Part I is organized into two chapters (see Figure 1.5). In Chapter 2, we address the first fundamental problem in an analysis of multiserver systems, namely ``How can we map a general distribution into a combination of exponential distributions?'' Specifically, we develop moment matching algorithms, which allow us to map a general probability distribution into a combination of exponential distributions. Chapter 3 is the heart of this thesis, and here we address the second fundamental problem in an analysis of multiserver systems, namely ``How can we analyze Markov chains on multidimensionally infinite state spaces?'' Specifically, we introduce dimensionality reduction (DR), which allows us to analyze a class of multidimensional Markov chains that can model many multiserver systems with resource sharing or job prioritization. The moment matching algorithm is also used in a key step of DR.