FIRST AND 2ND DERIVATIVE ESTIMATORS FOR CLOSED JACKSON-LIKE QUEUING-NETWORKS USING PERTURBATION ANALYSIS TECHNIQUES

We consider a closed Jackson-like queueing network with arbitrary serv ice time distributions and derive an unbiased second derivative estima tor of the throughput over N customers served at some node with respec t to a parameter of the service distribution at that node. Our approac h is based on observing a single sample path of this system, and evalu ating all second-order effects on interdeparture times as a result of the parameter perturbation. We then define an estimator as a condition al expectation over appropriate observable quantities, as in Smoothed Perturbation Analysis (SPA). This process recovers the first derivativ e estimator along the way (which can also be derived using other techn iques), and gives new insights into event order change phenomena which are of higher order, and on the type of sample path information we ne ed to condition on for higher-order derivative estimation. Despite the complexity of the analysis, the final algorithm we obtain is relative ly simple. Our estimators can be used in conjunction with other techni ques to obtain rational approximations of the entire throughput respon se surface as a function of system parameters.