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 $𝒮=\{m_1,\dots ,m_K\}$ , and with job types $𝒞=\{a,b,\dots \}$ . 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.