  We consider a polling system where a group of an infinite number of servers
visits sequentially a set of queues. When visited, each queue is attended for a
random time. Arrivals at each queue follow a Poisson process, and service time
of each individual customer is drawn from a general probability distribution
function. Thus, each of the queues comprising the system is, in isolation, an
M/G/$\infty$-type queue. A job that is not completed during a visit will have a
new service time requirement sampled from the service-time distribution of the
corresponding queue. To the best of our knowledge, this paper is the first in
which an M/G/$\infty$-type polling system is analysed. For this polling model,
we derive the probability generating function and expected value of the queue
lengths, and the Laplace-Stieltjes transform and expected value of the sojourn
time of a customer. Moreover, we identify the policy that maximises the
throughput of the system per cycle and conclude that under the Hamiltonian-tour
approach, the optimal visiting order is \emph{independent} of the number of
customers present at the various queues at the start of the cycle.
