The priority-driven scheduling od periodic and sporadic task systems upon identical multiprocessor platforms is considered, under the restrictions that (i) each job may be assigned exactly one priority throughout its lifetime, and (ii) each job may execute upon only a single processor. It is shown that the feasibility-analysis under these restrictions is intractable (NP-hard in the strong sense). A scheduling algorithm is presented that satisfies these restrictions, and that has a worst-case utilization bound comparable to the worst-case utilization bounds of partitioned scheduling algorithms, and of scheduling algorithms that retain the priority-assignment restriction but allow arbitrary interprocessor migration.