This thesis investigate two issues in operations research. The first one is the instantaneous Brownian control problem with a positive lead time, and the second one is the online resource allocation problem with general chaining structure.

In the first part, we consider a storage system where the content is driven by a Brownian motion in the absence of control. At any time, one may increase or decrease the content at a cost proportional to the amount of adjustment. A decrease of the content takes effect immediately, while an increase is realized after a fixed lead time ℓ. Holding costs are incurred continuously over time and are a convex function of the content. The objective is to find a control policy that minimizes the expected present value of the total costs. Due to the po...[ Read more ]

This thesis investigate two issues in operations research. The first one is the instantaneous Brownian control problem with a positive lead time, and the second one is the online resource allocation problem with general chaining structure.

In the first part, we consider a storage system where the content is driven by a Brownian motion in the absence of control. At any time, one may increase or decrease the content at a cost proportional to the amount of adjustment. A decrease of the content takes effect immediately, while an increase is realized after a fixed lead time ℓ. Holding costs are incurred continuously over time and are a convex function of the content. The objective is to find a control policy that minimizes the expected present value of the total costs. Due to the positive lead time for upward adjustments, one needs to keep track of all the outstanding upward adjustments as well as the actual content at time t as there may also be downward adjustments during [t,t+ℓ), i.e., the state of the system is a function on [0,ℓ]. We first extend the concept of L^♮-convexity to function spaces and establish the L^♮-convexity of the optimal cost function. We then derive various properties of the cost function and identify the structure of the optimal policy as a state-dependent two-sided reflection mapping making the minimum amount of adjustment necessary to keep the system states within a certain region.

In the second part, we consider a class of online resource allocation problems in unbalanced system. The resources are flexible in that each type of resources can serve more than one demand class. We focus on a special class of structures with a new flexibility condition called the Generalized Chaining Condition (GCC for short). GCC introduced by Shi et al. (2016) can be viewed as a generalization of the chaining idea put forth Jordan and Graves (1995), and includes the notion of long chain as a special case. We analyze the GCC in an online stochastic environment where the requests are drawn repeatedly and independently from a known probability distribution over the different demand classes. Also, the decision on how to address each request must be made immediately upon its arrival. We show that any partial flexibility structure that satisfies GCC is effective under a intuitive policy called the Load Deviation Fulfillment Policy (LDFP). In particular, we provide an upper bound on the expected total number of lost sales that is irrespective of how large the market size is.