Binomial Moments of the Distance Distribution and the Probability of Undetected Error

In [1] K. A. S. Abdel-Ghaffar derives a lower bound on the probability of undetected error for unrestricted codes. The proof relies implicitly on the binomial moments of the distance distribution of the code. We use the fact that these moments count the size of subcodes of the code to give a very simple proof of the bound in Abdel by showing that it is essentially equivalent to the Singleton bound. This proof reveals connections of the probability of undetected error to the rank generating function of the code and to related polynomials (Whitney function, Tutte polynomial, and higher weight enumerators). We also discuss some improvements of this bound.Finally, we analyze asymptotics. We show that an upper bound on the undetected error exponent that corresponds to the bound of Abdel improves known bounds on this function.     

Distributions of Runs and Consecutive Systems on Directed Trees

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.     

Equality in Distribution in a Convex Ordering Family

Let F and G be the respective distributions of nonnegative random variables X and Y satisfying the convex ordering. We investigate the class of functions h for which the equality E[h(X)] = E[h(Y)] guarantees F = G. It leads to extensions of some existing results and at the same time offers a somewhat simpler proof.     

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.     

利用鞍点近似法计算串音对阵列波导光栅复用器性能的影响

基于光的波动性,考虑信号和串音偏振态,改进了鞍点近似法的数学模型,计算了阵列波导光栅复用器的误码率和功率代价,并与高斯近似法和实验结果进行了比较,显示出较高的计算精度。     

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.     

Inference and prediction in bulk arrival queues and queues with service in stages

This paper deals with the statistical analysis from a Bayesian point of view, of bulk arrival queues where the batch size is considered as a fixed constant. The focus is on prediction of the usual measures of performance of the system in the steady state. The probability generating function of the posterior predictive distribution of the number of customers in the system and the Laplace transform of the posterior predictive distribution of the waiting time in the system are obtained. Numerical inversion of these transforms is considered. Inference and prediction of its equivalent single queue with service in stages is also discussed. © 1998 John Wiley & Sons, Ltd.     

Reliability of fault-tolerant systems with parallel task processing

The paper considers performance and reliability of fault-tolerant software running on a hardware system that consists of multiple processing units. The software consists of functionally equivalent but independently developed versions that start execution simultaneously. The computational complexity and reliability of different versions are different. The system completes the task execution when the outputs of a pre-specified number of versions coincide. The processing units are characterized by different availability and processing speed. It is assumed that they are able to share the computational burden perfectly and that execution of each version can be fully parallelized.     

A small generating set for the 3‐BD closed set of block sizes ≥ 6

Let v, k be positive integers and k ≥ 3, then K_k = : {v: v ≥ k} is a 3‐BD closed set. Two finite generating sets of 3‐BD closed sets K₄ and K₅ are obtained by H. Hanani [5] and Qiurong Wu [12] respectively. In this article we show that if v ≥ 6, then v ∈ B₃(K,1), where K = {6,7,…,41,45,46,47,51,52,53,83,84}\{22,26}; that is, we show that K is a generating set for K₆. Finally we show that v ∈ B₃(6,20) for all v ∈ K\{35,39,40,45}. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 128–136, 2008     

Combinatoric enumeration of two-dimensional proper arrays

An n×mproper array is a two-dimensional rectangular array composed of directed cubes that obey certain constraints. Because of these constraints, the n×m proper arrays may be classified via a schema in which each n×m proper array is associated with a particular n×1 column. For a fixed n, the goal is to enumerate, modulo symmetry, all possible edge configurations associated with n×m proper arrays. By varying n, one constructs four combinatoric sequences, each of which enumerates a particular class of edge configurations. Convolution arguments and resultant calculations associate these sequences with cubic equations. These cubic equations allow one to predict M_n, the number of edge configurations, modulo symmetry, associated with n×m proper arrays.