Polling systems in the critical regime

We study polling systems with Poisson arrival streams and exhaustive service discipline in the critical load case ρ=1. We show that the system may exhibit null-recurrence or transient behavior and give the corresponding conditions in terms of the first two moments of the distribution of service and switching times. For models with two nodes we establish existence of moments of small orders for the return time to the origin of the corresponding embedded Markov chain in the case of null-recurrence.