Roofline time is the max of a compute ceiling and a bandwidth ceiling

A machine has peak floating-point performance $\pi=200\ \mathrm{GFLOP/s}$ and peak DRAM bandwidth $\beta=50\ \mathrm{GB/s}$. Arithmetic intensity is $I=W/Q$ in $\mathrm{FLOP/B}$, where $W$ is work in floating-point operations and $Q$ is DRAM traffic in bytes. Attainable performance is \[ P=\min(\pi,I\beta), \] and a lower bound on running time is \[ T\ge\max\!\left(\frac{W}{\pi},\frac{Q}{\beta}\right)=\frac{W}{P}. \] The ridge point is $I_\star=\pi/\beta$. Use these two ceilings consistently: never add the two times, and never replace $P$ by $\max(\pi,I\beta)$. 1. Compute $I_\star$. For a kernel with $I=0.5\ \mathrm{FLOP/B}$, give $P$ and classify the kernel as compute-bound or memory-bound. 2. That same kernel performs $W=1\ \mathrm{GFLOP}$ of work and moves $Q=2\ \mathrm{GB}$. Bound its running time from below. 3. A second kernel performs the same $W=1\ \mathrm{GFLOP}$ at intensity $I=6\ \mathrm{FLOP/B}$. Classify it, give $P$, and bound its running time from below. Confirm that the two ceilings are consistent with the stated $P$. 4. Audit these claims: (i) $I=0.5$ still runs at $200\ \mathrm{GFLOP/s}$ because that is the peak; (ii) the two ceilings should be added, so the first kernel takes at least $45\ \mathrm{ms}$; (iii) the bound is Amdahl's law with the held serial fraction $1-p=0.08$ and coordination overhead $h(N)$. Do not rederive that Amdahl fraction, and do not convert the ridge into a speedup-versus-$N$ plot.

$I_\star=200/50=4\ \mathrm{FLOP/B}$. Intensity $0.5$ lies below the ridge.