Unlike uniprocessor scheduling, EDF (categorized into job-level fixed-priority (JFP) scheduling) shows relatively poor performance on global multiprocessor scheduling. As no other global JFP multiprocessor algorithms are illuminated beyond EDF, this work proposes one, called EQDF (earliest quasi-deadline first), as a generalization of EDF. We define the quasi-deadline of a job as a weighted sum of its absolute deadline (capturing "urgency") and its worst case execution time (capturing "parallelism") with a system-level control knob to balance urgency and parallelism effectively. This paper then seeks to explore how it can improve the schedulability of global JFP scheduling. In addition to providing a new schedulability analysis for EQDF scheduling, it addresses the problem of priority assignment under EQDF by controlling the system-level control knob. It presents optimal and heuristic solutions to the problem subject to our proposed EQDF analysis. Our empirical results show the proposed heuristic solution outperforms EDF significantly, giving close to optimal results.