| Two of the most widely used performance measures for queueing systems are the fraction of lost customers (in finite-buffer queues) and the probability of experiencing a long delay (in the infinite buffer context). However, in many applications the quality of service experienced by users depends strongly on the "clumpiness" of losses or long delays; for example, in video applications over the Internet, losing one isolated packet out of every 107 is very different from losing a group of 103 nearly consecutive packets out of every 1010 served. Surprisingly, the literature on such clumpiness issues is quite limited. In the first part of this dissertation we study the distribution of the loss-clumps in finite-buffer queues and that of exceedance-clumps in infinite-buffer queues. We are particularly interested in the impact of long-range dependence and heavy tails on these distributions, because of the relevance of such traffic models to various applications settings. We provide conditional limit theorems for several stylized traffic models that offer insight into both the size of the clumps and the most likely dynamics of the traffic and workload around the loss or exceedance event.; The last portion of this dissertation is concerned with the use of importance sampling to compute expectations of functionals of diffusion processes. It has long been known that the optimal (in the sense of minimal variance) sampling distribution weights outcomes in proportion to the value of the desired functional and their likelihood under the original distribution. However, under the said optimal distribution, the process will not in general be Markov, and hence may be very hard to simulate dynamically. Here we show that, for a class of expectations that can be characterized as positive solutions to linear differential equations, the optimal (zero-variance) importance sampling distribution preserves the Markovian nature of the underlying diffusion process. This suggests that good practical importance sampling distributions can be obtained simply by appropriately modifying (in a state-dependent manner) the drift coefficient of the underlying diffusion. The class of expectations considered includes, as a particular case, the cumulative discounted reward until hitting a set. |
