In this paper the Perturbation-Analysis-Robbins-Monro-Single-Run algorithm is applied to estimating the optimal parameter of a performance measure for the GI/G/1 queueing systems, where the algorithm is updated after every fixed-length observation period. Our aim is to analyze the limiting behavior of the algorithm. The almost sure convergence rate of the algorithm is established. It is shown that the convergence rate depends on the second derivative of the performance measure at the optimal point.