Volume 21, Issue 1

The paper presents a model of multistage degraded system subjected to random failures and partial repairs. A transient analysis is performed and transient probabilities are calculated to find the availability, the means of life time and operational life time. In the paper, constant state dependent transition rates for the degradation process as well as failure process are considered. On the other hand the partial repairs follow general distributions. This paper extends previous systems that can be considered as particular cases of this one. Numerical examples are provided to illustrate applicability of the expressions that are obtained throughout the paper.

The question of how to determine an appropriate sample size n for an attribute control chart has been the subject of a number of research papers. Unfortunately, this stream of research has neither produced tools that are easy to understand, nor it does provide a manager with a good understanding of the tradeoffs involved by the sample size decision. In this paper, we propose a practical method of determining an appropriate sample size that allows the manager to evaluate the tradeoffs between the ability to detect an out-of-control process, the probability of generating a “false alarm” and the sample size (which will determine the cost of implementing the sampling plan).

Counting processes are used in modelling of repairable systems. In Bayesian parameter inference the choice of fitted prior distributions and loss functions has great importance. This paper deals with studies of prior and posterior densities. Simulation enables comparisons of estimators obtained with different loss functions. Especially, in the case of a Weibull process, properties relative to prior and posterior distributions are investigated.

The detection of upward shifts in a process parameter using a CUSUM scheme can be improved by using an upper one-sided combined CUSUM–Shewhart scheme. Considerable advantage is to be gained since combined schemes take advantage of two well known facts: the Shewhart schemes behave well in case of a large shift, while CUSUM schemes allow a fast detection of small and moderate shifts. Having this in mind, upper one-sided combined CUSUM–Shewhart schemes for binomial data are discussed in detail in this paper. Numerical comparisons between upper one-sided combined CUSUM–Shewhart schemes and upper onesided CUSUM schemes with a 50% head start are also carried out, leading to – what we believe – surprising results.

Most of the approaches suggested during the last decades for solving life testing problems are markedly different from those used in the related but wider area of goodness of fit problems. In this paper, it is demonstrated that applying the goodness of fit approach makes sense also for solving life testing problems and that it results in simpler procedures that are asymptotically equivalent or better than standard ones. They may also have superior finite sample behavior. Testing exponentiality against the class of “new better than used renewal failure rate” (NBURFR) or “new better than used average renewal failure rate” (NBARFR) as proposed by Abouammoh and Ahmed [Statistcs & Probability Letter 14: 211-217, 1992] are addressed here. The asymptotic properties are derived for these tests in the case of complete data. The censored data case is also studied. A practical application of the tests to data taken from medical sciences is presented.

This paper deals with the log-logistic distribution as a life time model. Control charts and the corresponding control limits are developed analogous to Shewhart-charts for the process mean and process range. The proposed control limits are compared with those of Shewhart in detecting out of control signals for various sample sizes.

The article addresses the stability index of stochastic processes (time series) based on the statistical process control approach. The proposed stability index reveals the relation between assignable and common variabilities: the higher the proportion of the assignable variability, the lower the stability. It is calculated as a ratio of the average moving range to the maximum range between the time series mean and the means of the samples with the fixed end. To address sensitivity of the stability index to process changes such as mean and variance shifts, trends, seasonality, autocorrelation, and sudden changes (outliers), hundred of thousands of simulated time series are analyzed. Considering the impact of sample size, amplitude of process change, moment of process change, and type of distribution, the full factorial design of experiment has been applied. The proposed calculation of the stability index and suggested target levels serve as a guide for interpreting stability of the stochastic processes by index. A combination of stability and capability indexes emphasizes dynamic aspects of the process ability regarding specifications and are presented as a dynamic capability index.

A single control chart called Multivariate Maximum Control Chart (Max-Mchart) is proposed in this article that is capable of monitoring the process, with the quality of an item being determined by several correlated quality characteristics. This scheme simultaneously monitors the process location and process spread using only one chart. The chart quickly detects both small and large shifts in the process parameters and indicates the parameter that has shifted. Compared with other recently proposed single multivariate control charts, this chart quickly detects small shifts in the process mean and is also simple to use, as it applies the well known Shewhart control charts procedures to compute the average run lengths.

This paper proposes a condition based preventive maintenance (CBPM) policy for Markov deteriorating systems. The proposed model considers deterioration and random failures with minimal and major preventive maintenance (PM). Minimal repairs are carried out after every random failure, and the device is getting replaced after the occurrence of the deterioration failure. The system undergoes random inspections to assess the condition; mean time between inspections is exponentially distributed. Based upon the observed condition of the device, do nothing, minimal PM, and major PM may take place. Minimal PM makes the system one deterioration stage younger, while Major PM makes the system τ deterioration stages ( τ > 1) younger. The proposed models consider increasing intensity for the random failures. An exact recursive algorithm computes the steady-state probabilities of the system. Optimal solutions of the model are derived based on two criteria, viz., (a) availability maximization, and (b) total cost minimization.

There are many instances where the quality of a product is determined by the joint level of several characteristics. Under these types of situations, one wishes to detect quickly any departure from a satisfactory level, while at the same time it is of interest to know which characteristic(s) is(are) responsible for the deviation. The most common and popular solution to the first query is to use a T 2 control chart which is an aggregate monitoring function, when the mean vector of the correlated characteristics is the major concern and the normal approximation may be used. However, the answer to the second query is not so easy. The lack of a simple method of selecting the ‘out of control’ variable has made the routine use of T 2 control chart impractical so far. This paper provides a review of different techniques available to identify the culprit variable(s), when a T 2 control chart gives an out-of-control signal.

In 1933 A.N. Kolmogorov's famous essay Grundbegriffe der Wahrscheinlichkeitsrechnung was published by the publisher Julius Springer in Berlin. It contained an axiomatization of probability theory and led to an unimagined upturn of probability theory and its application. Especially the effect on application must surprise because based on the the axioms, probability theory had become a genuine part of mathematics without any concrete real-world meaning. Kolmogorov states himself about the field of probabilities : “What the elements of this set represent is of no importance in the purely mathematical development of the theory of probability.” In his treatise Kolmogorov also deals with the concept of independence and introduces it as a purely mathematical quantity without regarding any real-world meaning. In this paper it is shown that the mathematical concept of independence of events is inappropriate from a more realistic point of view.