## 5 Conclusions and Discussion  
This study focuses on quantifying uncertainties in modeling travel time variations in two distinct traffic scenarios--recurrent congestion and non-recurrent congestion--observed during four peak delivery periods in Casablanca's distribution logistics. The research highlights the critical importance of addressing uncertainties in travel times within Casablanca's dynamic traffic environment, necessitating reliable methods like collocation for effective management. The probability density functions (PDFs) and the cumulative density functions (CDFs) were utilized to analyze travel time distributions, offering valuable insights into logistics uncertainties. In cases of recurrent congestion during morning peak, inter-peak, evening peak, and off-peak times, fluctuations in travel time were observed. The PDFs showed some variability, but, overall, they indicated a relatively stable traffic pattern. Contrastingly, non-recurrent congestion scenarios, often triggered by incidents like accidents, exhibited more complex and multi-modal PDF distributions due to unpredictable traffic disruptions. The presence of multiple peaks in the PDFs during non-recurrent congestion scenarios underscored the diverse and distinct travel time variations caused by such incidents. Accidents and other unforeseen events have a profound impact on distribution logistics, causing delays and uncertainties in delivery times.  
The study emphasizes the importance of considering different traffic scenarios when modeling travel time variability for distribution logistics. By quantifying uncertainties and understanding the impact of both recurrent and non-recurrent congestion, logistics managers and planners can develop robust strategies to optimize routes, improve efficiency, and minimize disruptions' impact on delivery schedules. As for the limitations of this study, while it addresses the effects of recurrent and non-recurrent congestion on travel time variability, other factors may also play a role, such as weather conditions, unforeseeable incidents, and driver behaviors. These elements can introduce additional uncertainties that have not been fully explored within the scope of this research. Hence, future improvements  
\begin{table}
\begin{tabular}{l l l l} \hline
**Peak Time** & **Low** & **Medium** & **High** \\ \hline
**Moming peak** & 0.11765 & 0.058824 & 0.82353 \\ Inter-peak & 0.58824 & 0.058824 & 0.35294 \\ \hline Evening peak & 0.11765 & 0.11765 & 0.76471 \\ Off-peak & 0.76471 & 0.17647 & 0 \\ \hline \end{tabular}
\end{table}
Table 7: Travel time probability in non-recurrent congestion.  
might involve more accurate data collection and the consideration of additional data sources to validate the findings.  
Conceptualization, K.A.M., L.H., A.E.B., A.G.N.N. and E.S.d.C.; Formal analysis, K.A.M., L.H., A.E.B., A.G.N.N. and E.S.d.C.; Methodology, K.A.M., L.H. and E.S.d.C.; Software, K.A.M. and E.S.d.C.; Supervision, L.H.; Validation, L.H., A.E.B., A.G.N.N. and E.S.d.C.; Writing--original draft, K.A.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The supporting data for the results of this study can be obtained from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.