First- and second-derivative estimators for cyclic closed-queueing networks

We consider a cyclic closed-queueing network with arbitrary service time distributions and derive first- and second-derivative estimators of some finite horizon performance metrics with respect to a parameter of any one of the service distributions. Our approach is based on observing a single sample path of this system and evaluating first- and second-order effects on departure times as a result of the parameter perturbation. We then define an estimator as a conditional expectation over appropriate observable quantities, using smoothed perturbation analysis techniques. This process recovers the first-derivative estimator along the way and gives new insights into event order change phenomena which are of higher order. Despite the complexity of the analysis, the final algorithms we obtain are relatively simple. Further, we show that our estimators are unbiased and include some numerical examples. We also show the use of our estimators in obtaining approximations of the entire system response surface as a function of system parameters.