Recent growth of the Internet in both scale and complexity has imposed a number of difficult challenges on existing measurement techniques and approaches, which are essential for both network management and many ongoing research projects. For any measurement algorithm, achieving both accuracy and scalability is very challenging given hard resource constraints (e.g., bandwidth, delay, physical memory, and CPU speed). My dissertation research tackles this problem by first proposing a novel mechanism called residual sampling, which intentionally introduces a predetermined amount of bias into the measurement process. We show that such biased sampling can be extremely scalable; moreover, we develop residual estimation algorithms that can unbiasedly recover the original information from the sampled data. Utilizing these results, we further develop two versions of the residual sampling mechanism: a continuous version for characterizing the user lifetime distribution in large-scale peer-to-peer networks and a discrete version for monitoring flow statistics (including per-flow counts and the flow size distribution) in high-speed Internet routers. For the former application in P2P networks, this work presents two methods: ResIDual-based Estimator (RIDE), which takes single-point snapshots of the system and assumes systems with stationary arrivals, and Uniform RIDE (U-RIDE), which takes multiple snapshots and adapts to systems with arbitrary (including non-stationary) arrival processes. For the latter application in traffic monitoring, we introduce Discrete RIDE (D-RIDE), which allows one to sample each flow with a geometric random variable. Our numerous simulations and experiments with P2P networks and real Internet traces confirm that these algorithms are able to make accurate estimation about the monitored metrics and simultaneously meet the requirements of hard resource constraints. These results show that residual sampling indeed provides an ideal solution to balancing between accuracy and scalability.

A class of lifetime distributions which has received considerable attention in modelling and analysis of lifetime data is the class of lifetime distributions with bath-tub shaped failure rate functions because of their extensive applications. The purpose of this thesis was to introduce a new class of bivariate lifetime distributions with bath-tub shaped failure rates (BTFRFs). In this research, first we reviewed univariate lifetime distributions with bath-tub shaped failure rates, and several multivariate extensions of a univariate failure rate function. Then we introduced a new class of bivariate distributions with bath-tub shaped failure rates (hazard gradients). Specifically, the new class of bivariate lifetime distributions were developed using the method of Morgenstern’s method of defining bivariate class of distributions with given marginals. The computer simulations and numerical computations were used to investigate the properties of these distributions.