This thesis consists of three parts. In Part A, we study Gaussian queues: motivated by Section 1.3.2, we suppose that the input process is Gaussian. We focus on the steady-state buffer content and the steady-state (total) length of the busy period. First, we restrict ourselves to so-called logarithmic tail asymptotics and qualitative behavior of the queue. After that, we establish the (exact) tail asymptotics for the buffer content. The latter results are applied to examine reduced-load equivalence for Gaussian queues, i.e., the question when a subset of M independent Gaussian input processes dominates the tail asymptotics for the buffer content. Part B is motivated by the need to simulate the buffer content for Gaussian queues, as analytic results are often hard to obtain. Since the buffer-content distribution can be written as a so-called large-deviation probability, we first study how large-deviation probabilities can be simulated in general. To this end, we formulate sharp conditions under which a widely- used method, exponential twisting, works. These conditions are then applied to a random-walk setting, before we turn to the buffer content in a Gaussian queue. In Part C, we study L´evy-driven fluid systems, relying on path decompositions (so-called splitting properties). First, these are applied to analyze the transform of the buffer content in a queue with L´evy input and a special jump structure. Furthermore, splitting is an effective concept to investigate perturbed risk processes, a variant of the classical risk process discussed in Section 1.1.5. We also show that splitting is not only useful to obtain the exact tail asymp- totics for the buffer content in a single fluid queue, but that it is also a powerful method to study fluid networks driven by L´evy processes. For these networks, we find (joint) transforms of busy periods, idle periods, and buffer contents. Finally, some of these results are extended to queueing networks in a random environment, including the fluid-flow models of Section 1.3.1. This relies on an extensive analysis of Markov-additive processes. Each of the three parts starts with an introductory chapter, where fundamental results from the literature are discussed to put the material into the right context.

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M.R.H. Mandjes (Michel)
Universiteit van Amsterdam
hdl.handle.net/11245/1.260235
Stochastics

Dieker, T. (2006, March 9). Extremes and Fluid Queues. Retrieved from http://hdl.handle.net/11245/1.260235