We derive an expression for the Laplace-Stieltjes transform of the steady-state distribution of the queueing time for the M/G/1 finite capacity queue. The derivation proceeds in terms of a related 2-stage closed cyclic queueing network. The resulting expression is a rational function of the steady-state probabilities of the imbedded Markov chain at departure epochs and of the Laplace-Stieltjes transform of the service time distribution. The expression can be differentiated readily in order to obtain moments of the steady-state queueing time and some numerical results for the mean and coefficient of variation are presented.

This is a survey paper on the application of control variables to increase the efficiency of discrete event simulations. The emphasis is on the practical problems and potential of applying the method in the simulation of complex systems. The basic theory of control variables is reviewed and the equivalence of control variables and multiple estimators is discussed. Techniques for generating control variables are described. Inefficiencies resulting from the statistical estimation of control variable coefficients and the problem of confidence interval generation are treated. This is done both within the context of the method of independent replications and the regenerative method. The application literature is reviewed and the conditions under which control variables could be profitably applied in practical simulations are described. Finally, there is a set of recommended directions for future research.