We present a model that rapidly finds an approximation of the expected passenger flow on an airline network, given forecast data concerning 1) the distribution of the demand for each itinerary, seen as a random variable; 2) the time distribution of booking requests for each itinerary; and 3) the proportion of spill (from an itinerary) that is attracted to a given alternative itinerary. Solutions are found in a few seconds for a 30 000 itinerary network. Results differ from the expected passenger flow found by a simulation by about 0.1% for load factors below 80%.