Asymptotic optimality of constant capacity allocation policies for dynamic resource planning

We consider a stochastic capacity problem of dynamically matching the supply of resources and the uncertain demand for resources over a planning horizon of T periods. If demand exceeds the resource capacity in any time period, then such excess demand is lost and a penalty is incurred for the amount of lost demand. The discrete-time model reflects the relative time granularity at which resource capacity decisions can be made, whereas a lead time L > 0 represents in the model the relative time-scale difference between when a resource capacity decision is made and when this capacity decision takes effect. Our optimal control problem then is to determine the best policy for making resource capacity allocation decisions in each period, based on the demand realized up to that period, with the goal of maximizing net-benefit in expectation over the planning horizon.