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Global scheduling reduces the average response times as it can use the available computing cores more efficiently for scheduling ready tasks. However, this flexibility poses challenges in accurately quantifying interference scenarios, often resulting in either conservative response-time analyses or scalability issues. In this paper, we present a new response-time analysis for preemptive periodic tasks (or job sets) subject to release jitter under global job-level fixed-priority (JLFP) scheduling. Our analysis relies on the notion of schedule-abstraction graph (SAG), a reachability-based response-time analysis known for its potential accuracy and efficiency. Up to this point, SAG was limited to non-preemptive tasks due to the complexity of handling preemption when the number of preemptions and the moments they occur are not known beforehand. In this paper, we introduce the concept of time partitions and demonstrate how it facilitates the extension of SAG for preemptive tasks. Moreover, our paper provides the first response-time analysis for the global EDF(k) policy - a JLFP scheduling policy introduced in 2003 to address the Dhall’s effect. Our experiments show that our analysis is significantly more accurate compared to the state-of-the-art analyses. For example, we identify 12 times more schedulable task sets than existing tests for the global EDF policy (e.g., for systems with 6 to 16 tasks, 70% utilization, and 4 cores) with an average runtime of 30 minutes. We show that EDF(k) outperforms global RM and EDF by scheduling on average 24.9% more task sets (e.g., for systems with 2 to 10 cores and 70% utilization). Moreover, for the first time, we show that global JLFP scheduling policies (particularly, global EDF(k)) are able to schedule task sets that are not schedulable using well-known partitioning heuristics.