Y. L. Tong’s research while affiliated with Georgia Institute of Technology and other places

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

A consecutive k−out−of-n: F system is a coherent system of n-ordered components that functions if and only if there is no consecutive run of k failures among the components. Examples of such systems exist in telecommunications, oil pipelines, and integrated circuitry. Most of the research in reliability of consecutive k−out-of-n: F systems assumes that the components function independently of one another. In this paper, we develop a model incorporating positive dependence between adjacent components and show that for k ≥ (n + l)/2, the reliability of the system is a decreasing function of this dependence.

: This paper contains inequalities for the exceptions of permutation-invariant concave functions of the partial sums of nonnegative exchangeable random variables. Two majorization inequalities are derived, and an application in reliability theory is discussed. Keywords: Concave and Schurconcave functions.

We introduce the notion of comparison of the criticality of two nodes in a coherent system, and devlop a monotonicity property of the reliability function under component pairwise rearrangement. We use this property to find the optimal component arrangement. Worked examples illustrate the methods proposed.

Mixture distributions are a frequently used tool in modelling random phenomena. We consider mixtures of densities from a one-parameter exponenvial family of distributions. Using the tools of totally positive functions and the variation-diminishing property of such, we study the effect of sign-crossing properties of two mixing densities μ1 and μ2 on the resulting mixture distributions f1 and f2. The results enable us to make stochastic and variability cornparisons for binomial-beta, mixed Weibull, and mixed gamma distributions.

Majority systems are encountered in both decision theory and reliability theory. In decision theory for example a jury or committee employing a majority rule will make the ‘correct' decision if a majority of the individuals do so. In reliability theory some coherent systems function if and only if a majority of the components work properly. In this paper results concerning the reliability of majority systems are developed which are applicable in both areas. Two models incorporating dependence between individuals or components in majority systems are introduced, and various monotonicity results for their reliability functions are established. Comparisons are also made between direct (or simple) and indirect majority systems.

Majority systems are encountered in both decision theory and reliability theory. In decision theory for example a jury or committee employing a majority rule will make the ‘correct' decision if a majority of the individuals do so. In reliability theory some coherent systems function if and only if a majority of the components work properly. In this paper results concerning the reliability of majority systems are developed which are applicable in both areas. Two models incorporating dependence between individuals or components in majority systems are introduced, and various monotonicity results for their reliability functions are established. Comparisons are also made between direct (or simple) and indirect majority systems.

A real-valued function g of two vector arguments $\mathbf{x}$ and $\mathbf{y} \in R^n$ is said to be arrangement increasing if it increases in value as the arrangement of components in $\mathbf{x}$ becomes increasingly similar to the arrangement of components in $\mathbf{y}$. Hollander, Proschan and Sethuraman (1977) show that the convolution of arrangement increasing functions is arrangement increasing. This result is used to generate some interesting probability inequalities of a geometric nature for exchangeable random vectors. Other geometric inequalities for families of arrangement increasing multivariate densities are also given, and some moment inequalities are obtained.

Diversity of bugs or faults in a software system is a factor contributing to software unreliability which has not yet been appropriately emphasized. This paper is written with the intention of demonstrating the impact of fault diversity on the time to detection of software bugs. A new discrete software reliability model based on the multinomial distribution is introduced. It is shown that for models of this type, the more diverse the fault probabilities are, the longer (stochastically) it takes to detect or eliminate any n faults, while the smaller (stochastically) will be the number of faults detected or eliminated during a given amount of time (or during a given number of inputs to the system). The impact of fault diversity is also demonstrated for the Jelinski–Moranda model.