Polling systems with simultaneous batch arrivals

We study the delay in polling systems with simultaneous batch arrivals. Arrival epochs are generated according to a Poisson process. At any arrival epoch, batches of customers may arrive simultaneously at the different queues, according to a general joint batch-size distribution. The server visits the queues in cyclic order, the service times and the switch-over times are generally distributed, and the service disciplines are general mixtures of gated and exhaustive service. We derive closed-form expressions for the expected delay at each of the queues when the load tends to unity (under proper scalings), in a general parameter setting. The results are strikingly simple and reveal explicitly how the expected delay depends on the system parameters, and in particular, on the batch-size distributions and the simultaneity of the batch arrivals. Moreover, the results suggest simple and fast-to-evaluate approximations for the expected delay in heavily loaded polling systems. Numerical experiments demonstrate that the approximations are highly accurate in medium and heavily loaded systems.