A renewal system divides the slotted timeline into back to back time periods called renewal frames. At the beginning of each frame, it chooses a policy from a set of options for that frame. The policy determines the duration of the frame, the penalty incurred during the frame (such as energy expenditure), and a vector of performance metrics (such as instantaneous number of jobs served). The starting points of this line of research are Chapter 7 of the book [Nee10a], the seminal work [Nee13a], and Chapter 5 of the PhD thesis of Chih-ping Li [Li11]. These works consider stochastic optimization over a single renewal system. By way of contrast, this thesis considers optimization over multiple parallel renewal systems, which is computationally more challenging and yields much more applications. The goal is to minimize the time average overall penalty subject to time average overall constraints on the corresponding performance metrics. The main difficulty, which is not present in earlier works, is that these systems act asynchronously due to the fact that the renewal frames of different renewal systems are not aligned. The goal of the thesis is to resolve this difficulty head-on via a new asynchronous algorithm and a novel supermartingale stopping time analysis which shows our algorithms not only converge to the optimal solution but also enjoy fast convergence rates. Based on this general theory, we further develop novel algorithms for data center server provision problems with performance guarantees as well as new heuristics for the multi-user file downloading problems.
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