First and Second Derivative Estimators for Closed Jackson-Like Queueing Networks Using Perturbation Analysis Techniques

We consider a closed Jackson—like queueing network with arbitrary service time distributions and derive an unbiased second derivative estimator of the throughput over N customers served at some node with respect to a parameter of the service distribution at that node. Our approach is based on observing a single sample path of this system, and evaluating all second-order effects on interdeparture times as a result of the parameter perturbation. We then define an estimator as a conditional expectation over appropriate observable quantities, as in Smoothed Perturbation Analysis (SPA). This process recovers the first derivative estimator along the way (which can also be derived using other techniques), and gives new insights into event order change phenomena which are of higher order, and on the type of sample path information we need to condition on for higher-order derivative estimation. Despite the complexity of the analysis, the final algorithm we obtain is relatively simple. Our estimators can be used in conjunction with other techniques to obtain rational approximations of the entire throughput response surface as a function of system parameters.