  We consider polling models in the sense of Takagi (MIT Press, 1986). In our
case, the feature of the server is that it may be forced to wait idly for new
messages at an empty queue instead of switching to the next station. We propose
four different wait-and-see strategies that govern these waiting periods. We
assume Poisson arrivals for new messages and allow general service and
switchover time distributions. The results are formulas for the mean average
queueing delay and characterisations of the cases where the wait-and-see
strategies yield a lower delay compared to the exhaustive strategy.
