This study developed an approach to model network performance when its link capacities are subject to stochastic degradations, as in the form of day-to-day traffic incidents, which cause travel time variability. We postulate that drivers would select routes to lower their travel time variabilities, just as they would to lower their mean travel times. Over time, commuters learn the routes' travel time variabilities based on past experiences, factor such variabilities into their route choice considerations, and settle into a long-term equilibrium pattern. We characterize this route choice behavior in the face of uncertain travel times with the notion of probabilistic user equilibrium (PUE). This study then defined and formulated PUE with a reliability approach. We developed a nonlinear mathematical program to study the tradeoff between the maximum flow a network can carry and the extent of satisfying the PUE reliability constraints. As an analytical model, this formulation demonstrates certain interesting properties. The formulation can be used to analyze existing networks or to improve them by link capacity modifications. Numerical studies for a 19-link example are provided to show its performance and properties.