  We study a model of a polling system, that is, a collection of $d$ queues
with a single server that switches from queue to queue. The service time
distribution and arrival rates change randomly every time a queue is emptied.
This model is mapped to a mathematically equivalent model of a random walk with
random choice of transition probabilities, a model which is of independent
interest. All our results are obtained using methods from the constructive
theory of Markov chains. We determine conditions for the existence of
polynomial moments of hitting times for the random walk. An unusual phenomenon
of thickness of the region of null recurrence for both the random walk and the
queueing model is also proved.
