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

M/G/$\infty$ polling systems with random visit times