A Mean Value Analysis (MVA) approximation is presented for computing the average performance measures of closed multiclass queueing networks containing non pre-emptive Head Of Line (HOL) and Pre-empt Resume (PR) priority centers. The approximation has the same storage and computational requirements as MVA thus allowing computationally efficient solutions of large priority queueing networks. The accuracy of the MVA PR approximation is systematically investigated and presented in terms of error contour diagrams. The contour diagrams reveal that the approximation can compute the average performance measures of priority networks to within an accuracy of 5 percent for a large range of network parameter values. Accuracy of the method is also compared to Sevcik's shadow approximation and another MVA approximation recently proposed by Chandy and Lakshmi.