Joint Distributions of Runs in a Sequence of Multi-State Trials

In this paper we introduce a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variable of returnable type and discuss some of their properties. These concepts are extensions of the Markov chain imbeddable random variable of binomial type which was introduced and developed by Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743–766). By using the results, we obtain the distributions and the probability generating functions of numbers of occurrences of runs of a specified length based on four different ways of counting in a sequence of multi-state trials. Our results also yield the distribution of the waiting time problems.     

Joint Distributions of Numbers of Success-runs Until the First Consecutive k Successes in a Higher-Order Two-State Markov Chain

Let X-m+1, X-m+2,.., X0, X1, X2,.., be a time-homogeneous {0, 1}-valued m-th order Markov chain. Joint distributions of the numbers of trials, failures and successes, of the numbers of trials and success-runs of length l (m l k) and of the numbers of trials and success-runs of length l (l m k) until the first consecutive k successes are obtained in the sequence X1, X2,.., There are some ways of counting numbers of runs of length l. This paper studies the joint distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length l, the number of runs of length greater than or equal to l, the number of overlapping runs of length l and the number of runs of length exactly l. Marginal distributions of them can be obtained immediately, and surprisingly their distributions are very simple.     

On Waiting Time Problems Associated with Runs in Markov Dependent Trials

A general technique is developed to study the waiting time distribution for the r-th occurrence of a success run of length k in a sequence of Markov dependent trials. Sooner and later waiting time problems are also discussed.     

On Number of Occurrences of Success Runs of Specified Length in a Higher-Order Two-State Markov Chain

Let X^-m+1, X^-m+2,..., X⁰, X¹, X²,..., Xⁿ be a time-homogeneous {0, 1}-valued m-th order Markov chain. The probability distributions of numbers of runs of "1" of length k (k m) and of "1" of length k (k < m) in the sequence of a {0, 1}-valued m-th order Markov chain are studied. There are some ways of counting numbers of runs with length k. This paper studies the distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length k, the number of runs with length greater than or equal to k, the number of overlapping runs of length k and the number of runs of length exactly k.     

Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials

Let Z ₀, Z ₁,...,Z _n be a sequence of Markov dependent trials with state space Ω = {F ₁,...,F _λ, S ₁,...,S _ν}, where we regard F ₁,...,F _λ as failures and S ₁,...,S _ν as successes. In this paper, we study the joint distribution of the numbers of S _i -runs of lengths k _ij (i = 1,2,...,ν, j = 1,2,...,r _i ) based on four different enumeration schemes. We present formulae for the evaluation of the probability generating functions and the higher order moments of this distribution. In addition, when the underlying sequence is i.i.d. trials, the conditional distribution of the same run statistics, given the numbers of success and failure is investigated. We give further insights into the multivariate run-related problems arising from a sequence of the multistate trials. Besides, our results have potential applications to problems of various research areas and will come to prominence in the future. This research was partially supported by the ISM Cooperative Research Program (2004-ISM·CRP-2007).     

Start-Up Demonstration Tests Under Markov Dependence Model with Corrective Actions

A general probability model for a start-up demonstration test is studied. The joint probability generating function of some random variables appearing in the Markov dependence model of the start-up demonstration test with corrective actions is derived by the method of probability generating function. By using the probability generating function, several characteristics relating to the distribution are obtained.     

Joint Distributions of Numbers of Success-Runs and Failures Until the First Consecutive k Successes in a Binary Sequence

Joint distributions of the numbers of failures, successes andsuccess-runs of length less than k until the first consecutive k successesin a binary sequence were derived recently by Aki and Hirano (1995, Ann.Inst. Statist. Math., 47, 225-235). In this paper, we present an alternatederivation of these results and also use this approach to establish someadditional results. Extensions of these results to binary sequences of orderh are also presented.     

Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials

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.     

Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials

Let {Z _n , n 1} be a time-homogeneous {0, 1}-valued Markov chain, and let N _n be a random variable denoting the number of runs of "1" of length k in the first n trials. In this article we conduct a systematic study of N _n by establishing formulae for the evaluation of its probability generating function, probability mass function and moments. This is done in three different enumeration schemes for counting runs of length k, the "non-overlapping", the "overlapping" and the "at least" scheme. In the special case of i.i.d. trials several new results are established.     

On some applications of a discrete-time model of failure and repair

Research on a discrete-time model of failure and repair studied by Rocha-Martinez and Shaked (1995) is continued in this paper. Among various related results, we prove that if for one point x∈]0,1[ the probability generating function of a non-negative integer valued random variable S satisfies Φ_S(x)⩽x^m for some integer m⩾0, then E(S)⩾m. We use these results to show that for any M (the ‘input’ lifetime of a unit in the model) the R_m's (the allowed number of repairs on the unit at time m, m⩾0) can be chosen such that M_u (the ‘output’ lifetime of the unit through the model) is in hazard rate ordering (therefore in stochastic ordering) arbitrarily large and such that E(R_m) is a minimum in some sense. As a first application, we see how a low-quality item (car, computer, washing machine, etc.) might fulfil strict durability regulations under an appropriate imperfect repair strategy (and be able to compete against the existing leading brand in the market) in such a way that the mean number of repairs be a minimum in some sense. As a second application we show how it can be easily proven that if M is of class: NBU, NWE, DMRL, IMRL, NBUE or NWUE, then M_u is not necessarily of the same class. © 1998 John Wiley & Sons, Ltd.     

考虑n中连续取k失效准则的树状系统可靠性求解方法

树状网络系统在管道运输,网络通信中较为常见,对其进行可靠性评估对系统设计及优化具有重要意义。针对树状冗余系统,在n中连续取k失效准则下,通过有限马尔可夫嵌入法并对其进行变形,研究了树状系统可靠性求解方法。本文对树状系统建模加以定义,提出了基于层数参数,层-节点向量,父-子节点矩阵三元参数的树状系统表示方法,研究了变形有限马尔可夫嵌入法的树状系统n中连续取k失效准则下的可靠性求解方法,给出了三个数值算例应用并分析了算法的运算复杂度。最后,本文对比讨论了基于概率母函数法的树状系统在n中连续取k准则下系统可靠性求解方法的研究,得出结论本文算法针对树状冗余系统n中连续取k失效准则下系统可靠性求解应用范围更广,求解效率较高。     

Waiting time problems for a two-dimensional pattern

We consider waiting time problems for a two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. We deal with a two-dimensional pattern with a general shape in the two-dimensional lattice which is generated by the above sequence of random vectors. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given. Department of Statistical Science, School of Mathematical and Physical Science, The Graduate University for Advanced Studies This research was partially supported by the ISM Cooperative Research Program (2002-ISM-CRP-2007).     

Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths

Consider a sequence of n independent Bernoulli trials with the j-th trial having probability pj of success, 1 j n. Let M(n,K) and N(n, K) denote, respectively, the r-dimensional random variables (M(n, k1),..., M(n,kr) and (N(n,k1), ..., N(n, kr)), where K = (k1, k2, ..., kr) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K) and N(n, K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.     

Waiting Time Problems in a Two-State Markov Chain

Let F ₀ be the event that l ₀ 0-runs of length k ₀ occur and F ₁ be the event that l ₁ 1-runs of length k ₁ occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between F ₀ and F ₁ by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the distributions of the sooner and later waiting time problems.     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.     

多状态可靠性系统动态signature及其剩余寿命分析

系统signature是分析系统可靠性的一个强有力的工具。本文研究了多状态系统的动态signature。Signature概念被拓展到多维的情形,并将其应用于两状态元件下多态关联系统的分析,得到系统在每一状态下的联合生存函数的表达式,探究了系统在时刻的动态signature,为处于工作状态的多状态系统在时刻t接受检查,发现其处于l(t)状态并且恰好已发生了q(t)次元件失效,分别给出了两状态元件多态关联系统动态signature及其剩余寿命的表达式。     

Joint distributions of numbers of success-runs and failures until the first consecutivek successes

Joint distributions of the numbers of failures, successes and success-runs of length less thank until the first consecutivek successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of orderk. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.This research was partially supported by the ISM Cooperative Research Program (93-ISM-CRP-8).     

图论在离散事件动态系统中的应用

运用图论方法和极大代数方法,研究了非强连通图中的强连通分支的最大圈长平均值与该图的赋权邻接矩阵的特征值之间的关系,并进一步证明了其等价性.     

冠状系统的R-旋转图与(R)-旋转图

冠状系统Hc的R(R)-旋转变换是指对Hc的一个完美匹配M,同时将Hc中所有正常(非正常)M-交错的六边形变换为非正常(正常)M-交错的六边形,从而得到Hc的另一个完美匹配的变换.通过这两种旋转变换可分别建立Hc完美匹配集上的层次结构,分别称为R-旋转图和R-旋转图,记为R(Hc)和R(Hc).已经证明知道R(Hc)是有向森林,其每个分支都为有向根树.首先讨论了冠状系统的Z-变换有向图与其R-旋转图之间的关系,指出按连通分支对这两种图的顶点集进行划分,其结果一样.在此基础上,证明了R(Hc)的任一分支T(有向根树)都对应R(Hc)的一个分支T,且两者的顶点集相同,进而证明了T与(T)具有相同的高度和宽度.