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CONDITIONAL DISTRIBUTIONS AND WAITING TIMES IN MULTITYPE BRANCHING PROCESSES

ADVANCES IN APPLIED probability 2013年第3期45卷 692-718页

作者： Alexander, H. K. Queens Univ Kingston ON K7L 3N6 Canada ETH Inst Integrat Biol CH-8092 Zurich Switzerland

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes overtime, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

关键词： Multitype branching process probability generating function waiting time escape mutant evolutionary biology epidemiology

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Sooner and later waiting time problems for runs in Markov dependent bivariate trials

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1999年第1期51卷 17-29页

作者： Aki, S Hirano, K Osaka Univ Grad Sch Engn Sci Dept Informat & Math Sci Toyonaka Osaka 5608531 Japan Inst Stat Math Minato Ku Tokyo 1068569 Japan

In this paper we study exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We give systems of linear equations with respect to conditional probability generating functions of the waiting times. By considering bivariate trials, we can treat very general and practical waiting time problems for runs of two events which are not necessarily mutually exclusive. Numerical examples are also given in order to illustrate the feasibility of our results.

关键词： discrete distributions probability generating function run waiting time problem

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A polling access control with exhaustive service in wireless body area networks for mobile healthcare using the sleeping schema

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JOURNAL OF AMBIENT INTELLIGENCE AND HUMANIZED COMPUTING 2019年第10期10卷 3761-3774页

作者： He, Min Guan, Zheng Wu, Zhangyong Lu, Liang Zhou, Zhaoxu Anisetti, Marco Damiani, Ernesto Yunnan Univ Sch Informat Sci & Engn Kunming 650091 Yunnan Peoples R China Informat Technol Dev Ctr Yunnan Prov Kunming 650091 Yunnan Peoples R China Univ Milan Dipartimento Informat Via Bramante 65 I-26013 Crema CR Italy Khalifa Univ Dept Elect & Comp Engn POB 127788 Abu Dhabi U Arab Emirates

Wireless body area networks (WBANs) play an important role in human health monitoring for mobile healthcare. The improvement of service performance and low-power consumption are the two challenges for these medical WBANs, because those energy-limited wireless medical sensors must transmit the monitoring data to the personal server (PS) via intra-WBAN in time. Further, precise mathematical modeling combining with the sleeping state to measure this improvement theoretically is still lack. Therefore, a polling control with exhaustive service using the sleeping schema is proposed to address the problems in WBANs communication for smart health. Based on the managed access phase (MAP) specified in the IEEE 802.15.6 standard, it has been attempting to improve the energy efficiency through a self-managing sleeping schema for both the sensor nodes and the PS according to the load in intra-WBANs, in which the node will soon turn to sleep upon completion of transmitting all its current data packets. Additionally, by employing the embedded Markov chain and probability generating function, the proposed model is established, and the performance characteristics-the mean cyclic period and queue length at the polling moment-were accurately obtained to justify the service performance can be guaranteed. In addition, the expressions of the quantitative relationship among sleeping time, performance characteristics and network parameters were derived in closed form, which can easily evaluate the energy efficiency. Simulations were carried out to justify this model.

关键词： Mobile healthcare Polling control Exhaustive service Wireless body area networks (WBANs) probability generating function Energy efficiency

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Parameter estimation for strict arcsine distribution

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COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2024年

作者： Low, Yeh-Ching Phang, Yook-Ngor Khoo, Wooi-Chen Ong, Seng-Huat Sunway Univ Sch Engn & Technol Dept Comp & Informat Syst Bangi 47500 Selangor Malaysia Univ Teknol MARA Alor Gajah Melaka Malaysia UCSI Univ Inst Actuarial Sci & Data Analyt Kuala Lumpur Malaysia Univ Malaya Inst Math Sci Kuala Lumpur Malaysia

The two-parameter strict arcsine distribution as a member of the natural exponential family with cubic variance function has been shown to be a viable candidate for statistical analysis of count data. Efficient methods of parameter estimation will be essential in practical applications of the distribution. In this paper we examine some methods of parameter estimation. Due to the simple expression of the probability generating function, a probability generating function-based estimation procedure is considered and compared with other estimation procedures. Since the accuracy of the parameter estimation procedure affects the probability of correct selection in choosing the correct probability distribution, we extend the investigation by examining the discrimination between strict arcsine and generalized Poisson distributions in which both have cubic variance functions.

关键词： Cubic variance function Over dispersion Parameter estimation probability generating function probability of correct selection

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Extensions of the uncertainty quantification of common cause failure rates

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RELIABILITY ENGINEERING & SYSTEM SAFETY 2002年第1期78卷 63-69页

作者： Vaurio, JK Fortum Power & Heat Oy Lovisa Finland Lappeenranta Univ Technol Lappeenranta Finland

Uncertainties in common cause event observation, documentation and interpretation are taken into account by conditional probabilities and generalized impact vector weights that separate single and double events of a specific multiplicity in a single observation. Distributions and moments of common cause failure (CCF) rates of a system are obtained in terms of the weights by using probability generating functions, combining assessment uncertainties and statistical uncertainties. These results are then used to generate effective plant-specific input data to general empirical Bayes estimation methods to combine data from many plants. The posterior output yields CCF probabilities for standby safety system fault tree analysis or probabilistic safety assessments of a target plant. (C) 2002 Published by Elsevier Science Ltd.

关键词： common mode impact vector empirical Bayes fault tree probability generating function

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Modeling the random spreading of fake news through a two-dimensional time-inhomogeneous birth-death process

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2024年第18期47卷 13621-13650页

作者： Di Crescenzo, Antonio Paraggio, Paola Univ Salerno Dept Math Via Giovanni Paolo II 132 Fisciano SA Italy

We consider a two-dimensional time-inhomogeneous birth-death process to model the time-evolution of fake news in a population. The two components of the process represent, respectively: (i) the number of individuals (say spreaders) who know the rumor and intend to spread it and (ii) the number of individuals (say inactives) who have forgotten the rumor previously received. We employ the probability generating function-based approach to obtain the moments and the covariance of the two-dimensional process. We also analyze a new adimensional index to study the correlation between the two components. Some special cases are considered in which both the expected numbers of spreaders and inactives are equal to suitable sigmoidal curves, which are often adopted in modeling growth phenomena. Finally, we provide an application based on real data related to the diffusion of fake news, in which the optimal choice of the sigmoidal curve that fit the datasets is based on the minimization of the mean square error and of the relative absolute error.

关键词： adimensional correlation index fake news spreading mean square error probability generating function relative absolute error two-dimensional birth-death process

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Distributions of runs and consecutive systems on directed trees

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1999年第1期51卷 1-15页

作者： Aki, S Osaka Univ Dept Math Sci Osaka 5608531 Japan

In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutive-k-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.

关键词： probability generating function discrete distributions run directed tree graph reliability Markov tree lifetime consecutive system order statistics

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Performance Analysis of a Markovian Queue with Impatient Customers and Working Vacation

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Journal of the Operations Research Society of China 2023年第1期11卷 133-156页

作者： Shakir Majid Department of Mathematics University of LadakhLeh CampusLeh194101India

In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation *** customer’s impatience is due to its arrival during a working vacation period,where the service rate of the customer is lower than a normal busy *** the system is non-empty when the server returns from the working vacation,the server resumes the normal service ***,the server will take successive working vacations till it reaches the maximum number of K working vacations and then the server remains idle until the next ***-form probabilities are obtained by using the identities involving beta functions and degenerate hypergeometric functions,and the performance measures of the system are derived using generating *** stochastic decomposition structures of the mean queue length and mean waiting time are *** effects of the system parameters on some performance measures had been numerically illustrated.

关键词： Markovian queue Variant working vacations probability generating function

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Distributions of patterns of two successes separated by a string of k-2 failures

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STATISTICAL PAPERS 2012年第2期53卷 323-344页

作者： Dafnis, Spiros D. Philippou, Andreas N. Antzoulakos, Demetrios L. Univ Patras Dept Math Patras 26500 Greece Univ Piraeus Dept Stat & Insurance Sci Piraeus 18534 Greece

Let Z(1), Z(2),... be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Zt = 1) and q = Pr(Zt = 0) = 1 -p, respectively, t = 1, 2,.... For any given integer k = 2 we consider the patterns E1: two successes are separated by at most k -2 failures, E2: two successes are separated by exactly k -2 failures, and E3: two successes are separated by at least k -2 failures. Denote by N-n, k((i)) (respectively M-n, k,((i)) ) the number of occurrences of the pattern Ei, i = 1, 2, 3, in Z(1), Z(2),..., Zn when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T ((i)) r, k (resp. W ((i)) r, k) be the waiting time for the r -th occurrence of the pattern Ei, i = 1, 2, 3, in Z(1), Z(2),... according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N-n, k((i)), M-n, k,((i)) ,T-r,T-k ((i)) and W-r,W-k ((i)) (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.

关键词： Success runs Patterns Distributions of order k Waiting time distributions Markov chains Recurrence relations probability generating function probability mass function Moments Reliability

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Discrete distributions in the extended FGM family: The p.g.f. approach

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2009年第11期139卷 3891-3899页

作者： Piperigou, Violetta E. Univ Patras Dept Math Rion 26500 Greece

In this article the probability generating functions of the extended Farlie-Gumbel-Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Discrete data fit Discrete Farlie-Gumbel-Morgenstern family Geometric distribution Marginal-conditional distribution Multivariate distributions Negative correlation Poisson distribution probability generating function

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