## 3 Modeling of Interactions between Non-Simultaneous Vehicles
In this way, two new sets of \(h\) stress histories are obtained: \(X_{i}\) ordered according to the minima; \(Y_{i}\) ordered according to the maxima, obviously satisfying the expressions:  
\[i\ <\ j\ \Rightarrow\ \sigma_{X_{i}\min}\ \leq\ \ \sigma_{X_{j}\min}\ \ \ \ 1\ \leq\ i,\ \ \ \ \ j\ \leq\ h; \tag{49}\]  
\[i\ <\ j\ \Rightarrow\ \sigma_{Y_{i}\max}\ \geq\ \ \sigma_{Y_{j}\max}\ \ \ \ 1\ \leq\ i,\ \ \ \ \ j\ \leq\ h. \tag{50}\]  
From expressions (49) and (50) it clearly follows that:  
* When \(X_{1}\) coincides with \(Y_{1}\), \(X_{1}\) cannot combine with any other;
* otherwise, \(X_{1}\) can combine with \(Y_{1}\) and with the other \(k-1\) stress histories \(Y_{i}\) satisfying the inequality \(\sigma_{Y_{i}\max}\ >\ \sigma_{X_{1}\max},1\leq\ k\ <\ h\), that means that \(X_{1}\) takes part to \(k\) combinations, which are in principle the most aggressive ones;
* reiterating the procedure, it results that \(X_{2}\) combines with other stress histories as long as \(\sigma_{Y_{i}\max}\ >\ \sigma_{X_{2}\max}\); and, more generally, that \(X_{j}\) combines with other stress histories as long as \(\sigma_{Y_{i}\max}\ >\ \sigma_{X_{j}\max}\);  
So that all possible combinations \(Z_{i}Z_{j}\) (or \(Z_{j}Z_{i}\)) are identified, indeed, and sorted according to the maximum stress ranges in decreasing order. The number of occurrences of the combined stress histories and the residual number of occurrences of each individual stress history can be now derived by recursive formulae. Let \(N_{Z_{i}(0)}\) be the number of occurrences of \(Z_{i}\), \(1\ \leq\ i\ \leq\ h\), in the assigned non-interacting vehicle flow and \(Z_{q}Z_{s}\) be the combination giving the maximum stress range. Since the combined stress history \(Z_{q}Z_{s}\) is absolutely equivalent to \(Z_{s}Z_{q}\), in the following the first element of the combination is conventionally the one with the lower minimum, i.e., \(\sigma_{Z_{q}\min}\ <\ \sigma_{Z_{i}\min}\). According to what was previously remarked, the combination is not influenced by any occurrence of \(Z_{i}\), \(i\ \
eq\ q\ \wedge\ i\ \
eq\ s\) as, certainly, \(\sigma_{Z_{i}\min}\ >\ \sigma_{Z_{q}\min}\) and \(\sigma_{Z_{i}\max}\ <\ \sigma_{Z_{i}\max}\). Recalling Equations (47) and (48), the expected number of occurrences \(N_{Z_{q}Z_{s}}\) of the combined stress history \(Z_{q}Z_{s}\) results in:  
\[N_{Z_{q}Z_{s}}\ =\ \frac{N_{Z_{q}(0)}N_{Z_{s}(0)}}{N_{Z_{q}(0)}\ +\ N_{Z_{s}(0)}}; \tag{51}\]  
while the expected residual number of occurrences of \(Z_{q}\) after the first combination, \(N_{Z_{q}(1)}\), and the expected residual number of occurrences of \(Z_{s}\) after the first combination, \(N_{Z_{q}(1)}\), becomes \[N_{Z_{q}(1)}\ =\ \frac{N_{Z_{q}(0)}^{2}}{N_{Z_{q}(0)}\ +\ N_{Z_{s}(0)}};\ N_{Z_{s}(1)}\ =\ \frac{N_{Z_{s}(0)}^{2}}{N_{Z_{q}(0)}\ +\ N_{Z_{s}(0)}}. \tag{52}\]  
It must be highlighted that the \(N_{Z_{q}Z_{s}}\) occurrences of \(Z_{q}\) and the \(N_{Z_{q}Z_{s}}\) occurrences of \(Z_{s}\) spent in the combination do not take part in other combinations and that \(Z_{q}\) cannot further combine with \(Z_{s}\). Therefore, the number of occurrences associated to \(Z_{q}\) in the event of a second combination is \(N_{Z_{q}(1)}\), and the resulting residual \(N_{Z_{q}(2)}\) and so on; analogously for \(Z_{s}\).