Yu Hayakawa

Mixed poisson-type processes with application in software reliability☆

We introduce one generalization of the mixed Poisson process referred to as the mixed Poisson-type process. An approach taken here is to assume the l 1 -isotropy of interevent times and to define the parameter as a function of observable quantities. An inhomogeneous variant of the new process is studied as a software reliability model. As an illustration a numerical example is analyzed via the Gibbs sampler. The mixed Poisson-type process is constructed through probabilistic behaviour of observable quantities and includes the mixed Poisson process as the limiting case.

Warranty cost analysis: quasi‐renewal inter‐repair times

Purpose – To provide a brief introduction to warranty analysis and a classification of general repairs. To introduce the notion of accelerated probability distribution and use it to model imperfect warranty repairs.Design/methodology/approach – The notion of accelerated probability distribution is discussed and its similarity with quasi‐renewal and geometric processes is observed. An approach to modeling imperfect warranty repairs based on the accelerated probability distributions is presented, and the corresponding expected warranty cost over the warranty period under non‐renewing free replacement warranty policy is evaluated.Findings – It is observed that quasi‐renewal and the geometric processes are equivalent. Using data from an existing warranty database it is shown that the inter‐repair times form a quasi‐renewal process. The corresponding expected warranty cost over the warranty period under a non‐renewing free replacement warranty policy is evaluated.Research limitations/implications – This approa...

WARRANTY COST ANALYSIS: RENEWING WARRANTY WITH NON-ZERO REPAIR TIME

The main focus of this study is on the modeling of the warranty claims and evaluating the warranty expenses. The cost of each warranty claim depends on the repair time associated with the claim. Alternating renewal process is used to model the operating and repair times. The warranty costs over the warranty period under renewing free replacement policy are evaluated. Also, the expected warranty expenses over the life cycle of the product are studied. Numerical examples illustrate the ideas.

The Construction of New Bivariate Exponential Distributions from a Bayesian Perspective

Abstract We use an economic approach of Mendel to derive new bivariate exponential lifetime distributions. Features distinguishing this approach from the existing ones are (1) it makes use of the principle of indifference; (2) our parameter of interest is a measurable function of observable quantities; (3) the assessment of the probability measure for random lifetimes is performed by assessing that for random lifetime costs with a change of variables; and (4) characterization properties other than the bivariate loss-of-memory property are used to construct distributions. For the infinite population case, our distributions correspond to mixtures of existing bivariate exponential distributions such as the Freund distribution, the Marshall-Olkin distribution, and the Friday-Patil distribution. Moreover, a family of natural conjugate priors for Bayesian Freund (-type) bivariate exponential distributions is discussed.

Optimal two-dimensional warranty repair strategy

For repairable products, the warrantor has options in choosing the type of repair performed to an item that fails within the warranty period. The focus is on a particular warranty repair strategy, related to the degree of the warranty repair, under a non-renewing two-dimensional warranty policy that is free of charge to the consumer. A rectangular warranty region, as in the automotive industry, is considered and partitioned into disjoint subregions. Each of these subregions has a preassigned degree of repair for a faulty item. First, for a partition of size n, an expression is derived for the associated expected warranty servicing cost per item sold. Second, using an example, for a given discretization of the warranty period, the way in which the number of subregions and their shape can be determined, so that the expected warranty servicing cost per item sold is minimum, is demonstrated.

Capture–Recapture Estimation Using Finite Mixtures of Arbitrary Dimension

Reversible jump Markov chain Monte Carlo (RJMCMC) methods are used to fit Bayesian capture-recapture models incorporating heterogeneity in individuals and samples. Heterogeneity in capture probabilities comes from finite mixtures and/or fixed sample effects allowing for interactions. Estimation by RJMCMC allows automatic model selection and/or model averaging. Priors on the parameters stabilize the estimates and produce realistic credible intervals for population size for overparameterized models, in contrast to likelihood-based methods. To demonstrate the approach we analyze the standard Snowshoe hare and Cottontail rabbit data sets from ecology, a reliability testing data set.

Bayesian inference for a stochastic epidemic model with uncertain numbers of susceptibles of several types

Summary A stochastic epidemic model with several kinds of susceptible is used to analyse temporal disease outbreak data from a Bayesian perspective. Prior distributions are used to model uncertainty in the actual numbers of susceptibles initially present. The posterior distribution of the parameters of the model is explored via Markov chain Monte Carlo methods. The methods are illustrated using two datasets, and the results are compared where possible to results obtained by previous analyses.

Multicomponent Systems With Multiplicative Aging and Dependent Failures

We extend our earlier studies of a multicomponent system accumulating damage due to a series of fatal and nonfatal shocks. The model introduces statistical dependence among the system components by associating individual shock processes with potentially overlapping subsystems made up of groupings of components. We construct an aging and statistical dependence model where damage accumulates multiplicatively with each shock. We derive a representation of the systems joint survival function, and show that the Marshall-Olkin multivariate exponential model can be obtained as a special case of this model. We propose an approach to the simulation of the performance of the system, and provide several illustrative examples. We conclude by identifying possible further extensions of this model.

Warranty analysis: Estimation of the degree of imperfect repair via a bayesian approach

AbstractAn approach to modeling imperfect repairs under warranty settings is presented in Chukova, Arnold and Wang12. They model the imperfect repairs using the concepts of delayed and accelerated distribution functions. As an extension of their approach, we design a procedure for estimating the degree of repair as well as other modeling parameters by Markov chain Monte Carlo (McMC) methods.

Failure distributions in multicomponent systems with imperfect repairs

We present a model for correlated failures in multicomponent systems where failed components are repaired. Due to these repairs multiple failures of each component are observable. Repair is instantaneous, but imperfect: in general the component is not returned to the good-as-new condition. Correlation amongst failures and the overall ageing of the system are modelled by treating each component failure as a shock which damages the other components in the system. We construct the likelihood for observations generated by such a system, and show how to simulate from this model. We demonstrate inference of the model parameters with simulated data and two real-world examples.