Exponential Tail Bounds on Queues: A Confluence of Non-Asymptotic Heavy Traffic and Large Deviations

In general, obtaining the exact steady-state distribution of queue lengths is not feasible. Therefore, we establish bounds for the tail probabilities of queue lengths. Specifically, we examine queueing systems under Heavy-Traffic (HT) conditions and provide exponentially decaying bounds for the probability $\mathbb P(\epsilon q > x)$, where $\epsilon$ is the HT parameter denoting how far the load is from the maximum allowed load. Our bounds are not limited to asymptotic cases and are applicable even for finite values of $\epsilon$, and they get sharper as $\epsilon \to 0$. Consequently, we derive non-asymptotic convergence rates for the tail probabilities. Unlike other approaches such as moment bounds based on drift arguments and bounds on Wasserstein distance using Stein's method, our method yields sharper tail bounds. Furthermore, our results offer bounds on the exponential rate of decay of the tail, given by $-\frac{1}{x} \log \mathbb P(\epsilon q > x)$ for any finite value of $x$. These can be interpreted as non-asymptotic versions of Large Deviation (LD) results. We demonstrate our approach by presenting tail bounds for: (i) a continuous time Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete time single-server queue and (iii) an $M/M/n$ queue. We not only bridge the gap between classical-HT and LD regimes but also explore the large system HT regimes for JSQ and $M/M/n$ systems. In these regimes, both the system size and the system load increase simultaneously. Our results also close a gap in the existing literature on the limiting distribution of JSQ in the super-NDS (a.k.a. super slowdown) regime. This contribution is of an independent interest. Here, a key ingredient is a more refined characterization of state space collapse for JSQ system, achieved by using an exponential Lyapunov function designed to approximate the $\ell_{\infty}$ norm.Comment: 37 pages, 1 figur

ASYMPTOTIC ANALYSIS OF SINGLE-HOP STOCHASTIC PROCESSING NETWORKS USING THE DRIFT METHOD

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.Ph.D

Studies of Complex Biological Systems with Applications to Molecular Medicine: The Need to Integrate Transcriptomic and Proteomic Approaches

Omics approaches to the study of complex biological systems with potential applications to molecular medicine are attracting great interest in clinical as well as in basic biological research. Genomics, transcriptomics and proteomics are characterized by the lack of an a priori definition of scope, and this gives sufficient leeway for investigators (a) to discern all at once a globally altered pattern of gene/protein expression and (b) to examine the complex interactions that regulate entire biological processes. Two popular platforms in “omics” are DNA microarrays, which measure messenger RNA transcript levels, and proteomic analyses, which identify and quantify proteins. Because of their intrinsic strengths and weaknesses, no single approach can fully unravel the complexities of fundamental biological events. However, an appropriate combination of different tools could lead to integrative analyses that would furnish new insights not accessible through one-dimensional datasets. In this review, we will outline some of the challenges associated with integrative analyses relating to the changes in metabolic pathways that occur in complex pathophysiological conditions (viz. ageing and altered thyroid state) in relevant metabolically active tissues. In addition, we discuss several new applications of proteomic analysis to the investigation of mitochondrial activity

Cancer predisposing syndromes in childhood and adolescence pose several challenges necessitating interdisciplinary care in dedicated programs

Introduction: Genetic disposition is a major etiologic factor in childhood cancer. More than 100 cancer predisposing syndromes (CPS) are known. Surveillance protocols seek to mitigate morbidity and mortality. To implement recommendations in patient care and to ascertain that the constant gain of knowledge forces its way into practice specific pediatric CPS programs were established. Patients and methods: We retrospectively analyzed data on children, adolescents, and young adults referred to our pediatric CPS program between October 1, 2021, and March 31, 2023. Follow-up ended on December 31, 2023. Results: We identified 67 patients (30 male, 36 female, 1 non-binary, median age 9.5 years). Thirty-five patients were referred for CPS surveillance, 32 for features suspicious of a CPS including café-au-lait macules (n = 10), overgrowth (n = 9), other specific symptoms (n = 4), cancer suspicious of a CPS (n = 6), and rare neoplasms (n = 3). CPS was confirmed by clinical criteria in 6 patients and genetic testing in 7 (of 13). In addition, 6 clinically unaffected at-risk relatives were identified carrying a cancer predisposing pathogenic variant. A total of 48 patients were eventually diagnosed with CPS, surveillance recommendations were on record for 45. Of those, 8 patients did not keep their appointments for various reasons. Surveillance revealed neoplasms (n = 2) and metachronous tumors (n = 4) by clinical (n = 2), radiological examination (n = 2), and endoscopy (n = 2). Psychosocial counselling was utilized by 16 (of 45; 35.6%) families. Conclusions: The diverse pediatric CPSs pose several challenges necessitating interdisciplinary care in specified CPS programs. To ultimately improve outcome including psychosocial well-being joint clinical and research efforts are necessary