In the last several years monte-carlo simulation has become a major tool for the analysis of complex queuing systems which are no readily amenable to analysis by conventional mathematical methods. By a complex queueing system is mean a system composed of, physically or by analogy, a network of stations or servers with traffic units moving through all or some of the servers, into the system and out or around within the system. A traffic unit desiring service by a server may either have to enter a queue first or may be served immediately. Such systems have been simulated often with the main purpose of obtaining run estimates of the true or population means of certain variables whose probability density functions are not known. Two major problems are encountered when one desires to assure the statistical validity of these estimates. The first problem is the estimation of the time which takes the system to reach steady-state or equilibrium conditions. This time we refer to as the transient period. The second problem is that of calculating the required length of run for a desired accuracy of the estimates of the means of the variables of interest.