Mixed gated/exhaustive service in a polling model with priorities

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