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

TL;DR

This paper establishes the necessary and sufficient stability conditions for an N-model queueing system with a waiting time dependent threshold, using coupling and stochastic dominance techniques.

Contribution

It provides the first rigorous proof of stability conditions for this queueing model with a waiting time dependent threshold on the diagonal.

Findings

Derived necessary and sufficient stability conditions

Introduced coupling and stochastic dominance methods

Extended understanding of queue stability with waiting time thresholds

Abstract

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.