A product form solution to a system with multi-type jobs and multi-type servers

We consider a memoryless single station service system with servers S = {m1, . . . , mK}, and with job types C = {a, b, . . .}. Service is skill based, so that server mi can serve a subset of job types C(mi). Waiting jobs are served on a first come first served basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product form stationary distribution, and obtain explicit expressions for it. We also derive waiting time distributions in steady state.