The current trend in modeling and analyzing real-time systems is toward tighter yet safe timing constraints. Many practical real-time systems can de facto sustain a bounded number of deadline-misses, i.e., they have Weakly-Hard Real-Time (WHRT) constraints rather than hard real-time constraints. Therefore, we strive to provide tight Deadline Miss Models (DMMs) in complement to tight response time bounds for such systems. In this work, we bound the distribution of deadline-misses for task sets running on uniprocessors using the Earliest Deadline First (EDF) scheduling policy. We assume tasks miss their deadlines due to transient overload resulting from sporadic jobs, e.g., interrupt service routines. We use Typical Worst-Case Analysis (TWCA) to tackle the problem in this context. Also, we address the sources of pessimism in computing DMMs, and we discuss the limitations of the proposed analysis. This work is motivated by and validated on a realistic case study inspired by industrial practice (satellite on-board software) and on a set of synthetic test cases. The synthetic experiment is dedicated to extensively study the impact of EDF on DMMs by presenting a comparison between DMMs computed under EDF and Rate Monotonic (RM). The results show the usefulness of this approach for temporarily overloaded systems when EDF scheduling is considered. They also show that EDF is especially well for WHRT tasks.