  We consider a single-server cyclic polling system with three queues where the
server follows an adaptive rule: if it finds one of queues empty in a given
cycle, it decides not to visit that queue in the next cycle. In the case of
limited service policies, we prove stability and instability results under some
conditions which are sufficient but not necessary, in general. Then we discuss
open problems with identifying the exact stability region for models with
limited service disciplines: we conjecture that a necessary and sufficient
condition for the stability may depend on the whole distributions of the
primitive sequences, and illustrate that by examples. We conclude the paper
with a section on the stability analysis of a polling system with either gated
or exhaustive service disciplines.
