  We study a mutually enriching connection between response time analysis in
real-time systems and the mixing set problem. Thereby generalizing over known
results we present a new approach to the computation of response times in
fixed-priority uniprocessor real-time scheduling. We even allow that the tasks
are delayed by some period-constrained release jitter. By studying a dual
problem formulation of the decision problem as an integer linear program we
show that worst-case response times can be computed by algorithmically
exploiting a conditional reduction to an instance of the mixing set problem. In
the important case of harmonic periods our new technique admits a
near-quadratic algorithm to the exact computation of worst-case response times.
We show that generally, a smaller utilization leads to more efficient
algorithms even in fixed-priority scheduling. Worst-case response times can be
understood as least fixed points to non-trivial fixed point equations and as
such, our approach may also be used to solve suitable fixed point problems.
Furthermore, we show that our technique can be reversed to solve the mixing set
problem by computing worst-case response times to associated real-time
scheduling task systems. Finally, we also apply our optimization technique to
solve 4-block integer programs with simple objective functions.
