Probability Distribution Of Lifetime Research Articles

Finite weakest-link model of lifetime distribution of quasibrittle structures under fatigue loading

The design of various engineering structures, such as buildings, infrastructure, aircraft, ships, as well as microelectronic components and medical implants, must ensure an extremely low probability of failure during their service lifetime. Since such a low probability is beyond the means of histogram testing, we must rely on some physically based probabilistic model for the statistics of structural lifetime. Attention is focused on structures consisting of quasibrittle materials. These are brittle materials with inhomogeneities that are not negligible compared with the structure size, as exemplified by concrete, fiber composites, tough ceramics, rocks, sea ice, bone, wood, and many more at the micro- or nano-scale. This paper presents a finite weakest-link model of the fatigue lifetime of quasibrittle structures that fail at the fracture of one representative volume element (RVE). In this model, the probability distribution of critical stress amplitude is first derived by assuming a prescribed number of loading cycles and a fixed stress ratio. The probability distribution of fatigue lifetime is then deduced from the probability distribution of critical stress amplitude through the Paris law for fatigue crack growth. It is shown that the present theory matches well with the experimentally measured lifetime histograms of various engineering and dental ceramics, which systematically deviate from the two-parameter Weibull distribution. The theory indicates that the mean fatigue lifetime of quasibrittle structures must strongly depend on the structure size and geometry. Finally, the present model indicates that the probability distribution of fatigue lifetime can be determined from the mean size effect analysis.

Subcritical crack growth in SiNx thin-film barriers studied by electro-mechanical two-point bending

Mechanical failure resulting from subcritical crack growth in the SiNx inorganic barrier layer applied on a flexible multilayer structure was studied by an electro-mechanical two-point bending method. A 10 nm conducting tin-doped indium oxide layer was sputtered as an electrical probe to monitor the subcritical crack growth in the 150 nm dielectric SiNx layer carried by a polyethylene naphthalate substrate. In the electro-mechanical two-point bending test, dynamic and static loads were applied to investigate the crack propagation in the barrier layer. As consequence of using two loading modes, the characteristic failure strain and failure time could be determined. The failure probability distribution of strain and lifetime under each loading condition was described by Weibull statistics. In this study, results from the tests in dynamic and static loading modes were linked by a power law description to determine the critical failure over a range of conditions. The fatigue parameter n from the power law reduces greatly from 70 to 31 upon correcting for internal strain. The testing method and analysis tool as described in the paper can be used to understand the limit of thin-film barriers in terms of their mechanical properties.

Multidimensional smoothing by adaptive local kernel-weighted log-likelihood: Application to long-term care insurance

We are interested in modeling the mortality of long-term care (LTC) claimants having the same level of severeness (heavy claimant). Practitioners often use empirical methods that rely heavily on expert opinions. We propose approaches not depending on an expert’s advice. We analyze the mortality as a function of both the age of occurrence of the claim and the duration of the care. LTC claimants are marked by a relatively complex mortality pattern. Hence, rather than using parametric approaches or models with expert opinions, adaptive local likelihood methods allow us to extract the information from the data more pertinently. We characterize a locally adaptive smoothing pointwise method using the intersection of confidence intervals rule, as well as a global method using local bandwidth correction factors. The latter is an extension of the adaptive kernel method proposed by Gavin et al. (1995) to likelihood techniques. We vary the amount of smoothing in a location-dependent manner and allow adjustments based on the reliability of the data. Tests, and single indices summarizing the lifetime probability distribution are used to compare the graduated series obtained by adaptive local kernel-weighted log-likelihoods to p-spline and local likelihood models.

Some New Results on Uncertain Age Replacement Policy

Age replacement policy is a commonly policy in maintenance management of spare part. It means that a spare part is always replaced at failure or fixed time after its installation, whichever occurs first. An optimal age replacement policy of spare parts concerns with finding the optimal replacement time determined by minimizing the expected cost per unit time. The age of the part was generally assumed to be a random variable in the past literatures, but in many situations, there are few or even no observed data to estimate the probability distribution of part's lifetime. In order to solve this phenomenon, a new uncertain age replacement policy has been proposed recently, in which the age of the part was assumed to be an uncertain variable. This paper discusses the optimal age replacement policies by dealing with the parts' lifetimes as different distributed uncertain variables. Several results on the optimal age replacement time are provided when the lifetimes are described by the uncertain linear, zigzag and lognormal distributions.

Modified Inverse Weibull Distribution

A generalized version of four parameter modified inverse weibull distribution (MIWD) is introduced in this paper. This distribution generalizes the following distributions: (1) Modified Inverse exponential distribution, (2) Modified Inverse Rayleigh distribution, (3) Inverse weibull distribution. We provide a comprehensive description of the mathematical properties of the modified inverse weibull distribution along with its reliability behaviour. We derive the moments, moment generating function and examine the order statistics. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. The inverse weibull distribution is the life time probability distribution which is used in the reliability engineering discipline. The inverse weibull distribution can be used to model a variety of failure characteristics such as infant mortality, useful life and wear-out periods. Reliability and failure data both from life testing and in service records which is often modeled by the life time distributions such as the inverse exponential, inverse Rayleigh, inverse Weibull distributions. In this research we have developed a new reliability model called modified inverse weibull distribution. This paper focuses on all the properties of this model and presents the graphical analysis of modified inverse weibull reliability models. This paper present the relationship between shape parameter and other properties such as non- reliability function, reliability function, instantaneous failure rate, cumulative instantaneous failure rate models. This life time distribution is capable of modeling of various shapes of aging and failure criteria.The proposed model can be used as an alternative to inverse generalized exponential, inverse generalized Rayleigh, inverse generalized weibull distributions. The cumulative distribution function (CDF) of the Inverse weibull distribution is denoted by

Analysis of link lifetime in wireless mobile networks

Link lifetime (LL) analysis is crucially important in designing and evaluating wireless mobile network protocols. Although a considerable amount of research has been conducted on this issue, the current paper studies several aspects of link lifetime for the first time, e.g. a closed form expression for the probability distribution of the residual link lifetime (RLL) for the constant velocity mobility model is obtained, which facilitates marginal studies on the behavior of link lifetime. As a case, using a tail analysis of LL and RLL distributions, we analytically prove that they are heavy-tailed. Using this outcome, a self-similar behavior in the traffic pattern of a sample delay tolerant network is justified analytically. To extend the model, the effect of stationary nodes, the effect of buffer zone, and the effect of link establishment policies between different node categories on the link lifetime are investigated. Extensive simulation experiments are performed to validate the obtained analytical results and derivations precisely. Moreover, simulation results confirm that the analytic results can also be used for two well known mobility models (random waypoint and random direction) with a negligible error.

Size effect on strength and lifetime probability distributions of quasibrittle structures

Engineering structures such as aircraft, bridges, dams, nuclear containments and ships, as well as computer circuits, chips and MEMS, should be designed for failure probability < 10 − 6–10 − 7 per lifetime. The safety factors required to ensure it are still determined empirically, even though they represent much larger and much more uncertain corrections to deterministic calculations than do the typical errors of modern computer analysis of structures. The empirical approach is sufficient for perfectly brittle and perfectly ductile structures since the cumulative distribution function (cdf) of random strength is known, making it possible to extrapolate to the tail from the mean and variance. However, the empirical approach does not apply to structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared to structure size. This paper presents a refined theory on the strength distribution of quasibrittle structures, which is based on the fracture mechanics of nanocracks propagating by activation energy controlled small jumps through the atomic lattice and an analytical model for the multi-scale transition of strength statistics. Based on the power law for creep crack growth rate and the cdf of material strength, the lifetime distribution of quasibrittle structures under constant load is derived. Both the strength and lifetime cdf’s are shown to be size- and geometry-dependent. The theory predicts intricate size effects on both the mean structural strength and lifetime, the latter being much stronger. The theory is shown to match the experimentally observed systematic deviations of strength and lifetime histograms of industrial ceramics from the Weibull distribution.

Reliability Analysis of Zero‐Failure Data with Poor Information

For costly and dangerous experiments, growing attention has been paid to the problem of the reliability analysis of zero‐failure data, with many new findings in world countries, especially in China. The existing reliability theory relies on the known lifetime distribution, such as the Weibull distribution and the gamma distribution. Thus, it is ineffective if the lifetime probability distribution is unknown. For this end, this article proposes the grey bootstrap method in the information poor theory for the reliability analysis of zero‐failure data under the condition of a known or unknown probability distribution of lifetime. The grey bootstrap method is able to generate many simulated zero‐failure data with the help of few zero‐failure data and to estimate the lifetime probability distribution by means of an empirical failure probability function defined in this article. The experimental investigation presents that the grey bootstrap method is effective in the reliability analysis only with the few zero‐failure data and without any prior information of the lifetime probability distribution. Copyright © 2011 John Wiley &amp; Sons, Ltd.

A profit assessment model for equipment inspection and replacement under renewing free replacement warranty policy

Abstract For many types of equipment whose maintenance requires skilled labor or whose random failure may have serious consequences, such as aircraft engines, medical equipment and others, customers generally prefer to entrust maintenance activities to the manufacturer or any of its representatives. The latter must, in turn, provide the customer with a maintenance program that is economically viable for both parties. In this study, the customer and the manufacturer agree on what follows regarding a non-self-announcing failure equipment whose state is only known through inspection: during the warranty period, the manufacturer inspects the equipment according to a specified schedule which he establishes taking into account that the costs incurred for inspections and replacements are supported by him during the warranty period. After the expiration of the warranty, inspections and replacements are performed by the manufacturer and billed to the customer. Moreover, penalties related to inactivity periods between failures and their detection are always supported by the manufacturer. The warranty is applicable to any equipment replaced during the validity period of the service contract. This type of contract generates a profit for the manufacturer. In this paper, an analytical model taking into account the commitments of both parties has been developed. The model allows generating the instants (x1, x2,…, xn) at which the inspections must be performed and the corresponding expected profit for the manufacturer while considering the warranty period offered on the market for similar equipment. An algorithm has also been developed to generate the inspection instants given the costs structure and the lifetime probability distribution of the equipment. In a context where business models argue for a greater implication of suppliers towards their customers, the proposed decision model may be relevant and very useful.

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: II. Fatigue crack growth, lifetime and scaling

This paper extends the theoretical framework presented in the preceding Part I to the lifetime distribution of quasibrittle structures failing at the fracture of one representative volume element under constant amplitude fatigue. The probability distribution of the critical stress amplitude is derived for a given number of cycles and a given minimum-to-maximum stress ratio. The physical mechanism underlying the Paris law for fatigue crack growth is explained under certain plausible assumptions about the damage accumulation in the cyclic fracture process zone at the tip of subcritical crack. This law is then used to relate the probability distribution of critical stress amplitude to the probability distribution of fatigue lifetime. The theory naturally yields a power-law relation for the stress-life curve (S-N curve), which agrees with Basquin's law. Furthermore, the theory indicates that, for quasibrittle structures, the S-N curve must be size dependent. Finally, physical explanation is provided to the experimentally observed systematic deviations of lifetime histograms of various ceramics and bones from the Weibull distribution, and their close fits by the present theory are demonstrated.

Understanding bursty behavior in midlatitude geomagnetic activity

We provide a new example of a natural system exhibiting statistical signatures of complex scaling behavior predicted by the contemporary theory of nonequilibrium phase transitions. We examine statistical properties of bursty multiscale energy dissipation in the inner magnetosphere of Earth based on the dynamics of the SYM‐H index, a global marker of low‐latitude geomagnetic fluctuations. We show that on average, and for time scales shorter than 2 h, temporal development of SYM‐H bursts follows an algebraic form consistent with the predictions from the theory of nonequilibrium phase transitions. Probability distributions of sizes and lifetimes of the activity bursts reveal no characteristic scales other than the scales imposed by technical limitations of the analysis. This behavior is observed for a wide range of SYM‐H burst durations starting from about 5 min up to 10–15 days. The power law exponents describing the probability distributions suggest that the main energy dissipation in the inner magnetosphere takes place because of large activity bursts such as major space storms as opposed to smaller activations whose contribution is less significant despite their much higher relative occurrence. The results obtained provide statistical evidence that the energy dissipation mechanisms associated with magnetospheric activity in the inner magnetosphere are essentially “scale‐free,” displaying dynamical and statistical self‐similarity. Our results can also be used for validating existing and future ring current models in terms of their ability to correctly represent the cross‐scale coupling effects in this system. They show what could be the first quantitative evidence for the same universality class as directed percolation in a natural system.

Anomalies of water and hydrogen bond dynamics in hydrophobic nanoconfinement

Using molecular dynamic (MD) simulations of the TIP5P model of water, we investigate theeffect of hydrophobic confinement on the anomalies of liquid water. For confinement lengthLz = 1.1 nm, such that there are 2–3 molecular layers of water, we find the presence of the bulk-like densityand diffusion anomaly in the lateral directions. However, the lines of these anomalies in theP–T plane are shifted tolower temperatures (ΔT≈40 K) and pressures compared to bulk water. Furthermore, we introduce a method to calculatethe effective diffusion constant along the confinement direction and find that the diffusionanomaly is absent. Moreover, we investigate the hydrogen bond dynamics of confinedwater and find that the hydrogen bond dynamics preserves the characteristicsof HB dynamics in bulk water, such as a non-exponential behavior followed byan exponential tail of HB lifetime probability distributions and an Arrheniustemperature dependence of the average HB lifetime. The average number andlifetime of HBs decrease in confined water compared to bulk water at the sametemperature. This reduction may be the origin of the reasons for the different physicalproperties of confined water from bulk water, such as the 40 K temperature shift.

Subcritical crack growth law and its consequences for lifetime statistics and size effect of quasibrittle structures

For brittle failures, the probability distribution of structural strength and lifetime are known to be Weibullian, in which case the knowledge of the mean and standard deviation suffices to determine the loading or time corresponding to a tolerable failure probability such as 10−6. Unfortunately, this is not so for quasibrittle structures, characterized by material inhomogeneities that are not negligible compared with the structure size (as is typical, e.g. for concrete, fibre composites, tough ceramics, rocks and sea ice). For such structures, the distribution of structural strength was shown to vary from almost Gaussian to Weibullian as a function of structure size (and also shape). Here we predict the size dependence of the distribution type for the lifetime of quasibrittle structures. To derive the lifetime statistics from the strength statistics, the subcritical crack growth law is requisite. This empirical law is shown to be justified by fracture mechanics of random crack jumps in the atomic lattice and the condition of equality of the energy dissipation rates calculated on the nano-scale and the macro-scale. The size effect on the lifetime is found to be much stronger than that on the structural strength. The theory is shown to match the experimentally observed systematic deviations of lifetime histograms from the Weibull distribution.

Scaling of strength and lifetime probability distributions of quasibrittle structures based on atomistic fracture mechanics

The failure probability of engineering structures such as aircraft, bridges, dams, nuclear structures, and ships, as well as microelectronic components and medical implants, must be kept extremely low, typically <10(-6). The safety factors needed to ensure it have so far been assessed empirically. For perfectly ductile and perfectly brittle structures, the empirical approach is sufficient because the cumulative distribution function (cdf) of random material strength is known and fixed. However, such an approach is insufficient for structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared with the structure size. The reason is that the strength cdf of quasibrittle structure varies from Gaussian to Weibullian as the structure size increases. In this article, a recently proposed theory for the strength cdf of quasibrittle structure is refined by deriving it from fracture mechanics of nanocracks propagating by small, activation-energy-controlled, random jumps through the atomic lattice. This refinement also provides a plausible physical justification of the power law for subcritical creep crack growth, hitherto considered empirical. The theory is further extended to predict the cdf of structural lifetime at constant load, which is shown to be size- and geometry-dependent. The size effects on structure strength and lifetime are shown to be related and the latter to be much stronger. The theory fits previously unexplained deviations of experimental strength and lifetime histograms from the Weibull distribution. Finally, a boundary layer method for numerical calculation of the cdf of structural strength and lifetime is outlined.

Probability Models for Optimal Sparing of Distribution Network Transformers

A probabilistic approach for the determination of the optimum number of spare transformers is presented providing minimum annual cost consisting of capital cost for spares and transformer outage costs. The model suggested can cover any transformer unit lifetime and renewal time probability distribution. This makes it possible to account for the effects of transformer deterioration due to aging and to realistically, based upon empirical data, assess the impacts of the renewal time by treating it as a stochastic rather than as a deterministic variable lasting the entire planning period. The comparison of the model presented with the Poisson distribution based approach, widely used in sparing analyses in the past, has shown that the negligence of aging and of the stochastic nature of renewal times can lead to worse than optimal solutions.

Collective dynamics of bursty particle precipitation initiating in the inner and outer plasma sheet

Abstract. Using multiscale spatiotemporal analysis of bursty precipitation events in the nighttime aurora as seen by the POLAR UVI instrument, we report a set of new statistical signatures of high- and low-latitude auroral activity, signaling a strongly non-uniform distribution of dissipation mechanism in the plasma sheet. We show that small-scale electron emission events that initiate in the equatorward portion of the nighttime auroral oval (scaling mode A1) have systematically steeper power-law slopes of energy, power, area, and lifetime probability distributions compared to the events that initiate at higher latitudes (mode B). The low-latitude group of events also contain a small but energetically important subpopulation of substorm-scale disturbances (mode A2) described by anomalously low distribution exponents characteristic of barely stable thermodynamic systems that are prone to large-scale sporadic reorganization. The high latitude events (mode

Bayesian algorithm implementation in a real time exposure assessment model on benzene with calculation of associated cancer risks.

The objective of the current study was the development of a reliable modeling platform to calculate in real time the personal exposure and the associated health risk for filling station employees evaluating current environmental parameters (traffic, meteorological and amount of fuel traded) determined by the appropriate sensor network. A set of Artificial Neural Networks (ANNs) was developed to predict benzene exposure pattern for the filling station employees. Furthermore, a Physiology Based Pharmaco-Kinetic (PBPK) risk assessment model was developed in order to calculate the lifetime probability distribution of leukemia to the employees, fed by data obtained by the ANN model. Bayesian algorithm was involved in crucial points of both model sub compartments. The application was evaluated in two filling stations (one urban and one rural). Among several algorithms available for the development of the ANN exposure model, Bayesian regularization provided the best results and seemed to be a promising technique for prediction of the exposure pattern of that occupational population group. On assessing the estimated leukemia risk under the scope of providing a distribution curve based on the exposure levels and the different susceptibility of the population, the Bayesian algorithm was a prerequisite of the Monte Carlo approach, which is integrated in the PBPK-based risk model. In conclusion, the modeling system described herein is capable of exploiting the information collected by the environmental sensors in order to estimate in real time the personal exposure and the resulting health risk for employees of gasoline filling stations.

Scale‐free and scale‐dependent modes of energy release dynamics in the nighttime magnetosphere

Based on a spatiotemporal analysis of POLAR UVI images, we show for the first time that energy, power, area and lifetime probability distributions of electron precipitation events in the nighttime auroral oval have a significant latitudinal dependence. The low‐latitude group of the events contains a distinct subpopulation of strong auroral disturbances violating the uniform power‐law behavior reported in previous publications, while the high latitude group is described by nearly perfect power‐law statistics over the entire range of scales studied, in agreement with earlier findings. The results obtained indicate that the inner and outer portions of the plasma sheet may be characterized by substantially different scaling regimes of bursty energy dissipation suggestive of different universality classes and/or driving conditions associated with multiscale turbulence in these regions.

Algorithm for equalization of cluster lifetimes in a multi-level Beacon enabled 802.15.4 sensor network

We consider the problem of maintaining the prescribed event sensing reliability while maximizing cluster and network lifetime in a multi-cluster 802.15.4 sensor network. Clusters are connected through bridges which also act as cluster coordinators; both ordinary nodes and bridges resolve contention using the CSMA-CA algorithm. Cluster lifetime is maximized through the use of redundant sensors which are periodically sent to sleep using a simple distributed activity management algorithm. Network lifetime is maximized by equalizing energy consumption per backoff period in all clusters through the adjustment of the number of nodes. We model this problem analytically using the datasheet for tmote_sky ultra low power IEEE 802.15.4 compliant wireless sensor module [tmote sky lowpower wireless sensor module, Moteiv Corporation, San Francisco, CA, www.moteiv.com, tmote datasheet 802.15.4, 2006] and derive the probability distribution of the network lifetime. We also derive the expression for node count that compensates for the increased load due to contention caused by the bridge. Numerical results show that this technique easily equalizes cluster lifetimes.

Power law burst lifetime distribution of the SYM‐H index

We have examined the burst lifetime distribution functions of solar wind VBs and ɛ for 1995–2005 and compared them with the same for contemporaneous SYM‐H. The analysis yields clear power‐law exponents of the lifetime probability distributions. When we analyzed three years around solar minimum we found similar scaling behavior for all the variables, including SYM‐H. However, for the three years around solar maximum the difference between solar wind and SYM‐H scaling was very clear. The fact that power laws were consistently observed for different activity thresholds show that these features are robust and repeatable, although scaling can vary depending on solar cycle. During solar maximum it appears that the scaling properties of the low‐latitude magnetosphere, whose output is recorded by SYM‐H, are not purely a direct response to the scale‐free properties of the solar wind but are due to inherent properties of the magnetosphere. Since SYM‐H scaling is remarkably robust, irrespective of solar cycle, it could be that the solar wind never acts as a direct driver for the SYM‐H scaling.