Nonexistence of Brownian models for certain multiclass queueing networks

It has been demonstrated that Brownian models are effective for approximate analysis of queueing networks, particularly when no explicit analytical methods are available [3,4,1]. In 1988 Harrison [2] proposed Brownian models as approximations for multiclass open queueing networks. That work was expanded by Harrison and Nguyen in two recent papers [3,4] with the aim of developing a systematic method for performance analysis of open networks. The network models they consider have general routing and general service requirements. The corresponding Brownian model for a d-station network is a reflected Brownian motion in the d-dimensional non-negative orthant IRa+. Their work provides explicit formulas for calculating data of the reflected Brownian motion in terms of primitive network data. In this note we present two network examples where the Brownian model (reflected Brownian motion) proposed by Harrison and Nguyen does not exist. In