## 1 Introduction  
In recent years, the problem of modeling uncertain travel times in distribution logistics has gained increasing attention as a means to improve logistics operations and enhance overall efficiency. The efficient management of distribution logistics is of crucial importance for businesses operating in distribution logistics, especially in cities with high traffic congestion. The ability to accurately predict and account for uncertain travel times is essential for effective logistics planning and optimization. The unpredictable nature of travel times poses significant challenges for logistics professionals. Traditional logistics models that assume fixed travel times often fail to capture the dynamic and uncertain nature of transportation. This can lead to inefficient resource allocation, missed delivery deadlines, increased costs, and customer dissatisfaction. To address these challenges, researchers and practitioners have turned their attention to the modeling of uncertain travel times as a key factor in distribution logistics. The variability and duration of travel time are influenced by various factors, particularly congestion. Congestion can be categorized as recurrent or non-recurrent [1], each with its distinct characteristics and causes. Recurrent congestion arises from daily peak-hour delays caused by the imbalance between traffic demand and the capacity of the transportation infrastructure. On the other hand, non-recurrent congestion results from atypical events, including incidents [2, 3], disabled vehicles, road construction works, accidents, adverse weather conditions, and special events. Both categories of congestion introduce variability and uncertainty in travel times, leaving drivers uncertain about their precise arrival times at their destinations. Non-recurrent congestion can have dual effects on traffic conditions. It can generate new congestion during off-peak periods or increase delays during recurring congestion episodes. However, this type of congestion has often been overlooked in traffic engineering and modeling practices due to its irregular and unpredictable nature [4]. The modeling of the uncertain time variability has attracted the interest of several researchers. In a study by [PERSON] [5], the examination of travel time variability involved the assessment of various factors, including time of day, day of the week, weather conditions, and traffic accidents. To quantify the influence of these factors on travel time parameters, the author employed multiple linear regression models with two-way interactions. The researchers in [6] investigated the prediction of uncertainty in route travel times within urban road networks, utilizing an extensive dataset from floating taxis; in [7], the authors delved into the importance of precise travel time modeling for road-based public transport and provided an overview of the current advancements in statistically modeling bus travel time distributions. They emphasized the significance of aggregating spatial and temporal data and proposed potential avenues for future research in this domain. In [8], researchers investigated travel time variability within the context of Indian traffic conditions and assessed the effectiveness of travel time distributions when considering various temporal and spatial aggregations. Utilizing AVL data gathered from four transit routes in Mysore City, located in Karnataka, India, this research examined travel time distributions across peak and off-peak periods, as well as within different time intervals. The study employed the Anderson-Darling test to evaluate the goodness-of-fit of the distributions, taking into account segments featuring signalized intersections and a variety of land-use types. The findings underscored the effectiveness of the generalized extreme value (GEV) distribution in precisely characterizing travel time variations within public transit systems. Therefore, the quantification of uncertainty remains a pivotal component when modeling and estimating travel time variability.