This paper proposes link-based and approach-based variational inequality (VI) formulations for the stochastic frequency-based transit assignment with soft capacity constraints. An augmented route-section network representation is developed to address the covariance of in-vehicle travel costs between different sections of the same transit line. The augmented route-section network allows us applying the dynamic programming method to compute the mapping function of the VI, even if the correlation of travel times between route sections exists. To solve the problem, an extragradient method that only requires a mild assumption for convergence is proposed. A capacity paradox, a phenomenon in which the network maximum throughput may be reduced after providing new transit lines to a transit network or increasing the frequency of an existing line, is discussed. Numerical examples are provided to illustrate the effects of different parameters
on the occurrence of the capacity paradox. Moreover, it is found that the capacity paradox may not occur simultaneously with the Braess-like paradox, a phenomenon in which providing new transit lines to a network may deteriorate the network performance in terms of the total effective travel cost.