Asymptotic Product-form Steady-state for Multiclass Queueing Networks with SBP Service Policies in Multi-scale Heavy Traffic

In this work, we study the stationary distribution of the scaled queue length vector process in multiclass 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. A major assumption in proving the desired product-form limit is the uniform moment bound for scaled queue lengths. We prove this assumption holds if the unscaled high-priority queue lengths have uniform moment bound and a certain reflection matrix is a P-matrix.