The Stochastic Sequential Assignment Problem With Random Deadlines

Abstract

Resources are to be allocated sequentially to activities to maximize the total expected return, where the return from an allocation is the product of the value of the resource and the value of the activity. The set of activities and their values are given ahead of time, but the resources arrive according to a Poisson process and their values are independent random variables that are observed upon arrival. It is assumed that either there is a single random deadline for all activities, which is the same as discounting the returns, or the activities have independent random deadlines. The model has applications machine scheduling, packet switching, and kidney allocation for transplant. It is known that the optimal policy in the discounted case has a very simple form that does not depend on the activity values. We show that this is also true when the deadlines are independent and in this case the solution can expressed in terms of solutions to single activity models. These results also hold when there are batch arrivals of resources. The effects of pooling separate identical systems with a single activity into a combined system is investigated for both models. When activities have independent deadlines it is optimal to reject a resource in the combined system if and only if it is optimal to reject it in the single activity system. However, when returns are discounted, it is sometimes optimal to accept a resource in the combined system that would be rejected in the single activity system.