  This report considers a sporadic real-time task system with $n$ sporadic
tasks on a uniprocessor platform, in which the lowest-priority task is a
segmented self-suspension task and the other higher-priority tasks are ordinary
sporadic real-time tasks. This report proves that the schedulability analysis
for fixed-priority preemptive scheduling even with only one segmented
self-suspending task as the lowest-priority task is $co{\cal NP}$-hard in the
strong sense. Under fixed-priority preemptive scheduling, Nelissen et al. in
ECRTS 2015 provided a mixed-integer linear programming (MILP) formulation to
calculate an upper bound on the worst-case response time of the lowest-priority
self-suspending task. This report provides a comprehensive support to explain
several hidden properties that were not provided in their paper. We also
provide an input task set to explain why the resulting solution of their MILP
formulation can be quite far from the exact worst-case response time.
