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={m_1, ..., m_K}, and with job types C={a, b, ...}. Service is skill-based, so that server m_i can serve a subset of job types C(m_i). 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. Keywords: Service system – First-come-first-served policy – Multi-type jobs – Multi-type servers – Partial balance – Product form solution