Regenerative Simulation for Estimating Extreme Values

Let X(t) denote the regenerative process being simulated and assume that it converges in distribution to a steady state random variable. This paper considers estimating the extreme values of the regenerative process. Suppose we are interested in the largest value attained in the interval [0, t], call it X*(t). The paper develops a method for estimating the distribution of X*(t). When the regenerative process is either the GI/G/1 queue or a birth-death process, theoretical results are available for the distribution of X*(t). Our development simulated the waiting time, queue length, and virtual waiting time for an M/M/1 queue, employed the method for estimating the distribution of X*(t), and compared the simulation results with the theoretical results.