Simulating Stable Stochastic Systems, I: General Multiserver Queues

A technique is introduced for analyzing simulations of stochastic systems in the steady state. From the viewpoint of classical statistics, questions of simulation run duration and of starting and stopping simulations are addressed. This is possible because of the existence of a random grouping of observations which produces independent identically distributed blocks from the start of the simulation. The analysis is presented in the context of the general multiserver queue, with arbitrarily distributed interarrival and service times. In this case, it is the busy period structure of the system which produces the grouping mentioned above. Numerical illustrations are given for the M/M/1 queue. Statistical methods are employed so as to obtain confidence intervals for a variety of parameters of interest, such as the expected value of the stationary customer waiting time, the expected value of a function of the stationary waiting time, the expected number of customers served and length of a busy cycle, the tail of the stationary waiting time distribution, and the standard deviation of the stationary waiting time. Consideration is also given to determining system sensitivity to errors and uncertainty in the input parameters.