  In this paper we study a two-queue polling model with zero switch-over times
and $k$-limited service (serve at most $k_i$ customers during one visit period
to queue $i$, $i=1,2$) in each queue. The arrival processes at the two queues
are Poisson, and the service times are exponentially distributed. By increasing
the arrival intensities until one of the queues becomes critically loaded, we
derive exact heavy-traffic limits for the joint queue-length distribution using
a singular-perturbation technique. It turns out that the number of customers in
the stable queue has the same distribution as the number of customers in a
vacation system with Erlang-$k_2$ distributed vacations. The queue-length
distribution of the critically loaded queue, after applying an appropriate
scaling, is exponentially distributed. Finally, we show that the two
queue-length processes are independent in heavy traffic.
