This paper proposes a novel stochastic mean-excess traffic equilibrium model that considers both reliability and unreliability aspects of travel time variability and perception errors within the travelers’ route choice decision processes. In the model, each traveler not only considers a travel time budget for ensuring on-time arrival at a confidence level α, but also accounts for the impact of encountering worst travel times in the (1-α) quantile of the distribution tail. Furthermore, due to the imperfect knowledge of the travel time variability, the travelers’ route choice decisions are based on the perceived travel time distribution rather than the actual travel time distribution. In order to compute the perceived mean-excess travel time, an approximation method based on moment analysis is developed. The proposed model is formulated as a variational inequality (VI) problem, and solved by a route-based solution algorithm with the use of the modified alternating direction method. Numerical examples are also provided to illustrate the application of the proposed model and solution method.