Transit assignment models represent the stochastic nature of waiting times, but usually adopt a deterministic representation route flows and costs. Especially in cities where transit vehicles are small and not operating to timetables, there is a need to represent the variability in flows and costs to enable planners make more informed decisions. Stochastic process (SP) models consider the day-to-day dynamics of the transit demand-supply system, explicitly modelling passengers’ information acquisition and decision processes. A Monte Carlo simulation based SP model that includes strict capacity constraints is presented in this paper. It uses micro-simulation to constrain passenger flows to capacities and obtain realistic cost estimates. Applications of the model and its comparison with the De Cea and Fernandez (Transp Sci, 27:133–147, 1993) model are presented using a small network.