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

TL;DR

This work analyzes the stability of the two-type, two-server N-model with a waiting-time dependent diagonal threshold $T>0$. It introduces a coupled upper-bound system and uses stochastic dominance to prove that the stability region under the threshold coincides with the classical region $\lambda_1+\lambda_2<\mu_1+\mu_2$ and $\lambda_2<\mu_2$, valid for all $T\ge 0$. The proofs combine coupling arguments with known results on the standard N-model and an $M/D/\infty$-type bound for the delayed type-1 queue, resolving a gap in the literature for non-Markovian, waiting-time dependent service disciplines. The approach provides a rigorous basis for analyzing similar skill-based queues and suggests directions for extending these results to more complex topologies, though certain models (e.g., the X-model) may require new techniques.

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.

Figures (1)

• Figure 1: Schematic overview of the $N$-model under two service disciplines. Here, the $\lambda_i$ denote the arrival rates, $\mu_j$ the service rates, and $T$ the deterministic threshold.

Theorems & Definitions (1)

• Theorem 1