We focus on the problem of computing tight deadline miss models for real-time systems, which bound the number of potential deadline misses in a given sequence of activations of a task. In practical applications, such guarantees are often sufficient because many systems are in fact not hard real-time. Our major contribution is a general formulation of that problem in the context of systems where some tasks occasionally experience sporadic overload. Based on this new formulation, we present an algorithm that can take into account fine-grained effects of overload at the input of different tasks when computing deadline miss bounds. Finally, we show in experiments with synthetic as well as industrial data that our algorithm produces bounds that are much tighter than in previous work, in sufficiently short time.