M/G/∞ polling systems with random visit times

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 the 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/∞-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 article is the first in which an M/G/∞-type polling system is analyzed. 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 maximizes the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle.