ASYMPTOTIC ANALYSIS OF SINGLE-HOP STOCHASTIC PROCESSING NETWORKS USING THE DRIFT METHOD
Hurtado Lange, Daniela
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Today’s era of cloud computing and big data is powered by massive data centers. The focus of my dissertation is on resource allocation problems that arise in the operation of these large-scale data centers. Analyzing these systems exactly is usually intractable, and a usual approach is to study them in various asymptotic regimes with heavy traffic being a popular one. We use the drift method, which is a two-step procedure to obtain bounds that are asymptotically tight. In the first step, one shows state-space collapse, which, intuitively, means that one detects the bottleneck(s) of the system. In the second step, one sets to zero the drift of a carefully chosen test function. Then, using state-space collapse, one can obtain the desired bounds. This dissertation focuses on exploiting the properties of the drift method and providing conditions under which one can completely determine the asymptotic distribution of the queue lengths. In chapter 1 we present the motivation, research background, and main contributions. In chapter 2 we revisit some well-known definitions and results that will be repeatedly used in the following chapters. In chapter 3, chapter 4, and chapter 5 we focus on load-balancing systems, also known as supermarket checkout systems. In the load-balancing system, there are a certain number of servers, and jobs arrive in a single stream. Once they come, they join the queue associated with one of the servers, and they wait in line until the corresponding server processes them. In chapter 3 we introduce the moment generating function (MGF) method. The MGF, also known as two-sided Laplace form, is an invertible transformation of the random variable’s distribution and, hence, it provides the same information as the cumulative distribution function or the density (when it exists). The MGF method is a two-step procedure to compute the MGF of the delay in stochastic processing networks (SPNs) that satisfy the complete resource pooling (CRP) condition. Intuitively, CRP means that the SPN has a single bottleneck in heavy traffic. A popular routing algorithm is power-of-d choices, under which one selects d servers at random and routes the new arrivals to the shortest queue among those d. The power-of-d choices algorithm has been widely studied in load-balancing systems with homogeneous servers. However, it is not well understood when the servers are different. In chapter 4 we study this routing policy under heterogeneous servers. Specifically, we provide necessary and sufficient conditions on the service rates so that the load-balancing system achieves throughput and heavy-traffic optimality. We use the MGF method to show heavy-traffic optimality. In chapter 5 we study the load-balancing system in the many-server heavy-traffic regime, which means that we analyze the limit as the number of servers and the load increase together. Specifically, we are interested in studying how fast the number of servers can grow with respect to the load if we want to observe the same probabilistic behavior of the delay as a system with a fixed number of servers in heavy traffic. We show two approaches to obtain the results: the MGF method and Stein’s method. In chapter 6 we apply the MGF method to a generalized switch, which is one of the most general single-hop SPNs with control on the service process. Many systems, such as ad hoc wireless networks, input-queued switches, and parallel-server systems, can be modeled as special cases of the generalized switch. Most of the literature in SPNs (including the previous chapters of this thesis) focuses on systems that satisfy the CRP condition in heavy traffic, i.e., systems that behave as single-server queues in the limit. In chapter 7 we study systems that do not satisfy this condition and, hence, may have multiple bottlenecks. We specify conditions under which the drift method is sufficient to obtain the distribution function of the delay, and when it can only be used to obtain information about its mean value. Our results are valid for both, the CRP and non-CRP cases and they are immediately applicable to a variety of systems. Additionally, we provide a mathematical proof that shows a limitation of the drift method.