Tight matrices and heavy traffic steady state convergence in queueing networks

We are interested to prove that the stationary distribution of a multiclass queueing network converges to the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in heavy traffic. A key condition for this convergence is that the sequence of the pre-limit stationary distributions under appropriate scaling is tight. In Braverman et al.(2025), a sufficient condition for this tightness is introduced in the term of the reflection matrix $R$ of the SRBM, which is coined for $R$ to be ``tight''. In this paper, we study how we can verify this tightness of $R$ of an SRBM. For a $2$-dimensional SRBM, we give necessary and sufficient conditions for $R$ to be tight, while, for a general dimension, we only give sufficient conditions. We then apply these results to the SRBMs arising from the diffusion approximations of multiclass queueing networks with static buffer priority service disciplines that are studied in Braverman et al.(2025). It is shown that $R$ is always tight for this network with two stations if $R$ is completely-$\sr{S}$. For the case of more than two stations, it is shown that $R$ is tight for reentrant lines with last-buffer-first-service (LBFS) discipline, but it is not always tight for reentrant line with first-buffer-first-service (FBFS) discipline.