Providing guaranteed quantification of properties of soft real-time systems is important in practice to ensure that a system performs correctly most of the time. We study utility accrual for real-time systems, in which the utility of a real-time job is defined as a time utility function with respect to its response time. Essentially, we answer the fundamental questions: Does the utility accrual of a periodic real-time task in the long run converge to a single value? If yes, to which value? We first show that concrete problem instances exist where evaluating the utility accrual by simulating the scheduling algorithm or conducting scheduling experiments in a long run is erroneous. Afterwards, we show how to construct a Markov chain to model the interactions between the scheduling policy, the probabilistic workload of a periodic real-time task, the service provided by the system to serve the task, and the effect on the utility accrual. For such a Markov chain, we also provide the theoretical fundamentals to determine whether the utility accrual converges in the long run and the derivation of the utility accrual if it converges.