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

TLDR: It is shown that there exist assignment probabilities under which the system has a product-form stationary distribution, and it is derived that waiting time distributions in steady state are derived.

Abstract: We consider a memoryless single station service system with servers $\mathcal{S}=\{m_{1},\ldots,m_{K}\}$ , and with job types $\mathcal{C}=\{a,b,\ldots\}$ . Service is skill-based, so that server m i can serve a subset of job types $\mathcal{C}(m_{i})$ . 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.