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

Authors:
Gang Bao;ChristosG. Cassandras;Michael A. Zazanis
Affiliations:
Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003;Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003;Dept. of Industrial Engineering and Operations Research, University of Massachusetts, Amherst, MA 01003
Venue:
Discrete Event Dynamic Systems
Year:
1997

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Abstract

Weconsider a closed Jackson—like queueing network with arbitraryservice time distributions and derive an unbiased second derivativeestimator of the throughput over N customers servedat some node with respect to a parameter of the service distributionat that node. Our approach is based on observing a single samplepath of this system, and evaluating all second-order effectson interdeparture times as a result of the parameter perturbation.We then define an estimator as a conditional expectation overappropriate observable quantities, as in Smoothed PerturbationAnalysis (SPA). This process recovers the first derivative estimatoralong the way (which can also be derived using other techniques),and gives new insights into event order change phenomena whichare of higher order, and on the type of sample path informationwe need to condition on for higher-order derivative estimation.Despite the complexity of the analysis, the final algorithm weobtain is relatively simple. Our estimators can be used in conjunctionwith other techniques to obtain rational approximations of theentire throughput response surface as a function of system parameters.