A brushless-DC commutation computer runs two independent implicit-deadline periodic…

A brushless-DC commutation computer runs two independent implicit-deadline periodic tasks on a fully preemptive uniprocessor under either rate-monotonic or earliest-deadline-first (EDF) scheduling. Zero jitter, zero blocking, and zero overhead are assumed. Times: \[ \tau_1:\ C_1=2.00\,\mathrm{ms},\ T_1=5.00\,\mathrm{ms},\qquad \tau_2:\ C_2=4.00\,\mathrm{ms},\ T_2=7.00\,\mathrm{ms}. \] For implicit-deadline independent periodic tasks, EDF on a uniprocessor is schedulable if and only if \(U\le 1\). RM uses the Liu--Layland sufficient bound \(U_{\mathrm{lub}}(2)=2(\sqrt{2}-1)\) and the exact recurrence \[ R_i^{(0)}=C_i,\qquad R_i^{(k+1)}=C_i+\sum_{h\in hp(i)}\Bigl\lceil\frac{R_i^{(k)}}{T_h}\Bigr\rceil C_h, \] with a pass only if the least fixed point satisfies \(R_i\le T_i\). (a) Compute \(U\) as an exact rational and as a decimal to four places. Compute \(U_{\mathrm{lub}}(2)\) to four places. Decide the Liu--Layland RM test and the EDF utilization test. (b) Under RM, with \(\tau_1\) strictly higher priority, compute \(R_1\) and the iterates of \(R_2\). Decide both RM deadlines. (c) Under EDF, construct the synchronous schedule over one hyperperiod \(H=\mathrm{lcm}(T_1,T_2)\). For every job, report release time, absolute deadline, execution intervals, and completion time. Decide every deadline. (d) A third implicit-deadline task \(\tau_3\) with \(C_3=0.20\,\mathrm{ms}\) and \(T_3=35.0\,\mathrm{ms}\) is added. Recompute \(U\). Decide EDF. Under RM with priorities by period (\(\tau_1>\tau_2>\tau_3\)), decide whether \(\tau_2\) can become schedulable merely because a lower-priority task was added, and recompute \(R_2\) to confirm.

Implicit-deadline EDF is a utilization test; RM needs either Liu--Layland or an exact response-time fixed point.