On some mixtures of distributions of order k (original) (raw)

On Comparison of Multiserver Systems with Multicomponent Mixture Distributions

Journal of Mathematical Sciences

In this paper, we introduce and study the relations between parameters of the m-component mixture distributions which imply the stochastic and failure rate comparisons. Then we apply the failure rate and stochastic ordering techniques to construct the upper and lower bounds for the steady-state performance indexes in a multiserver queueing system with multicomponent exponential-Pareto mixture service time distribution. The uniform distance between multicomponent mixture distribution and its parent distribution is discussed. The obtained theoretical results are then illustrated by a few numerical examples based on the regenerative simulation multiserver queueing systems with mixture service time distributions.

On finite 3-component mixture of exponential distributions: Properties and estimation

Cogent Mathematics, 2016

To study reliability problems, life time and survival analysis, a new mixture model, called the 3-component mixture of the Exponential distributions, is introduced. This study is mainly concerned with the problem of investigating the different statistical properties of the newly developed 3-component mixture of Exponential distributions. Firstly, some basic properties of the 3-component mixture model are discussed. Secondly, we discuss hazard rate function, cumulative hazard rate function, reversed hazard rate function, mean residual life function, and mean waiting time function. Different measures of entropy and inequality indices are also discussed. Closed form expressions of the density functions of order statistics and their statistical properties are derived. Finally, the parameters of the proposed mixture distribution are estimated by making use of the maximum likelihood approach under complete and censored sampling. The results on maximum likelihood estimation are also supported through a simulation study and a real-life data.

On mixture failure rate ordering

2005

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually this property can be observed asymptotically as

On discrete failure-time distributions

Reliability Engineering & System Safety, 1989

Different classes of discrete life distributions are defined parallel to the continuous life distributions. Some implications known to be present in continuous life distributions are shown to exist among discrete life distributions. Some test statistics of geometric distribution property versus other classes o f failure-time distribution are proposed. Critical points of these statistics, based on simulations are presented. Powers of these tests against increasing failure rate (IFR) and decreasing failure rate (D FR) distributions are also calculated.

On a new class of probability distributions

Applied Mathematics Letters, 2011

Multicomponent systems are widely used in computer science. The reliability of these systems plays a very important role in efficient working. These systems are not always supposed to follow the standard probability distributions and so pseudo-distributions can be thought of as suitable alternatives. In this work we have defined a new bivariate pseudo-Weibull distribution. Some standard properties of the distribution have been studied. The distributions of the order statistics and concomitants have also been obtained.

DISTRIBUTIONS OF ORDER K WITH APPLICATIONS

INTERNATIONAL CONFERENCE ON MATHEMATICS EDUCATION AND MATHEMATICS IN ENGINEERING AND TECHNOLOGY (ICMET 13), 2015

The distributions of order k are infinite families of probability distributions indexed by a positive integer k, which reduce to the respective classical probability distributions for k = 1, and they have many applications in Statistics, Engineering, Meteorology, etc. A few of the most applicable ones, namely the geometric, the negative binomial, the binomial, and the Poisson distributions of order k, are briefly discussed presently and an application is given in Reliability. 1. The Geometric Distribution of Order k For any positive integer k, denote by Tk the number of independent trials with success probability p until the occurrence of the k th consecutive success, and set q = 1 – p. Philippou and Muwafi (1982) found that, for n = k, k + 1, … , (í µí¿. í µí¿) í µí²‡ í µí²Œ (í µí²) = í µí±·(í µí±» í µí²Œ = í µí²) = í µí²‘ í µí² ∑ (í µí² í µí¿ + ⋯ + í µí² í µí²Œ í µí² í µí¿ , … , í µí² í µí²Œ) (í µí²’ í µí²‘) í µí² í µí¿ +⋯+í µí² í µí²Œ and 0 otherwise, where the summation is taken over all k-tuples of non-negative integers n1, n2, …, nk such that n1 +2n2 + …+ knk = n – k. The proof is based on the observation that a typical element of the event (í µí±» í µí²Œ = í µí²) is an arrangement (í µí¿. í µí¿) í µí±¨= í µí²™ í µí¿ í µí²™ í µí¿ … í µí²™ í µí² í µí¿ +í µí² í µí¿ +⋯+í µí² í µí²Œ í µí±ºí µí±º … í µí±º (í µí²Œ í µí±º ′ í µí²”), such that n1 of the x's are E1 = F, n2 of the x's are E2 = SF, …, nk of the x's are Ek = SS…SF (k-1 S's), and n1 + 2n2 + …+ knk = n – k. Fix n1, …, nk. Then the number of the A's is (í µí¿. í µí¿‘) (í µí² í µí¿ + ⋯ + í µí² í µí²Œ í µí² í µí¿ , … , í µí² í µí²Œ) and each one has probability (í µí¿. í µí¿’) í µí±·(í µí±¨) = [í µí±·(í µí±¬ í µí¿)] í µí² í µí¿ [í µí±·(í µí±¬ í µí¿)] í µí² í µí¿ … [í µí±·(í µí±¬ í µí²Œ)] í µí² í µí²Œ í µí±·(í µí±ºí µí±º … í µí±º) (í µí²Œ í µí±º ′ í µí²”) = í µí²’ í µí² í µí¿ (í µí²‘í µí²’) í µí² í µí¿ … (í µí²‘ í µí²Œ−í µí¿ í µí²’) í µí² í µí²Œ í µí²‘ í µí²Œ = í µí²‘ í µí² (í µí²’ í µí²‘) í µí² í µí¿ +⋯+í µí² í µí²Œ .

On a Distribution of the Process Describing a Service System with Unreliable Devices

2019

In the paper, the distribution is found for the process {ηt, ξt}, t ≥ 0, in the terms of Laplace transformation. The considered process describes the queuing system with nonhomogeneous Poisson stream of demands and n unreliable devices. It is essential that the process {ηt, ξt}, t ≥ 0, for ξt ≥ n is a homogeneous with respect to the second component Markov process. The results obtained in the paper are based on the theory of matrices and solution of the system of linear integral equations.

A GENERAL CLASS OF NEW CONTINUOUS MIXTURE DISTRIBUTION AND APPLICATION

SCIK Publishing Corporation, 2020

A generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an investigation is conducted on a distribution and its generalization. Several available generalizations of the distribution are reviewed and recent trends in the construction of generalized classes with a generalized mixing parameter are discussed. To check the suitability and comparability, real data set have been used.

Classes of Probability Distributions and Their Applications

The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation x = b G(µ µx), where b G(s) is the Laplace-Stieltjes transform of an unknown probability distribution function G(x) of a positive random variable having the first two moments g1 and g2, and µ is a positive parameter satisfying the condition µg1 > 1. The addi- tional information characterizing G(x) is that it belongs to the special class of distributions such that the difference between two elements of that class in the Kolmogorov (uniform) metric is not greater than �. The obtained result is then used to establish the lower and upper bounds for loss probabilities in certain loss queueing systems with large buffers as well as continuity theorems in large M/M/1/n queueing systems.

Mixture Model on Development of Bivariate Product Distribution and its Properties

Asian Journal of Probability and Statistics

In the study, some bivariate distributions were developed from mixture model offspring, using the Independent (Product) distribution approach. These developments are categorized under the IID and IInD: where the Bivariate Exponential distribution, Bivariate Lindley distribution and Bivariate Juchez distribution are constructed as IIDs; and Bivariate Exponential-Lindley distribution, Bivariate Exponential-Juchez distribution and Bivariate Lindley-Juchez distribution as (IInDs). The properties of these distributions which involve: the shape of the bivariate PDFs, moments, moment generating function, mean, covariance and coefficient of correlation, maximum likelihood estimator, reliability analysis, renewal property and probability patterns; are studied across the distributions. Finally, under renewal properties, functions are derived which can model two-dimensional queuing and renewal processes, for events where the arrival and service times are dependent.