Stability conditions of the $N$-model with a waiting time dependent threshold on the diagonal

We consider the $N$-model queueing system with a waiting time dependent threshold on the diagonal: the service discipline is First–Come–First–Served, but type-1 jobs can only be served by server 2 if their waiting time exceeds a deterministic threshold. We prove the necessary and sufficient stability conditions for this model – an intuitive result that has not been established in literature up to this point. Our proof relies on coupling the queue length process to a carefully constructed upper (and lower) bound system, and establishing stochastic dominance for the queue length process.