## 3 Materials and Methods
### Justification of the Expression for Estimating \(\mu_{p}\)  
Equation (9) encompasses several terms, each in charge of one task. Let us review them in reverse order.  
The maximum of \(n\) independent and identically distributed (iid) normal random variables has no analytical expression, but it is upper bounded by \(\sqrt{2\cdot LN(n)}\). This is important as, in our context, \(n\approx N_{p}\), the random variables represent the path durations (which converge to normal distributions as there are more activities in series by the Central Limit theorem), and their maximum coincides with the average project duration (\(\mu_{p}\)).  
This upper bound, however, largely overestimates \(\mu_{p}\) for small values of \(n\) (for instance, when \(n<50\), the deviations are still above 20%). Most real projects have far fewer than 50 critical and subcritical parallel paths. That is why, from the upper bound \(\sqrt{2\cdot LN(n)}\), we kept the \(LN(n)\) term but reformulated it as \(K_{1}\cdot LN(S\cdot N_{p})\). More precisely, the contribution of \(S\) was discussed in the paper, whereas \(K_{1}\) was adjusted to maximize the fit for \(N_{p}<5\) paths (which approximately corresponds to projects with a serial-parallel indicator \(SP>0.2\), that is, most real projects as justified earlier). Consequently, in cases when \(N_{p}>5\), Equation (9) slowly (but progressively) loses accuracy.  
Next, the term \(\sum\sigma_{j}/\sum d_{j}\) in Equation (9) represents the average activity duration coefficient of variation, that is, the average \(CV_{j}\). In particular, this term is obtained from a weighted average of all activity coefficients of variation (\(CV_{j}\)) using the deterministic activity durations (\(d_{j}\)) as weights.  
\[\frac{\frac{\sum\bigl{(}d_{j}\cdot CV_{j}\bigr{)}}{n_{j}}}{\frac{\sum d_{j}}{n _{j}}}=\frac{\sum\bigl{(}d_{j}\cdot CV_{j}\bigr{)}}{\sum d_{j}}=\frac{\sum \sigma_{j}}{\sum d_{j}}.\]The ratio above is eventually multiplied by \(PD\) in Equation (9) which converts it into a proper standard deviation (in time units). The _raison d'etre_ of this term is that, following the previous analogy, the maximum of \(n\) iid normally distributed random variables is always proportional to the duration standard deviation of the parallel paths to which those activities belong. This justifies why we need to include a term that represents the average activity duration standard deviation, or, in this case, the average activity coefficient of variation.  
There is just one term left to justify from Equation (9); this is \(\sqrt{N_{p}/n_{j}}\). This term provides an estimate of the average number of activities per critical and subcritical path. When \(n\) iid activities are in series, then \(\sqrt{\sum\sigma_{j}^{2}}=\sqrt{n\cdot\sigma_{j}^{2}}\). However, since we are working with the coefficient of variation here instead of variances, it means we need to divide the latter expression by the average activity duration, that is, \(\frac{\sqrt{n\cdot\sigma_{j}^{2}}}{n\cdot d_{j}}=\frac{\sqrt{n\cdot d_{j}^{2} +CV_{j}^{2}}}{n\cdot d_{j}}=\frac{CV_{j}}{\sqrt{n}}\). In this expression, \(CV_{j}\) represents the term \(\sum\sigma_{j}/\sum d_{j}\) explained earlier. Hence, \(1/\sqrt{n}\) represents the reduction of \(CV_{j}\) as more activities are located in the same path. In our case, the average number of critical and subcritical activities per path can be approximated by \(n\approx\frac{n_{j}}{N_{p}}\), then \(\frac{1}{\sqrt{n}}\approx\sqrt{\frac{N_{p}}{n_{j}}}\), exactly as in Equation (9).