Design heuristic for parallel many server systems under FCFS-ALIS

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 $\mathcal{G}$, where arc $(c_i, s_j)$ indicates that server type $s_j$ can serve customer type $c_i$, and service policy of first come first served FCFS, assign longest idle server ALIS. 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 type $c_i$ by servers of type $s_j$, 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 FCFS bipartite infinite matching model. We present simulation evidence to support this heuristic argument, and show how this can be used to design systems for given performance requirements