  Occasional deadline misses are acceptable for soft real-time systems.
Quantifying probabilistic and deterministic characteristics of deadline misses
is therefore essential to ensure that deadline misses indeed happen only
occasionally. This is supported by recent research activities on probabilistic
worst-case execution time, worst-case deadline failure probability, the maximum
number of deadline misses, upper bounds on the deadline miss probability, and
the deadline miss rate. This paper focuses on the deadline miss rate of a
periodic soft real-time task in the long run. Our model assumes that this soft
real-time task has an arbitrary relative deadline and that a job can still be
executed after a deadline-miss until a dismiss point. This model generalizes
the existing models that either dismiss a job immediately after its deadline
miss or never dismiss a job. We provide mathematical notation on the
convergence of the deadline miss rate in the long run and essential properties
to calculate the deadline miss rate. Specifically, we use a Markov chain to
model the execution behavior of a periodic soft real-time task. We present the
required ergodicity property to ensure that the deadline miss rate in the long
run is described by a stationary distribution.
