Continuous-Time Airline Overbooking with Time-Dependent Fares and Refunds

We analyze a model of airline overbooking in which customer cancellations and no-shows are explicitly considered. We model the reservations process as a continuous-time birth-and-death process with rewards representing the fares received and refunds paid and a terminal-value function representing the bumping penalty. The airline controls the reservation acceptance (birth) rate by declining reservation requests. Assuming that the fares and refunds are piecewise-constant functions of the time to flight, we demonstrate that a piecewise-constant booking-limit policy is optimal, i.e., at all times, the airline accepts reservation requests up to a booking limit if the current number of reservations is less than that booking limit, and declines reservation requests otherwise. When the fare is constant over time or falls toward flight time, the optimal booking limit falls toward flight-time.