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Research accomplished during the period 3/2/08-9/2/11 in ten specific areas of study is reported. This work includes the treatment of the problems of (1) comparing coherent or mixed systems of different sizes, and obtaining representation results for system reliability under relaxed assumptions on component lifetimes (e.g., exchangeability), (2) deriving new representations of system reliability for used systems known to be working at an inspection time t, (3) developing extensions of the notion of system signatures to dynamic reliability settings, with applications to nonparametric models in reliability and to the engineering practice of burn-in, (4) inference about a common component distribution F from system failure time data, (5) the derivation and application of the joint signature of pairs of systems with shared components, (6) a comparison of the Bayesian and frequentist approaches to estimation (a research monograph), (7) skewness and dispersion among convolutions of independent gamma variables, (8) signature-based representations for the reliability of systems with heterogeneous components, (9) network reliability (10) a proof of the no internal zeros property of system signatures.

The diffusion approximation for queueing networks is proved via the Trotter-kato Theorem. This involves delicate calculations involving the domains of certain operators some of which have been successful and some not. One tries t solve the martingale problem instead of characterizing the domain and hopes to use the Stroock-Varadham approach in order to prove the corresponding limit theorem. For example, we solve the martingale problem for a class of Markov processes whose infinite simal generators are integro-differential operators. Extensions of these results to more complicated queueing systems are currently in progress. Publications: (1) On the Accuracy of Kingman's Heavy Traffic Approximation in the Theory of Queues; (2) Limit theorems for Markov processes via a variant of the Trotter-Kato theorem; and (3) On a integro-differential equation occurring in Queueing and Storage theory.

In this paper we discuss a notion which is the dual of the failure rate and is defined as f(t)/F(t). This concept has come up several times in the literature, but has never been investigated in detail. In this paper the name survival rate is used. We show that in many models of practical interest the survival rates of certain random variables have nice properties. This makes it possible to obtain bounds on the distribution functions of these random variables. Moreover, we show that a form of stochastic dominance based on the survival rates of the random variables has some interesting applications. (Author)