Rare events, heavy tails, and simulation

We explore current and our own new developments for rare events simulation for stochastic processes where the rare event is caused by an underlying heavy-tailed distribution. These distributions fail to have the exponential moments required for standard existing algorithms that apply to light-tailed distributions. Simulation of heavy-tailed distributions for estimation of steady-state measures is not easy, as the simulation must run exceptionally long in order to capture the effect of the distribution tail, i.e., the rare events, which even with small probability of occurrence can significantly affect the given system performance.; Much of the research focus for heavy-tailed data is on the analysis of heavy-tailed data and on the estimation of extreme value indices. A much smaller portion of work has been devoted to the detection of heavy tails in data. Therefore, an efficient method to detect whether or not sample data is from a heavy-tailed distribution is becoming necessary for modern research. We develop a formal hypothesis test involving maximum-to-sum ratios in order to test whether or not a data set was generated by an underlying heavy-tailed distribution. We combine the test with bootstrap estimation in order to increase the power of the test. This appears to work remarkably well for even small sample sizes.; We specifically consider Jackson networks with heavy-tailed service time distribution and propose an approximation for the tail of the sojourn time distributions. Very little is known in the literature about these cases of generalized Jackson networks. Most work has been limited to tandem queues where the system is not complicated by feedback. We compare our approximation to a long-standing conjecture in the literature about the tail of the sojourn time distribution. We find that our approximation generally outperforms the existing conjecture and often offers a vast improvement. We develop a method to improve our simulation result by decreasing the relative error, and compare the improved simulated result to both our approximation and the existing conjecture.