Optimal control of continuous‐time terminal‐value birth‐and‐death processes and airline overbooking

Consider a continuous‐time airline overbooking problem that relates to a single‐leg flight and a single service class with a stationary fare. Passengers may cancel their reservations at any time and receive a full refund. Therefore fares can be thought of as being paid at flight time. At that time, the airline bumps passengers in excess of flight capacity and pays a penalty for so doing. The wflight‐time revenue, that is, fares received less bumping penalties paid, is quasiconcave in the number of reservations at that time. We model the reservations process as a continuous‐time terminal‐value birth‐and‐death process. A more general model than is necessary for an airline reservations system is considered, in which the airline controls both the reservation acceptance (birth) and the cancellation (death) rates. In current practice airlines do not control cancellation rates (though other industries do exercise such control, e.g., hotels) and control reservation acceptance rates by declining reservation requests. The more general model might be applied to other targeting applications, such as steering a vehicle through space toward a target location. For the general model a piecewise‐constant booking‐limit policy is optimal; that is, 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 airline is allowed to decline all reservation requests, as is the case in practice, the booking‐limit optimal policy defined by using the greatest optimal booking limit at all times is piecewise constant. Moreover, these booking limits fall toward flight time. © 1996 John Wiley & Sons, Inc.