  We consider a general polling model with $N$ stations. The stations are
served exhaustively and in cyclic order. Once a station queue falls empty, the
server does not immediately switch to the next station. Rather, it waits at the
station for the possible arrival of new work ("wait-and-see") and, in the case
of this happening, it restarts service in an exhaustive fashion. The total time
the server waits idly is set to be a fixed, deterministic parameter for each
station. Switchover times and service times are allowed to follow some general
distribution, respectively. In some cases, which can be characterised, this
strategy yields strictly lower average queueing delay than for the exhaustive
strategy, which corresponds to setting the "wait-and-see credit" equal to zero
for all stations. This extends results of Pek\"oz (Probability in the
Engineering and Informational Sciences 13 (1999)) and of Boxma et al. (Annals
of Operations Research 112 (2002)). Furthermore, we give a lower bound for the
delay for {\it all} strategies that allow the server to wait at the stations
even though no work is present.
