We study dynamic routing in a parallel server queueing network with a
single Poisson arrival process and two servers with exponential processing
times of different rates. Each customer must be routed at the time of
arrival to one of the two queues in the network. We establish that this
system operating under a threshold routing policy exhibits complete
resource pooling; i.e., it can be well approximated by a one-dimensional
reflected Brownian motion when the arrival rate to the network is close to
the processing capacity of the two servers. For complete resource pooling
to occur, the threshold should grow at a logarithmic rate as the heavy
traffic limit is approached. Second order growth terms then determine the
behavior of the limiting Brownian diffusion. We provide necessary and
sufficient conditions for (i) complete resource pooling (ii) positive
recurrence of the limiting Brownian diffusion and (iii) asymptotic
optimality of the threshold policy.

This is joint work with Yih-Choung Teh.