This paper is concerned with dynamic scheduling in a queueing system
that has two independent Poisson input streams, two servers, deterministic
service times and linear holding costs. One server can process both classes of
incoming jobs, but the other can process only one class, and the service time
for the shared job class is different depending on which server is involved. A
bound on system performance is developed in terms of a single pooled resource,
or super-server, whose capabilities combine those of the original two servers.
Thereafter, attention is focused on the heavy traffic regime, where the
combined capacity of the two servers is approximately equal to the total input
rate. We construct a discrete-review control policy and show that if its
parameters are chosen correctly as one approaches the heavy traffic limit, then
its cost performance approaches the bound associated with a single pooled
resource. Thus the discrete-review policy is proved to be asymptotically
optimal in the heavy traffic limit. Although resource pooling in heavy traffic
has been observed to occur in other network scheduling problems, there have
been very few studies that rigorously proved the pooling phenomenon, or that
proved the asymptotic optimality of a specific policy. Our discrete-review
policy is obtained by applying a general method, called the BIGSTEP method in
an earlier paper, to the parallel-server model.