Abstract: To study the route choice behavior of travelers under travel time uncertainty, a probability density function of truncated normal distribution with respect to route travel time was proposed by taking the route free-flow travel time as the lower bound of route travel time for the degradable transport network. Subsequently, the travel time budget model considering route travel time boundary was developed. Based on boundedly rational decision-making characteristics of travelers, the maximum acceptable travel time was defined as the sum of the shortest travel time budget and in difference threshold, and the concept of boundedly rational confidence level was further presented, which refers to the probability that travelers can arrive within the indifference threshold of the shortest travel time budget. Then, the perceived boundedly rational confidence level of travelers was seen as the travel cost based on the random utility maximization theory, and the reliability-based boundedly rational traffic equilibrium model was developed. Additionally, its equivalent variational inequality problem was formulated. A route-based algorithm was designed to solve the proposed model. Finally, a numerical example was used to verify the proposed model and algorithm. Results show that ignoring the route free-flow travel time will reduce the on-time arrival possibility of travelers, and the scales of acceptable indifference threshold has a great impact on travelers' route choice behavior.