Load-capacity interference and the bathtub curve

Load-capacity (stress-strength) interference theory is used to derive a heuristic failure rate for an item subjected to repetitive loading which is Poisson distributed in time. Numerical calculations are performed using Gaussian distributions in load and capacity. Infant mortality, constant failure rate (Poisson failures), and aging are shown to be associated with capacity variability, load variability, and capacity deterioration, respectively. Bathtub-shaped failure rate curves are obtained when all three failure types are present. Changes in load or capacity distribution parameters often strongly affect the quantitative behavior of the failure-rate curves, but they do not affect the qualitative behavior of the bathtub curve. Neither is it likely that the qualitative behavior will be affected by the use of nonGaussian distributions. The numerical results, however, indicate that infant mortality and wear-out failures interact strongly with load variability. Thus bathtub curves arising from this model cannot be represented as simple superpositions of independent contributions from the three failure types. Only if the three failure types arise from independent failure mechanisms or in different components is it legitimate simply to sum the failure rate contributions     

Statistical Analysis of a Compound Power-Law Model for Repairable Systems

A compound (mixed) Poisson distribution is sometimes used as an alternative to the Poisson distribution for count data. Such a compound distribution, which has a negative binomial form, occurs when the population consists of Poisson distributed individuals, but with intensities which have a gamma distribution. A similar situation can occur with a repairable system when failure intensities of each system are different. A more general situation is considered where the system failures are distributed according to nonhomogeneous Poisson processes having Power Law intensity functions with gamma distributed intensity parameter. If the failures of each system in a population of repairable systems are distributed according to a Power Law process, but with different intensities, then a compound Power Law process provides a suitable model. A test, based on the ratio of the sample variance to the sample mean of count data from s-independent systems, provides a convenient way to determine if a compound model is appropriate. When a compound Power Law model is indicated, the maximum likelihood estimates of the shape parameters of the individual systems can be computed and homogeneity can be tested. If equality of the shape parameters is indicated, then it is possible to test whether the systems are homogeneous Poisson processes versus a nonhomogeneous alternative. If deterioration within systems is suspected, then the alternative in which the shape parameter exceeds unity would be appropriate, while if systems are undergoing reliability growth the alternative would be that the shape parameter is less than unity.     

A two non-identical three-state units redundant system with common-cause failures and one standby unit

This paper presents a mathematical model for predicting a two non-identical three-state active units redundant system with common-cause failures and one standby unit. The units may fail in either of two mutually exclusive failure modes or by the occurrence of common-cause failures. System is only repaired when all the units fail (including the standby unit). The failure rates of units are constant and system repair times are arbitrarily distributed. Laplace transforms of the state probabilities are derived.     

Yield modeling of acoustic charge transport transversal filters

This paper presents a yield model for acoustic charge transport transversal filters. This model differs from previous IC yield models in that it does not assume that individual failures of the nondestructive sensing taps necessarily cause a device failure. A redundancy in the number of taps included in the design is explained. Poisson statistics are used to describe the tap failures, weighted over a uniform defect density distribution. A representative design example is presented. The minimum number of taps needed to realize the filter is calculated, and tap weights for various numbers of redundant taps are calculated. The critical area for device failure is calculated for each level of redundancy. Yield is predicted for a range of defect densities and redundancies. To verify the model, a Monte Carlo simulation is performed on an equivalent circuit model of the device. The results of the yield model are then compared to the Monte Carlo simulation. Better than 95% agreement was obtained for the Poisson model with redundant taps ranging from 30% to 150% over the minimum     

The limit of sum of Markov Bernoulli variables in systemreliability evaluation

For 2-state maintainable and repairable systems modeled by nonstationary Markov chains, a limiting compound Poisson distribution is derived for the sum of Markov Bernoulli random variables. The result is useful for estimating the distribution of the sum of negative-margin hours in a boundary-crossing scenario involving any physical system with interarrival times of system failures that are negative-exponentially distributed, where the positive- and negative-margin states denote desirable and undesirable operating conditions. three test cases from the IEE Reliability Test system are analyzed. The mean and variance/mean ratio are generated for each case. The results of compound Poisson distribution estimation for the sum of Markov Bernoulli random variables with varying probabilities can be used to solve the problem of estimating the distribution of the popular reliability index (cumulated loss-of-load hours) in large electric power generation systems where the hourly load demand varies     

Software-reliability growth model: primary-failures generatesecondary-faults under imperfect debugging

Little work has been done on extending existing models with imperfect debugging to the more realistic situation where new faults are generated from unsuccessful attempts at removing faults completely. This paper presents a software-reliability growth model which incorporates the possibility of introducing new faults into a software system due to the imperfect debugging of the original faults in the system. The original faults manifest themselves as primary failures and are assumed to be distributed as a nonhomogeneous Poisson process (NHPP). Imperfect debugging of each primary failure induces a secondary failure which is assumed to occur in a delayed sense from the occurrence time of the primary failure. The mean total number of failures, comprising the primary and secondary failures, is obtained. The authors also develop a cost model and consider some optimal release-policies based on the model. Parameters are estimated using maximum likelihood and a numerical example is presented     

Probabilistic analysis of a pulverizer system with common-cause failures

This paper presents a newly developed mathematical stochastic model to represent ‘n’ number of active redundant pulverizers with a cold standby pulverizer. The active redundant pulverizers may also fail due to common-cause failures. Failed system repair times are arbitrarily distributed. Laplace transforms of the state probability and point-wise availability equations are developed. In addition, the Laplace transforms of the state probability equations of three special case models for n=2, n=4 and n=5 are presented. For constant failure and repair rates and n=2 (with one standby) steady state system availability plots are shown.     

Reliability Analysis of Regular Distributed Chain Structures

This paper considers the reliability analysis of chain structures of identical sections. The sections are distributed in the sense that there is more than one input and output node to each section. System failure occurs if no path exists between system input and output. (System branch failures are by open circuit only.) This model is particularly applicable to redundant communication systems. It is shown that the distributed chain structure is asymptotically equivalent to a series structure of identical equivalent sections, assuming that the number of sections is large and that the branch reliabilities are equal. The reliability of the equivalent section is obtained as the subdominant eigenvalue of a transition matrix. The error in this representation is negligible for system branch reliabilities large.     

A recurrence relation for calculating a reliability increment in aseries-parallel system

A recurrence relation is given for the easy calculation of an increment in the reliability of a series-parallel system if the number of redundant elements is increased by one. This relation is used for cold or standby redundancy when the number of failures obeys the Poisson distribution. The formulas have been implemented in various computer programs for system reliability calculation and optimization. The experience with many real and hypothetical examples has shown no computational problems. There is no need to care either about the magnitude of the number of redundancies or about the accuracy of computation. This shows the advantage of this algorithm over the approximation proposed by M. Messinger and M. Shooman (1970)     

System Reliability in the Presence of Common-Cause Failures

This paper presents a method for calculating the reliability of a system depicted by a reliability block diagram, with identically distributed components, in the presence of common-cause failures. To represent common-cause failures, we use the Marshall & Olkin formulation of the multivariate exponential distribution. That is, the components are subject to failure by Poisson failure processes that govern simultaneous failure of a speciflc subset of the components. The method for calculating system relability requires that a procedure exist for determining system reliability from component reliabilities under the statistically-independent-component assumption. The paper includes several examples to illustrate the method and compares the reliability of a system with common-cause failures to a system with statistically-independent components. The examples clearly show that common-cause failure processes as modeled in this paper materially affect system reliability. However, inclusion of common-cause failure processes in the system analysis introduces the problem of estimating the rates of simultaneous failure for multiple components in addition to their individual failure rates.     

Multi-state device redundant systems with common-cause failures and one standby unit

This paper presents two mathematical models. Model I represents a two identical unit active redundant system whose units may fail in either of the two mutually exclusive failure modes. Similarly, model II represents a two non-identical unit active parallel system whose units either fail or survive. In addition, models I and II comprise the occurrence of common-cause failures and one standby unit. Systems are only repaired when all the system units fail (including the standby unit). System repair times are arbitrarily distributed. Laplace transforms of the state probability equations are developed.     

A reliability model for a k-out-of-N:G redundant system with multiple failure modes and common cause failures

The paper presents a reliability model of a k-out-of-N:G redundant system with M mutually exclusive failure modes and common cause failures. Failed system repair times are arbitrarily distributed. The system is in a failed state when (N−k+1) units failed or a common cause failure occurred. Laplace transforms of the state probabilities and the availability of the system are derived. Finally, the system steady-state availability is also reported.     

An optimal real‐time pricing algorithm under load uncertainty and user number variation in smart grid

The progress of real‐time communication systems for smart grid has led to the importance of real‐time pricing becoming more highlighted. There are many investigations that have already been done. Real‐time pricing frameworks have proposed an implemented distributed algorithm with or without consideration of effect of load uncertainty. In some existing literature the effect of different types of load uncertainty models on average consumption and generated capacity is considered. However, the number of users is considered to be constant. In this paper, it is assumed that the number of users is varying independently and randomly. In this case the effect of variation of number of users on the basis of Poisson process and uniform distribution is compared with results from previous works, when bounded uncertainty model was applied for added noise to the consumption. Simulation results indicate that when users vary on the basis of Poisson distribution, the waste of energy decreases and the welfare increases.     

On Predicting Failures in a Future Time Period From Known Observations

The problem of a prediction interval for the number of failures of a system during a future time period knowing the failures observed during a time interval in the past is considered. It is assumed that the failures follow a Poisson process. Now, if the failures in both time intervals were known, they could be used to test the hypothesis that the two observations were generated by the same Poisson law. By appropriately inverting the inequalities for the critical region of the above test, it is possible to generate the prediction limits for the number of failures in one time interval by observing the failures in the other time interval, provided both observations are subject to the same Poisson law.     

Spurious exponentiality observed when incorrectly fitting adistribution to nonstationary data

Failure data for a repairable system can be represented either by a set of chronologically ordered arrival times at which the system failed, or by a set of interarrival times defined as the times observed between successive failures (ignoring repair times in both cases). The two representations are mathematically equivalent if the chronological order of the interarrival times is maintained. Methods aimed at describing the distribution of the observed interarrival times are meaningful only if the interarrival times are identically distributed. In all other cases, such analyses are meaningless and often result in maximally misleading impressions about the system behavior, as demonstrated here by several examples. That is, when the information in the chronological order of interarrival times is ignored, they often appear spuriously exponential, leading to the impression that the system can be modeled using a homogeneous Poisson process. Misunderstandings of this nature can be avoided by applying an appropriate test for trend before attempting to fit a distribution to the interarrival times. If evidence of trend is determined, then a nonstationary model such as the nonhomogeneous Poisson process should be fitted using the chronologically ordered data     

Multi-inverter UPS system with redundant load sharing control

The concept of a redundant multi-inverter UPS (uninterruptible power supply) system which includes extended monitoring of the status and the operating conditions of all power electronic equipment is described. Each block of the UPS system is monitored by two independent microcomputers that process the same data. The microcomputers are part of a redundant distributed monitoring system that is separately interlinked by two serial data buses through which they communicate. They establish a hierarchy among the participating blocks by defining one of the healthy inverter blocks as the master. The actual master runs the central synchronizing unit for the entire system, whereas the slave units perform the control of equal active and reactive load sharing. Operation and fault detection are experimentally illustrated in a dual inverter system with a rating of 10 kVA of redundant power     

A k-out-of-N:G redundant system with cold standby units and common-cause failures

This paper presents a k-out-of N:G redundant system with M cold standby units, r repair facilities and common-cause failures. The constant failure rates of the operating and cold standby units are different. Failed system repair times are arbitrarily distributed. The system is in a failed state when (N+M?k+1) units failed or a common-cause occurred. Laplace transforms of the state probabilities, the availability of the system and the system steady-state availability are derived.     

Applications and extensions of the chains-of-rare-events model

The chains-of-rare-events model (ChRE) is extended. The ChRE was originally introduced in order to analyze occurrences which can be produced with simple, double, triple, etc., multiplicity. In the original ChRE, each occurrence of multiplicity (i) is independently distributed according to a Poisson law with parameter λ_i ; and a simple relation for these parameters is considered. In this way, ChRE can be applied to analyze outcomes produced in occurrences with multiple events, such as failures, queuing, automobile accidents, telephone calls, and accidents in a factory. The original ChRE is extended to analyze the total number of outcomes in which a given total number of occurrences of different multiplicity occur. The model can be analyzed as a compound Poisson distribution where the compounding distribution is Poisson truncated at zero. Applications to reliability and queuing processes data are presented. The results compare favorably with those from other models     

Sensitivity of Bayes Estimates of Reciprocal MTBF and Reliability to an Incorrect Failure Model

A unit is placed on test for a fixed time, and the number of failures is observed. The stochastic process generating the failures is assumed to have s-independent, Erlang distributed times between failures. Bayes estimates of reciprocal MTBF (RMTBF) and reliability are given where the loss function is squared error and the prior distribution for RMTBF is gamma. We investigate what happens to the Bayes estimates when the shape parameter in the failure model is incorrectly specified (e.g., the failure model is assumed to be Poisson when it is not). This question is answered for parameters which are typical of a wide range of actual military equipment failure data. As the shape parameter in the failure model changes 1) there is only a small to moderate change in the estimates of RMTBF; 2) there is a small to moderate change in the estimate of reliability for small numbers of failures but a larger change for an unusually large number of failures; 3) there is little change in the s-efficiencies of the estimates as measured by s-expected squared error loss. For the range of parameters in this study, not much is lost in s-efficiency by restricting attention to the mathematically tractable Erlang failure model instead of using a more general gamma failure model.