  In this paper, we analyze a two-queue random time-limited Markov modulated
polling model. In the first part of the paper, we investigate the fluid
version: Fluid arrives at the two queues as two independent flows with
deterministic rate. There is a single server that serves both queues at
constant speeds. The server spends an exponentially distributed amount of time
in each queue. After the completion of such a visit time to one queue, the
server instantly switches to the other queue, i.e., there is no switchover
time.
  For this model, we first derive the Laplace-Stieltjes Transform (LST) of the
stationary marginal fluid content/workload at each queue. Subsequently, we
derive a functional equation for the LST of the two-dimensional workload
distribution that leads to a Riemann-Hilbert boundary value problem (BVP).
After taking a heavy-traffic limit, and restricting ourselves to the symmetric
case, the boundary value problem simplifies and can be solved explicitly.
  In the second part of the paper, allowing for more general (L\'evy) input
processes and server switching policies, we investigate the transient
process-limit of the joint workload in heavy traffic. Again solving a BVP, we
determine the stationary distribution of the limiting process. We show that, in
the symmetric case, this distribution coincides with our earlier solution of
the BVP, implying that in this case the two limits (stationarity and heavy
traffic) commute.
