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In this article we present estimates for the gradients of the cycle variables with respect to a thinning parameter m the arrival process of G\G\ 1 queuing systems. Our estimates belong to the category of the likehhood ratio method, smoothed-perturbation analysm, and rare-perturbation analysls. In particular, we obtain unbiased estimates of derivatives of any order m the random-horizon case and the stationary and ergodic estimates for the first derivatives. Moreover, by inserting the stationary estimates mto the Palm distribution framework we obtain the regenerative type of derivative estimates without requiring regenerate structure. Our theoretical setting is general, and the methods used here could find applications in other derivative estimation problems as well.

Derivatives of likelihood ratios and smoothed perturbation analysis for the routing problem