Consider a production system where two types of jobs, $a$ and $b$ arrive at a central buffer according to independent Poisson processes with rates $lambda_a$ and $lambda_b$, respectively. Machines $m_1$ and $m_2$ select jobs from the central buffer when they are idle. When machine $m_1$ becomes idle, it takes the job from the head of the line, regardless of type, and serves it at rate $mu_1$. When machine $m_2$ becomes idle, it takes the earliest arrival among the type $a$ jobs and serves it at rate $mu_2$. Thus, the queue discipline is FCFS subject to the restriction that machine $m_2$ only processes type $a$ jobs. We derive rough and exact asymptotic results for the steady state behavior of this production system. For some parameter values, the fluid limit of the most likely approach to the rare event is nonlinear.

Joint work with Ivo Adan and David McDonald