Before revision: We study a parallel service queueing system with servers of types s_1,\ldots,s_J, customers of types c_1,\ldots,c_I, bipartite compatibility graph \mathcal{G . 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 r_{c_i,s_j , the fraction of services of customers of typec_i by servers of type s_j , is intractable.
After revision: We study a parallel service 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 r_{c_i,s_j , the fraction of services of customers of typec_i by servers of type s_j , is intractable.