Transient analysis of cycle lengths in cyclic polling models

We consider cyclic polling models with gated or globally gated service, and study the transient behavior of all cycle lengths. Our aim is to analyze the dependency structure between the different cycles, as this is an intrinsic property making polling models challenging to analyze. Moreover, the cycle structure is related to the output of a polling model and the current analysis may be useful to study networks of polling models. In addition, transient performance is of great interest in systems where disruptions or breakdowns may occur, leading to excessive cycle lengths. The time to recover from such events is a primary performance measure. For the analysis we assume that the distribution of the first cycle (globally gated) or NN residence times (gated), where NN is the number of queues, is known and that the arrivals are Poisson. The joint Laplace–Stieltjes transform (LST) of all x subsequent cycles (globally gated) or all x>N subsequent residence times (gated) is expressed in terms of the LST of the first cycle. From this joint LST, we derive first and second moments and correlation coefficients between different cycles. Finally, a heavy-tailed first cycle length or the heavy-traffic regime provides additional insights into the time-dependent behavior.