be a robust test and a burn-in period prior to testing makes the classical MTBF test even more robust.

On Some Stochastic Ordering Comparisons for Renewal Processes

After a brief introduction to ordinary renewal process, we have defined compound renewal process generated by an ordinary renewal process followed by the distributional properties of the corresponding random variables. A few results are obtained by comparing independent renewal processes with respect to several stochastic orderings between the generating inter-arrival time random variables, like, stochastic order, hazard rate order, likelihood ratio order and variability order, as well as some ageing classes of the generating random variables. Some numerical illustrations are given. The results obtained here appear to be new.

STOCHASTIC MODEL FOR TIME TO RECRUITMENT IN A SINGLE GRADE MANPOWER SYSTEM WITH TWO TYPES OF INTERDECISION TIMES WHEN THE BREAKDOWN THRESHOLD HAS THREE COMPONENTS

In this paper, the problem of time to recruitment is analyzed for a single grade manpower system in which attrition takes place due to two types of policy decisions where this classification is done according to intensity of attrition, it form an ordinary renewal process. Assuming (i) policy decisions and exits occur at different epochs (ii) wastage of manpower due to exits and wastage due to frequent breaks taken by the personnel working in the manpower system separately form a sequence of independent and identically distributed exponential random variables with different means and (iii) breakdown threshold for the cumulative wastage of manpower in the system has three components which are independent exponential random variables. A stochastic model is constructed and the variance of the time to recruitment is obtained using an univariate CUM policy of recruitment. Employing a different probabilistic analysis, analytical results in closed form for system characteristics are derived.

Point and Interval Estimators of Reliability Indices for Repairable Systems Using the Weibull Generalized Renewal Process

The generalized renewal process considers repair efficiency of imperfect repair in reliability assessment of repairable systems; therefore, its evaluation results are close to real repair circumstance than the ordinary renewal process or the non-homogeneous Poisson process. Based on the asymptotic distribution of maximum likelihood estimation, a calculating method of reliability indices for repairable systems with imperfect repair is proposed. The point and approximate interval estimators of model parameters for Kijima type Weibull generalized renewal processes Models I and II, as well as reliability indices of repairable systems, such as reliability, cumulative failure number and failure intensity, etc., are all presented. Two real cases are studied using generalized renewal processes Models I and II respectively to show the validity of our method. The results show that imperfect repair makes the instantaneous failure intensity of a repairable system discretely jump either up or down at the time of each failure, and the method proposed in this paper agrees well with the other exiting methods, and can also reduce the complexity of calculation.

Steady-state imperfect repair models

Imperfect maintenance models are widely used in reliability engineering. This paper discusses relevant asymptotic properties for the steady-state virtual age processes. It is shown that the limiting distributions of age, the residual lifetime and the spread that describe an ordinary renewal process can be generalized to the stable virtual age process, although the cycles of the latter are not independent. Asymptotic distributions of the virtual age at time t, as well as of the virtual ages at the start and the end of a cycle containing t (as t tends to infinity) are explicitly derived for two popular in practice imperfect maintenance models, namely, the Arithmetic Reduction of Age (ARA) and the Brown–Proschan (BP) models. Some applications of the obtained results to maintenance optimization are discussed.

Numerical Estimation of Expected Number of Failures for Repairable Systems Using a Generalized Renewal Process Model

The reliability of repairable systems is important for many engineering applications such as warranty forecasting, maintenance strategies, and durability, among others. A generalized renewal process (GRP) approach, considering the effectiveness of repairs, is commonly used, modeling the concepts of minimal repair, perfect repair, and general repair. The effect of the latter is between the effects of a minimal and a perfect repair. The GRP models the sequence of recurrent failure/repair events for repairable systems by solving the so-called g-renewal equation, which has no analytical solution. This paper proposes a data-driven numerical estimation of the expected number of failures (ENF) for the GRP model without solving the complicated g-renewal equation directly. Instead, it uses an empirical relationship among the cumulative intensity function (CIF) of the GRP, ordinary renewal process (ORP), and nonhomogeneous Poisson process (NHPP). The ORP and NHPP are limiting cases of the generalized renewal process. For practical reasons, it is common to observe only a few units of a repairable system population for only a short time. Using the observed data, the proposed approach creates a reliability metamodel of a renewal process, which is then used to predict the expected number of failures, and assess the average effectiveness of each repair. This increases the usefulness of the method for many practical reliability problems where the collection of a large amount of data is not possible or economical. The good accuracy of the proposed approach is demonstrated with three examples using simulated data, and a real-life example of locomotive braking grids using actual data.

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.

On Reliability of a Multi‐Socket Repairable System

Consider a set of the so‐called sibling components in a multi‐socket repairable system. In the case of an automobile, for example, these siblings would be spark plugs, light bulbs, tires, that is, identical components that are coded with the same part number. When field data are analyzed, a dilemma arises as to how to interpret a recurrent replacement of a sibling component: as a secondary failure of the component that has already been replaced once, or as the first failure of the component's sibling(s)? From the stand point of root‐cause analysis, the task is to understand whether recurrent failures are related to (i) a particular sibling, which might be operating in inauspicious conditions relative to other siblings, or (ii) to all siblings on the vehicle. One could attribute Scenario 1 to a system‐level (e.g. system interaction) problem, and Scenario 2 to a component‐level (supplier quality) problem. We first review a statistical procedure that solves the above‐mentioned dilemma in the framework of ordinary renewal process (ORP) and then extend the discussion to the non‐homogeneous Poisson process (NHPP) and the g‐renewal process (GRP). We also propose advanced Monte Carlo procedure for estimating GRP in this context. Copyright © 2016 John Wiley & Sons, Ltd.

Phase Diagram in Stored-Energy-Driven Lévy Flight

Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven L\'evy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.

Numerical method for Weibull generalized renewal process and its applications in reliability analysis of NC machine tools

In generalized renewal process (GRP) reliability analysis for repairable systems, Monte Carlo (MC) simulation method instead of numerical method is often used to estimate model parameters because of the complexity and the difficulty of developing a mathematically tractable probabilistic model. In this paper, based on the conditional Weibull distribution for repairable systems, using negative log-likelihood as an objective function and adding inequality constraints to model parameters, a nonlinear programming approach is proposed to estimate restoration factor for the Kijima type GRP model I, as well as the model II. This method minimizes the negative log-likelihood directly, and avoids solving the complex system of equations. Three real and different types of field failure data sets with time truncation for NC machine tools are analyzed by the proposed numerical method. The sampling formulas of failure times for the GRP models I and II are derived and the effectiveness of the proposed method is validated with MC simulation method. The results show that the GRP model is superior to the ordinary renewal process (ORP) and the power law non-homogeneous Poisson process (PL-NHPP) model.

An Approximate Solution to the G–Renewal Equation With an Underlying Weibull Distribution

An important characteristic of the g-renewal process, of great practical interest, is the g-renewal equation, which represents the expected cumulative number of recurrent events as a function of time. Just like in an ordinary renewal process, the problem is that the g-renewal equation does not have a closed form solution, unless the underlying event times are exponentially distributed. The Monte Carlo solution, although exhaustive, is computationally demanding. This paper offers a simple-to-implement (in an Excel spreadsheet) approximate solution, when the underlying failure-time distribution is Weibull. The accuracy of the proposed solution is in the neighborhood of 2%, when compared to the respective Monte Carlo solution. Based on the proposed solution, we also consider an estimation procedure of the g-renewal process parameters.

Renewal stochastic processes with correlated events: Phase transitions along time evolution

We consider renewal stochastic processes generated by nonindependent events from the perspective that their basic distribution and associated generating functions obey the statistical-mechanical structure of systems with interacting degrees of freedom. Based on this fact we look briefly into the less-known case of processes that display phase transitions along time. When the density distribution ψ{n}(t) for the occurrence of the nth event at time t is considered to be a partition function, of a "microcanonical" type for n "degrees of freedom" at fixed "energy" t, one obtains a set of four partition functions of which that for the generating function variable z and Laplace transform variable ε, conjugate to n and t, respectively, plays a central role. These partition functions relate to each other in the customary way and in accordance to the precepts of large deviations theory, while the entropy, or Massieu potential, derived from ψ{n}(t) satisfies an Euler relation. We illustrate this scheme first for an ordinary renewal process of events generated by a simple exponential waiting-time distribution ψ(t). Then we examine a process modeled after the so-called Hamiltonian mean-field model that is representative of agents that perform a repeated task with an associated outcome, such as an opinion poll. When a sequence of (many) events takes place in a sufficiently short time the process exhibits clustering of the outcome, but for larger times the process resembles that of independent events. The two regimes are separated by a sharp transition, technically of the second order. Finally we point out the existence of a similar scheme for random-walk processes.

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered. In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.

Ruin probabilities for a risk model with two classes of claims

In this paper we consider a risk model with two kinds of claims, whose claims number processes are Poisson process and ordinary renewal process respectively. For this model, the surplus process is not Markovian, however, it can be Markovianized by introducing a supplementary process. We prove the Markov property of the related vector processes. Because such obtained processes belong to the class of the so-called piecewise-deterministic Markov process, the extended infinitesimal generator is derived, exponential martingale for the risk process is studied. The exponential bound of ruin probability in infinite time horizon is obtained.

Supply chain inventory optimization with two customer classes in discrete time

In this paper we consider a single-item inventory system where two demand classes with different service requirements are satisfied from a common inventory. A critical level, reorder point, order quantity or ( s, q, k) policy is in use. The time axis is divided into discrete time units, which is a common characteristic of many real-life supply-chain processes. The inventory process within the lead time of a replenishment order is modelled as a sequence of (1) an ordinary renewal process and (2) two alternating renewal processes. Approximations are developed for the demand class-specific fill rates and the probability distribution of the waiting time of low priority customer orders. This waiting time distribution is used for the inventory allocation in a two-stage supply chain.

Some results on the compound Markov binomial model

This paper considers the compound Markov binomial risk model proposed by Cossette et al. (20032004). Two discrete-time renewal (ordinary renewal and delayed renewal) risk processes associated with the compound Markov binomial risk model are analyzed. Based on the associated ordinary renewal process, a defective renewal equation for the conditional Gerber–Shiu expected discounted penalty function is obtained. The relationship between the conditional expected discounted penalty function in the ordinary renewal case and that in the delayed renewal case is then established. From these results, the conditional ultimate probability of ruin as well as the conditional joint distribution of the surplus just prior to ruin and the deficit at ruin are studied. Finally, it is shown that a modified version of the compound Markov binomial risk model is a special case of the discrete-time semi-Markov risk model introduced by Reinhard and Snoussi (20012002).

Frequency analysis of binary oscillators triggered by a random noise

The frequency analysis of a bistable binary oscillator triggered by a stationary random signal is performed by analyzing the expression for the power spectrum of its output. The oscillator state changes when the input noise exceeds a fixed threshold θ, which occurs in correspondence of the events produced by an ordinary renewal process. After each state change a fixed refractory time T is to be passed before the output performs a new transition. A theoretical study shows that a coherence resonance (CR) effect appears in the output square wave if the threshold θ is lower than a positive real value θ . When the overcoming of the input threshold occurs according to a Poisson process with exponential parameter α, the regularity is present at the output only if the product αT exceeds 2 −1 . The value of the characteristic frequency increases rapidly with αT towards its asymptotic value 1/(2 T). The same probabilistic approach is then adopted to perform the spectral analysis of a set of uncoupled binary oscillators, sharing the same refractory time T and the same input random signal. Under mild assumptions the whole system behaves as a single oscillator, whose output is amplified proportionally to the number of elements involved.

The Trend-Renewal Process for Statistical Analysis of Repairable Systems

The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes, corresponding to minimal repairs, and renewal processes, corresponding to perfect repairs. This article introduces and studies a more general model for recurrent events, the trend-renewal process (TRP). The TRP is a time-transformed renewal process having both the ordinary renewal process and the nonhomogeneous Poisson process as special cases. Parametric inference in the TRP model is studied, with emphasis on the case in which several systems are observed in the presence of a possible unobserved heterogeneity between systems.

Pairs of renewal processes whose superposition is a renewal process

A renewal process is called ordinary if its inter-renewal times are strictly positive. S.M. Samuels proved in 1974 that if the superposition of two ordinary renewal processes is an ordinary renewal process, then all processes are Poisson. This result is generalized here to the case of processes whose inter-renewal times may be zero. We show that, besides the Poisson processes, there are two pairs of binomial-like processes whose superposition is a renewal process. A new proof of Samuels's theorem is included, which, unlike the original, does not require the renewal theorem. If the two processes are assumed identical, then a very simple proof is possible.

Generalized Einstein relation: A stochastic modeling approach

For anomalous random walkers, whose mean square displacement behaves like $〈{x}^{2}(t)〉\ensuremath{\sim}{t}^{\ensuremath{\delta}}$ $(\ensuremath{\delta}\ensuremath{\ne}1),$ the generalized Einstein relation between anomalous diffusion and the linear response of the walkers to an external field $F$ is studied, using a stochastic modeling approach. A departure from the Einstein relation is expected for weak external fields and long times. We investigate such a departure using the Scher-Lax-Montroll model, defined within the context of the continuous time random walk, and which describes electronic transport in a disordered system with an effective exponent $\ensuremath{\delta}<1.$ We then consider a collision model which for the force free case may be mapped on a L\'evy walk $(\ensuremath{\delta}>1).$ We investigate the response in such a model to an external driving force and derive the Einstein relation for it both for equilibrium and ordinary renewal processes. We discuss the time scales at which a departure from the Einstein relation is expected.