In this paper, we develop a stochastic, multiclass network equilibrium model of airline passenger transportation. The model explicitly incorporates the behavior of the passengers in regard to the decision whether or not to travel, and route selection, and formalizes the passenger's decision as an abstract network. The equilibrium between the realized demand for the routes of travel and the supply of the seats is shown to satisfy a system of nonlinear equations. A Gauss-Seidel algorithm is then proposed for the computation of the equilibrium and conditions for convergence are established. Finally, the algorithm is applied to compute the solution to the national Air Canada airline passenger network, and the output is compared to the actual observed flows.