In this paper, we propose a procedure of selecting samples from a set of samples coming from Markovian processes of finite order and finite alphabet. Under the assumption of the existence of a law that prevails in at least q% of the samples of the collection, we show that the procedure allows to identify samples governed by the predominant law. The approach is based on a local metric between samples, which tends to zero when we compare samples of identical law and tends to infinity when comparing samples with different laws. The local metric allows to define a criterion which takes arbitrarily large values when the previous assumption about the existence of a predominant law does not hold. By means of this procedure, we map similarities and dissimilarities of some Brazilian stocks' daily trading volume dynamic.

In this paper, we address the problem of deciding if two independent samples coming from discrete Markovian processes are governed by the same stochastic law. We establish a local metric between samples based on the Bayesian information criterion. In addition, we derive the bound that must be used in this metric to take the decision. In the case on which is decided that the laws are not the same, the metric allows to detect the specific elements of the state space where the discrepancies are manifested. We prove that the metric is statistically consistent to detect if the samples follow the same law, tending to zero when the sample sizes increase. Moreover, we show that the metric assumes arbitrarily large values when the sample sizes increase and the stochastic laws are different. This concept is applied to analyze two lines of production of alcohol fuel, described by five variables each. We identify the variables that most contribute to the discrepancy and, using the local nature of the metric, we list the realizations in which the processes behave differently. KEYWORDSBayesian information criterion, Markov processes, proximity between processes, relative entropy 868

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Waiting lines or queues are commonly occurred both in everyday life and in a variety of business and industrial situations. The various arrival rates, service rates and processing times of jobs/tasks usually assumed are exact. However, the real world is complex and the complexity is due to the uncertainty. The queuing theory by using vague environment is described in this paper. To illustrate, the approach analytical results for M/M/1/8 and M/M/s/8 systems are presented. It optimizes queuing models such that the arrival rate and service rate are vague numbers. This paper results a new approach for queuing models in the vague environment that it can be more effective than deterministic queuing models. A numerical example is illustrated to check the validity of the proposed method.

SUMMARY The fuzzy Bayesian system reliability assessment based on prior two‐parameter exponential distribution under squared error symmetric loss function and precautionary asymmetric loss function is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. Because the goal of the paper is to obtain fuzzy Bayes point estimators of system reliability assessment, prior distributions of location‐scale family has been changed to scale family with change variable. On the other hand, also the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability have been provided. In order to achieve this purpose, we transform the original problem into a non‐linear programming problem. This non‐linear programming problem is then divided into four sub‐problems for the purpose of simplifying computation. Finally, the sub‐problems can be solved by using any commercial optimizers, e.g. GAMS or LINGO. Copyright © 2010 John Wiley & Sons, Ltd.

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This paper seeks to focus on the study and Bayesian and non-Bayesian estimators for the shape parameter, reliability and failure rate functions of the Kumaraswamy distribution in the cases of progressively type II censored samples. Maximum likelihood estimation and Bayes estimation, reliability and failure rate functions are obtained using symmetric and asymmetric loss functions. Comparisons are made between these estimators using Monte Carlo simulation study. With prior information on the parameter of the Kumaraswamy distribution, Bayes approach under squared error loss function in the reliability function has been suggested based on the pervious observations, this approach can be used for both progressively type II censorings. The study is useful for researchers and practitioners in reliability theory and quality also for scientists in physics and chemistry special hydrological literatare, where Kumaraswamy distribution is widely used.

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