Ergodicity, moment stability and central limit theorems of station times in polling systems

The station times are an important measure of performance in polling systems, and are often used to determine efficiently other performance measures, such as waiting times. In this paper we give sufficient and necessary conditions for the existence of all moments of station times in steady state, for polling systems with Gated and Globally-Gated disciplines. Moreover, we show that the moments converge geometrically fast to the steady state ones under these conditions. We then address the question of the rate of convergence of the sample averages of functions of the station times. We establish the applicability of central limit theorems (CLT) and the law of iterated logarithm (LIL) for all moments of the station times. In particular, we compute explicitly the constants involved in the CLT and LIL for the cycle durations.