A closed-form estimation of the travel time percentile function for characterizing travel time reliability

Travel time reliability (TTR) has received great attention in the past decades. The majority of TTR measures rely on the travel time percentile function as a basic element for performance evaluation. There are two main approaches for deriving the travel time percentile function: simple unimodal probability distribution models and mixture/nonparametric models. Despite the tractability of the former approach, they cannot sufficiently capture the travel time distributions (TTDs) due to their heterogeneity, and also often encounters many issues such as the failure of significance tests and the indecisiveness among multiple fitted distributions. On the other hand, the latter approach possesses greater flexibility for capturing diverse TTDs, but it does not have a simple and closed-form travel time percentile function. Motivated by the above drawbacks, this paper proposes a closed-form and flexible approach for estimating the travel time percentile function of diverse TTDs based on the Cornish–Fisher expansion without the need to assume/fit a certain distribution type. To ensure a high-quality estimation, we introduce and integrate two improvements with theoretically proven foundation into the Cornish–Fisher expansion while guaranteeing a closed-form expression of the travel time percentile function. Specifically, the first improvement, logarithm transformation, increases the probability of satisfying the validity domain of the Cornish–Fisher expansion; while the second improvement, rearrangement, guarantees a monotone travel time percentile function when travel time datasets cannot satisfy the validity domain after the logarithm transformation. Realistic travel time datasets are used to examine the accuracy and robustness of the proposed method. Compared to five widely-used probability distributions, the proposed method is sufficiently adaptable to estimating percentile function of diverse TTDs with lower estimation error. More importantly, it has a closed-form expression of the travel time percentile function, which would facilitate characterizing TTR in large-scale network applications.