Studies in heavy traffic and in production systems

Queueing theory as a subject of research started at the beginning of this century, when A. K. Erlang conducted his pioneering studies on individual queues, in the context of telephone switching. Since the 1950\u27s, this area of study has experienced a rapid development and has found enormous amounts of application in industry and business. Today, more general queueing systems or queueing networks, rather than individual queues, have become a fashion as people have studied various types of interrelated service systems in real life. The rapidly developing of computer and communication technology has been a major factor in solving and creating many new problems;This research addresses both of these types of problems. The first one, consisting of chapter two only, is of classic queueing theory fashion, and investigates the moment convergence of the M/G/1 model and establishes its normalized moment convergence under heavy traffic. The second one, which consists of chapters three and four, addresses lineal and confluent production systems. Steady state performance measures of such systems are discussed under various disciplines and service time distributions. Some design issues also are discussed, with emphasis on the comparison of two-level confluent systems with their lineal counterparts, a topic apparently not discussed heretofore in the literatures

Large deviations analysis for the M/H2/n+MM/H_2/n + M queue in the Halfin-Whitt regime

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems

Extended Entropy Maximisation and Queueing Systems with Heavy-Tailed Distributions

Numerous studies on Queueing systems, such as Internet traffic flows, have shown to be bursty, self-similar and/or long-range dependent, because of the heavy (long) tails for the various distributions of interest, including intermittent intervals and queue lengths. Other studies have addressed vacation in no-customers’ queueing system or when the server fails. These patterns are important for capacity planning, performance prediction, and optimization of networks and have a negative impact on their effective functioning. Heavy-tailed distributions have been commonly used by telecommunication engineers to create workloads for simulation studies, which, regrettably, may show peculiar queueing characteristics. To cost-effectively examine the impacts of different network patterns on heavy- tailed queues, new and reliable analytical approaches need to be developed. It is decided to establish a brand-new analytical framework based on optimizing entropy functionals, such as those of Shannon, Rényi, Tsallis, and others that have been suggested within statistical physics and information theory, subject to suitable linear and non-linear system constraints. In both discrete and continuous time domains, new heavy tail analytic performance distributions will be developed, with a focus on those exhibiting the power law behaviour seen in many Internet scenarios. The exposition of two major revolutionary approaches, namely the unification of information geometry and classical queueing systems and unifying information length theory with transient queueing systems. After conclusions, open problems arising from this thesis and limitations are introduced as future work

Resource Management in Computing Systems

Resource management is an essential building block of any modern computer and communication network. In this thesis, the results of our research in the following two tracks are summarized in four papers. The first track includes three papers and covers modeling, prediction and control for multi-tier computing systems. In the first paper, a NARX-based multi-step-ahead response time predictor for single server queuing systems is presented which can be applied to CPU-constrained computing systems. The second paper introduces a NARX-based multi-step-ahead query response time predictor for database servers. Both mentioned predictors can predict the dynamics of response times in the whole operation range particularly in high load scenarios without changes having to be applied to the current protocols and operating systems. In the third paper, queuing theory is used to model the dynamics of a database server. Several heuristics are presented to tune the parameters of the proposed model to the measured data from the database. Furthermore, an admission controller is presented, and its parameters are tuned to control the response time of queries which are sent to the database to stay below a predefined reference value.The second track includes one paper, covering a problem formulation and optimal solution for a content replication problem in Telecom operator's content delivery networks (Telco-CDNs). The problem is formulated in the form of an integer programming problem trying to minimize the communication delay and cost according to several constraints such as limited content replication budget, limited storage size and limited downlink bandwidth of each regional content server. The solution of this problem is a performance bound for any distributed content replication algorithm which addresses the same problem

Spontaneous Resonances and the Coherent States of the Queuing Networks

We present an example of a highly connected closed network of servers, where the time correlations do not go to zero in the infinite volume limit. This phenomenon is similar to the continuous symmetry breaking at low temperatures in statistical mechanics. The role of the inverse temperature is played by the average load.Comment: 3 figures added, small correction

Computing performability measures in Markov chains by means of matrix functions

We discuss the efficient computation of performance, reliability, and availability measures for Markov chains; these metrics, and the ones obtained by combining them, are often called performability measures. We show that this computational problem can be recasted as the evaluation of a bilinear forms induced by appropriate matrix functions, and thus solved by leveraging the fast methods available for this task. We provide a comprehensive analysis of the theory required to translate the problem from the language of Markov chains to the one of matrix functions. The advantages of this new formulation are discussed, and it is shown that this setting allows to easily study the sensitivities of the measures with respect to the model parameters. Numerical experiments confirm the effectiveness of our approach; the tests we have run show that we can outperform the solvers available in state of the art commercial packages on a representative set of large scale examples

On the Price of Anarchy for flows over time

Dynamic network flows, or network flows over time, constitute an important model for real-world situations where steady states are unusual, such as urban traffic and the Internet. These applications immediately raise the issue of analyzing dynamic network flows from a game-theoretic perspective. In this paper we study dynamic equilibria in the deterministic fluid queuing model in single-source single-sink networks, arguably the most basic model for flows over time. In the last decade we have witnessed significant developments in the theoretical understanding of the model. However, several fundamental questions remain open. One of the most prominent ones concerns the Price of Anarchy, measured as the worst case ratio between the minimum time required to route a given amount of flow from the source to the sink, and the time a dynamic equilibrium takes to perform the same task. Our main result states that if we could reduce the inflow of the network in a dynamic equilibrium, then the Price of Anarchy is exactly e/(e − 1) ≈ 1.582. This significantly extends a result by Bhaskar, Fleischer, and Anshelevich (SODA 2011). Furthermore, our methods allow to determine that the Price of Anarchy in parallel-link networks is exactly 4/3. Finally, we argue that if a certain very natural monotonicity conjecture holds, the Price of Anarchy in the general case is exactly e/(e − 1)

Applied Probability

[no abstract available

A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering

Routing mechanisms for stochastic networks are often designed to produce state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the limiting process to a lower-dimensional subset of its full state space. In a fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding mode control that forces the fluid trajectories to a lower-dimensional sliding manifold. Within deterministic dynamical systems theory, it is well known that sliding-mode controls can cause the system to chatter back and forth along the sliding manifold due to delays in activation of the control. For the prelimit stochastic system, chattering implies fluid-scaled fluctuations that are larger than typical stochastic fluctuations. In this paper we show that chattering can occur in the fluid limit of a controlled stochastic network when inappropriate control parameters are used. The model has two large service pools operating under the fixed-queue-ratio with activation and release thresholds (FQR-ART) overload control which we proposed in a recent paper. We now show that, if the control parameters are not chosen properly, then delays in activating and releasing the control can cause chattering with large oscillations in the fluid limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even when the arrival rates are smaller than the potential total service rate in the system, a phenomenon referred to as congestion collapse. We show that the fluid limit can be a bi-stable switching system possessing a unique nontrivial periodic equilibrium, in addition to a unique stationary point