Timeliness is an important feature for many embedded systems. Although soft real-time embedded systems can tolerate and allow certain deadline misses, it is still important to quantify them to justify whether the considered systems are acceptable. In this paper, we provide a way to safely over-approximate the expected deadline miss rate for a specific sporadic real-time task under fixed-priority preemptive scheduling in uniprocessor systems. Our approach is compatible with the existing results in the literature that calculate the probability of deadline misses either based on the convolution-based approaches or analytically. We demonstrate our approach by considering randomly generated task sets with an execution behavior that simulates jobs that are subjected to soft errors incurred by hardware transient faults under a given fault rate. To empirically gather the deadline miss rates, we implemented an event-based simulator with a fault-injection module and release the scripts. With extensive simulations under different fault rates, we evaluate the efficiency and the pessimism of our approach. The evaluation results show that our approach is effective to derive an upper bound of the expected deadline miss rate and efficient with respect to the required computation time.