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 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 customer-server type , is intractable.