  We study a parallel queueing system with multiple types of servers and
customers. A bipartite graph describes which pairs of customer-server types are
compatible. We consider the service policy that always assigns servers to the
first, longest waiting compatible customer, and that always assigns customers
to the longest idle compatible server if on arrival, multiple compatible
servers are available. For a general renewal stream of arriving customers and
general service time distributions, the behavior of such systems is very
complicated. In particular, the calculation of matching rates, the fraction of
services of customer-server type, is intractable. We suggest through a
heuristic argument that if the number of servers becomes large, the matching
rates are well approximated by matching rates calculated from the tractable
bipartite infinite matching model. We present simulation evidence to support
this heuristic argument, and show how this can be used to design systems with
desired performance requirements.
