Since classical approaches in queueing theory lead to complex expressions or do not apply for complex (multiserver) systems, at least two main streams of research emerged: computational probability and heavy traffic approximations. The theme of computational probability is to give numbers efficiently. Computational probability provides ways (algorithms) to compute performance measures rather than explicit (closed form) expressions. The need for computational probability was advocated by M. F. Neuts and others. On the other hand, the theme of heavy traffic approximations is to provide explicit expressions that are accurate in the heavy traffic limit (when the servers are almost always busy). Heavy traffic approximations originates from the work of J. F. C. Kingman in 1961 , where he provided a heavy traffic approximation for a single-server system via the central limit theorem. Since computational approaches are subject to inaccuracy and/or slow convergence in the heavy traffic limit, heavy traffic approximations complement computational probability.
However, even computational probability usually does not apply directly to multiserver systems with resource sharing or job prioritization. In this thesis, we will address two fundamental problems towards an analysis of multiserver systems with resource sharing or job prioritization via computational probability. The first problem involves approximating a general probability distribution by a collection of exponential distributions. This is an important step in an analysis of (multiserver) systems via computational probability. The second problem involves a multidimensionally infinite state space (multidimensional state space) that is needed to capture the behavior of a multiserver system with resource sharing or prioritization. The difficulty of a performance analysis via computational probability stems from the multidimensional state space.