  We investigate the function $L(h,p,q)$, called here the threshold function,
related to periodicity of partial words (words with holes). The value
$L(h,p,q)$ is defined as the minimum length threshold which guarantees that a
natural extension of the periodicity lemma is valid for partial words with $h$
holes and (strong) periods $p,q$. We show how to evaluate the threshold
function in $O(\log p + \log q)$ time, which is an improvement upon the best
previously known $O(p+q)$-time algorithm. In a series of papers, the formulae
for the threshold function, in terms of $p$ and $q$, were provided for each
fixed $h \le 7$. We demystify the generic structure of such formulae, and for
each value $h$ we express the threshold function in terms of a piecewise-linear
function with $O(h)$ pieces.
