  In this work, we study the stationary distribution of the scaled queue length
vector process in multi-class queueing networks operating under static buffer
priority service policies. We establish that when subjected to a multi-scale
heavy traffic condition, the stationary distribution converges to a
product-form limit, with each component in the product form following an
exponential distribution. This convergence is achieved under the assumption of
uniform moment bounds. As an example, the uniform moment bound condition is
shown to hold in a two-station five-class re-entrant line. Whether such a bound
holds for general multi-class queueing networks remains an open problem.
