Thread Efficiency

Thread Efficiency considers two sources of inefficiency:

Serial computation on the master outside OpenMP, i.e. reflects Amdahl's law
Inefficiencies within threads, e.g. serialisation across threads

These two can be measured with Amdahl's Efficeincy and OpenMP region Efficiency respectively. Thread Efficeincy can be computed directly or as a sum of these two sub-metrics minus one:

$\begin{eqnarray*}Thread\; Efficiency=\frac{max(runtime) - avg(time\; in\; OpenMP) + avg(time\; in\; useful\; computation\; within\; OpenMP) - avg(idling\; time\; of\; OpenMP\; threads)}{max(runtime)} \end{eqnarray*}$

= Amdahl's Efficiency + OpenMP Region Efficiency - 1

Where average idling time of OpenMP threads considers that threads are idling if only master thread is working and can be computed by following formula

$\begin{eqnarray*}average\; idling\; time\; of\; OpenMP\; threads=\sum_{process=0}^{num\; of\; processes} \frac{serial\; computation \cdot (number\; of\; threads\; per\; process - 1)}{number\; of\; all\; available\; threads} \end{eqnarray*}$

Moreover, average time in OpenMP computed as weighted arithmetic mean. If number of threads is equal across processes average time in OpenMP can be computed as ordinary arithmetic mean.