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

Abstract

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.