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.