We consider an ATM multiplexer with several FIFO buffers operating in the non-preemptive priority regime. Each connection or type of traffic (e.g. connectionless services) is allocated to one priority characterized by its grade of service constraints. Under very general assumptions we derive a model for buffer dimensioning and effective bandwidth allocation. The approach is based on upper bounds for grade of service measures (cell blocking, average cell delay) expressed in terms of performance of independent non-priority multiplexers (one for each priority). It is assumed that a model for buffer dimensioning and effective bandwidth allocation in a non-priority multiplexer with one FIFO queue is available. The important features of the approach are illustrated by means of numerical examples. The efficiency of the approach is also discussed.