On a queueing-inventory system with advanced reservation and cancellation for the next K time frames ahead: the case of overbooking

Abstract

We analyse the evolution of a system designed for reservation of some items in advance (for example, seats in aircrafts or trains or bus) by customers arriving at random moments. The reservation has to be done by the server in one of the K time frames. At the beginning of the pth time frame, the inventoried items in it (as well as those sold from it earlier) have life time distribution which is a p-fold convolution of a phase-type distribution with itself, for $$1 \le p \le K.$$ Cancellation of reserved items is possible before the expiry of their life. Distributions characterizing the service and inter-cancellation times are assumed to be independent exponential random variables and the customer arrivals are according to a Markovian arrival process. The number of items for reservation, available at the beginning of each time frame, is finite. If, at the commencement of service of a customer, the item in the required time frame is not available, the reservation may still be possible, through overbooking. Overbooking up to a maximum fixed level is permitted for each time frame. If, for the required day, the overbooked item is available, the customer is served the same. If this too is not available, he is asked to give alternatives. If none of his alternatives can be met, he is provided with a reservation for the time frame (day) for which one is available. If that too is not available, then he will have to wait until the expiry of one time frame; in the last case all remaining customers will have to wait. On expiry of one phase distribution, the time frames are renumbered and a new time frame with a K-fold convolution of the phase-type distribution is added $$(0 \leftarrow 1 \leftarrow 2 \leftarrow \cdots \leftarrow K-1 \leftarrow K \leftarrow K+1).$$ All overbooked customers present in the recently expired time frame are provided with a reservation in the newly added time frame (which has, at that epoch, a life time of a K-fold convolution of the phase-type distribution). This system is analysed and illustrated through numerical experiments. The special case of Poisson arrival, coupled with blocking of arrivals when all time frames $$1, 2, \ldots , K$$ are overbooked, is shown to yield a product form solution. For this case, an appropriate cost function is constructed and its properties investigated numerically.