In this paper, we study non-preemptive uniprocessor realtime scheduling using the non-preemptive RM (npRM) and EDF (npEDF) scheduling algorithms. We discuss the limitations of existing studies, identifying pessimism in current schedulability analysis and inefficiencies in existing processor speedup results. Focusing on harmonic task sets, we show that even with restrictions placed on the execution times of the tasks, npRM and npEDF are not able to schedule all feasible task sets. We obtain necessary conditions for the feasibility of the harmonic tasks with arbitrary integer period ratios. Then we derive sufficient conditions for the schedulability of npRM and npEDF upon harmonic task sets. Based on these conditions, a superior speedup factor which guarantees the schedulability in cases where there are fewer restrictions on the execution times is derived. Results from simulation experiments show an average speedup factor three times less than the only existing feasible method to obtain speedup factor.