In this paper we consider a single-server polling system with switch-overtimes. We introduce a new service discipline, mixed gated/exhaustive service,that can be used for queues with two types of customers: high and low prioritycustomers. At the beginning of a visit of the server to such a queue, a gate isset behind all customers. High priority customers receive priority in the sensethat they are always served before any low priority customers. But highpriority customers have a second advantage over low priority customers. Lowpriority customers are served according to the gated service discipline, i.e.only customers standing in front of the gate are served during this visit. Incontrast, high priority customers arriving during the visit period of the queueare allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for eachcustomer type, the joint queue length distribution of all priority classes atall queues at polling epochs, and the steady-state marginal queue lengthdistributions for each customer type. Through numerical examples we illustratethat the mixed gated/exhaustive service discipline can significantly decreasewaiting times of high priority jobs. In many cases there is a minimal negativeimpact on the waiting times of low priority customers but, remarkably, it turnsout that in polling systems with larger switch-over times there can be even apositive impact on the waiting times of low priority customers.
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