The sojourn time and related characteristics of the age-dependent branching process

An age-dependent branching process where disasters occur as a renewal process leading to annihilation or survival ofall the cells, is considered. For such a process, the total mean sojourn time of all the cells in the system is analysed using the regeneration point technique. The mean number of cells which die in timet and its asymptotic behaviour are discussed. When the disasters arrival as a Poisson process and the lifetime of the cells follows exponential distribution, elegant inter-relationships are found among the means of (i) the total number of cells which die in timet (ii) the total sojourn time of all cells in the system upto timet and (iii) the number of living cells at timet. Some of the existing results are deduced as special cases for related processes.     

Plans accélérés optimaux : estimation de la vie moyenne d'un systèmeOptimal designs for accelerated runs: estimation of life time

An optimal design is presented for the estimation of the extrapolation of the mean lifetime of a system, observed under stressed conditions. Optimality is intended as a function of the MSE. To cite this article: G. Celant, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 69–72.     

Distributions of lifetime after explosion for birth and death processes

How long can a process live after explosion for a birth and death process with explosion and lifespan σ,X ={X(t),t<σ}?Distribution of lifetime from explosion to the end of life, expected lifetime and probability of explosion which means the end of life are found.     

On the mean residual life of a k-out-of-n:G system with a single cold standby component

The concept of mean residual life is one of the most important characteristics that has been widely used in dynamic reliability analysis. It is a useful tool for investigating ageing properties of technical systems. In this paper, we define and study three different mean residual life functions for k-out-of-n:G system with a single cold standby component. In particular, we obtain explicit expressions for the corresponding functions using distributions of order statistics. We also provide some stochastic ordering results associated with the lifetime of a system. We illustrate the results for various lifetime distributions.     

Evaluating the lifetime performance index for the exponential lifetime products

Process capability indices (PCIs) are used to measure process potential and performance. Since the lifetime of products generally may possess an exponential, gamma or Weibull distribution, etc., so under a two-parameter exponential distribution, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of the lifetime performance index based on the right type II censored sample. Then the UMVUE of the lifetime performance index is utilized to develop the new hypothesis testing procedure in the condition of known L. Finally, a practical example is illustrated to employ the testing procedure to determine whether the process is capable.     

Noise-induced effects in nonlinear relaxation of condensed matter systems

Noise-induced phenomena characterise the nonlinear relaxation of nonequilibrium physical systems towards equilibrium states. Often, this relaxation process proceeds through metastable states and the noise can give rise to resonant phenomena with an enhancement of lifetime of these states or some coherent state of the condensed matter system considered. In this paper three noise induced phenomena, namely the noise enhanced stability, the stochastic resonant activation and the noise-induced coherence of electron spin, are reviewed in the nonlinear relaxation dynamics of three different systems of condensed matter: (i) a long-overlap Josephson junction (JJ) subject to thermal fluctuations and non-Gaussian, Lévy distributed, noise sources; (ii) a graphene-based Josephson junction subject to thermal fluctuations; (iii) electrons in a n-type GaAs crystal driven by a fluctuating electric field. In the first system, we focus on the switching events from the superconducting metastable state to the resistive state, by solving the perturbed stochastic sine-Gordon equation. Nonmonotonic behaviours of the mean switching time versus the noise intensity, frequency of the external driving, and length of the junction are obtained. Moreover, the influence of the noise induced solitons on the mean switching time behaviour is shown. In the second system, noise induced phenomena are observed, such as noise enhanced stability (NES) and stochastic resonant activation (SRA). In the third system, the spin polarised transport in GaAs is explored in two different scenarios, i.e. in the presence of Gaussian correlated fluctuations or symmetric dichotomous noise. Numerical results indicate an increase of the electron spin lifetime by rising the strength of the random fluctuating component. Furthermore, our findings for the electron spin depolarization time as a function of the noise correlation time point out (i) a non-monotonic behaviour with a maximum in the case of Gaussian correlated fluctuations, (ii) an increase up to a plateau in the case of dichotomous noise. The noise enhances the coherence of the spin relaxation process.     

Existence of quasi-stationary states at the long range threshold

In this paper the lifetime of quasi-stationary states (QSS) in the α–HMF model are investigated at the long range threshold (α equals to one). It is found that QSS exist and have a diverging lifetime with system size which scales logaritmically with the number of constituents. This contrast to the exhibited power law below the long range threshold (α smaller than one) and the observed finite lifetime beyond. Also even beyond this long range threshold the long range nature of the system is displayed, namely the existence of a phase transition. As a consequence of our findings the definition of a long range system is discussed.     

Asymmetric non-Gaussian effects in a tumor growth model with immunization

The dynamical evolution of a tumor growth model, under immune surveillance and subject to asymmetric non-Gaussian α-stable Lévy noise, is explored. The lifetime of a tumor staying in the range between the tumor-free state and the stable tumor state, and the likelihood of noise-induced tumor extinction, are characterized by the mean residence time and the escape probability, respectively. For various initial densities of tumor cells, the mean residence time and the escape probability are computed with different noise parameters. It is observed that unlike the Gaussian noise or symmetric non-Gaussian noise, the asymmetric non-Gaussian noise plays a constructive role in the tumor evolution in this simple model. By adjusting the noise parameters, the mean residence time can be shortened and the escape probability can be increased, simultaneously. This suggests that a tumor may be mitigated with higher probability in a shorter time, under certain external environmental stimuli.     

Characterization of lifetime distributions in the Ls-sense by a generalized spread

We characterize partial orderings between lifetime distributions by L_s-tailweight. We obtain some ageing properties by comparing distributions with the negative exponential distribution with unit mean.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

Assessing the lifetime performance index of Rayleigh products based on the Bayesian estimation under progressive type II right censored samples

Lifetime performance assessment is important in service (or manufacturing) industries. Hence, lifetime performance index C_L is used to measure the potential and performance of a process, where L is the lower specification limit. In this paper, assuming the conjugate prior distribution and squared-error loss function, this study constructs a Bayes estimator under the Rayleigh distribution with the progressive type II right censored sample. The Bayes estimator of C_L is then utilized to develop a credible interval in the condition of known L. Moreover, we also propose a Bayesian test to assess the lifetime performance of products. Finally, we give two examples and the Monte Carlo simulation to assess the behavior of the lifetime performance index C_L. Moreover, the purchasers can then employ the credible interval and the Bayesian test to determine whether the product performance adheres to the required level.     

A Bayesian approach to the triage problem with imperfect classification

A collection of jobs (or customers, or patients) wait impatiently for service. Each has a random lifetime during which it is available for service. Should this lifetime expire before its service starts then it leaves unserved. Limited resources mean that it is only possible to serve one job at a time. We wish to schedule the jobs for service to maximise the total number served. In support of this objective all jobs are subject to an initial triage, namely an assessment of both their urgency and of their service requirement. This assessment is subject to error. We take a Bayesian approach to the uncertainty generated by error prone triage and discuss the design of heuristic policies for scheduling jobs for service to maximise the Bayes’ return (mean number of jobs served). We identify problem features for which a high price is paid in number of services lost for poor initial triage and for which improvements in initial job assessment yield significant improvements in service outcomes. An analytical upper bound for the cost of imperfect classification is developed for exponentially distributed lifetime cases.     

Non-parametric estimation of lifetime and repair time criteria for a semi-Markov process

In this Note, we model an industrial system by a semi-Markov process where failure and repair phenomena are in mutual competition. A non-parametric estimation method for system component lifetime and repair time distributions and for associated hazard rate functions is proposed. The lifetime and repair time empirical distributions are reduced to two Kaplan–Meier estimators. A numerical example from an industrial system with three components and one repair man modeled by a birth and death process is provided to illustrate the previous results. To cite this article: A.-L. Afchain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions

A system with n independent components which has a k-out-of-n: G structure operates if at least k components operate. Parallel systems are 1-out-of-n: G systems, that is, the system goes out of service when all of its components fail. This paper investigates the mean residual life function of systems with independent and nonidentically distributed components. Some examples related to some lifetime distribution functions are given. We present a numerical example for evaluating the relationship between the mean residual life of the k-out-of-n: G system and that of its components.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.     

Adaptive age replacement strategies based on nonparametric predictive inference

We consider an age replacement problem using nonparametric predictive inference (NPI) for the lifetime of a future unit. Based on n observed failure times, NPI provides lower and upper bounds for the survival function for a future lifetime X_n+1, which are lower and upper survival functions in the theory of interval probability, and which lead to upper and lower cost functions, respectively, for age replacement based on the renewal reward theorem. Optimal age replacement times for X_n+1 follow by minimizing these cost functions. Although the renewal reward theorem implicitly assumes that the corresponding optimal strategy will be used for a long period, we study the effect on this strategy when the observed value for X_n+1, which is either an observed failure time or a right-censored observation, becomes available. This is possible due to the fully adaptive nature of our nonparametric approach, and the next optimal strategy will be for X_n+2. We compare the optimal strategies for X_n+1 and X_n+2 both analytically and via simulation studies. Our NPI-based approach is fully adaptive to the data, to which it adds only few structural assumptions. We discuss the possible use of this approach, and indeed the wider importance of the conclusions of this study to situations where one wishes to combine the statistical aspects of estimating a lifetime distribution with the more traditional operational research approach of determining optimal replacement strategies for lifetime distributions that are assumed to be known.     

Opportunity-based age replacement with a one-cycle criterion

In this paper age replacement (AR) and opportunity-based age replacement (OAR) for a unit are considered, based on a one-cycle criterion, both for a known and unknown lifetime distribution. In the literature, AR and OAR strategies are mostly based on a renewal criterion, but in particular when the lifetime distribution is not known and data of the process are used to update the lifetime distribution, the renewal criterion is less appropriate and the one-cycle criterion becomes an attractive alternative. Conditions are determined for the existence of an optimal replacement age T^* in an AR model and optimal threshold age T_opp^* in an OAR model, using a one-cycle criterion and a known lifetime distribution. In the optimal threshold age T_opp^*, the corresponding minimal expected costs per unit time are equal to the expected costs per unit time in an AR model. It is also shown that for a lifetime distribution with increasing hazard rate, the optimal threshold age is smaller than the optimal replacement age. For unknown lifetime distribution, AR and OAR strategies are considered within a nonparametric predictive inferential (NPI) framework. The relationship between the NPI-based expected costs per unit time in an OAR model and those in an AR model is investigated. A small simulation study is presented to illustrate this NPI approach.     

Inference for constant-stress accelerated degradation test based on Gamma process

This paper proposes a Gamma constant-stress accelerated degradation model based on the principle of the degradation mechanism invariance. The maximum likelihood estimators of the parameters of the proposed model are derived. Based on Cornish–Fisher expansion, the approximate confidence interval for the shape parameter of the Gamma degradation process is developed. Since it is difficult to obtain the exact confidence intervals for other model parameters and some quantities such as the mean degradation in unit time, the quantile and the reliability function of the lifetime at the normal stress level, the generalized confidence intervals for these quantities are proposed. The percentiles of the proposed generalized pivotal quantities can be obtained by the simulation. The performances of the proposed confidence intervals are evaluated by the Monte Carlo simulation method. In the simulation study, the proposed confidence intervals are compared with the Wald and the bootstrap-p confidence intervals. The simulation results show that the proposed confidence intervals outperform the Wald and the bootstrap-p confidence intervals in terms of the coverage percentage. Finally, a real example is used to illustrate the proposed procedures.     

Bondesson's functions in reliability theory

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. Copyright © 1999 John Wiley & Sons, Ltd.     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.