Polling systems have been extensively studied, and have had many applications. They have often been used for studying wired local areas networks (such as token passing rings) and wireless local area networks (such as bluetooth). In this work we relax one of the main restrictions on the statistical assumptions under which polling systems have been analyzed. Namely, we allow correlation between walking times. We consider (i) the gated regime where a gate closes whenever the server arrives at a queue. He then serves at that queue all customers who were present when the gate closes. (ii) the exhaustive service in which the server remains at a queue till it empties. Our analysis is based on stochastic recursive equations related to branching processes with migration with a random environment. In addition to our derivation of expected waiting times for polling systems with correlated vacations, we set the foundations for computing second order statistics of the general multi-dimensional stochastic recursions.