  We consider a two-queue polling model with switch-over times and $k$-limited
service (serve at most $k_i$ customers during one visit period to queue $i$) in
each queue. The major benefit of the $k$-limited service discipline is that it
- besides bounding the cycle time - effectuates prioritization by assigning
different service limits to different queues. System performance is studied in
the heavy-traffic regime, in which one of the queues becomes critically loaded
with the other queue remaining stable. By using a singular-perturbation
technique, we rigorously prove heavy-traffic limits for the joint queue-length
distribution. Moreover, it is observed that an interchange exists among the
first two moments in service and switch-over times such that the HT limits
remain unchanged. Not only do the rigorously proven results readily carry over
to $N$($\geq2$) queue polling systems, but one can also easily relax the
distributional assumptions. The results and insights of this note prove their
worth in the performance analysis of Wireless Personal Area Networks (WPAN) and
mobile networks, where different users compete for access to the shared scarce
resources.
