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.