Numerical methods for queues with shared service

A queueing system is a mathematical abstraction of a situation where elements, called customers, arrive in a system and wait until they receive some kind of service. Queueing systems are omnipresent in real life. Prime examples include people waiting at a counter to be served, airplanes waiting to take off, traffic jams during rush hour etc. Queueing theory is the mathematical study of queueing phenomena. As often neither the arrival instants of the customers nor their service times are known in advance, queueing theory most often assumes that these processes are random variables. The queueing process itself is then a stochastic process and most often also a Markov process, provided a proper description of the state of the queueing process is introduced. This dissertation investigates numerical methods for a particular type of Markovian queueing systems, namely queueing systems with shared service. These queueing systems differ from traditional queueing systems in that there is simultaneous service of the head-of-line customers of all queues and in that there is no service if there are no customers in one of the queues. The absence of service whenever one of the queues is empty yields particular dynamics which are not found in traditional queueing systems. These queueing systems with shared service are not only beautiful mathematical objects in their own right, but are also motivated by an extensive range of applications. The original motivation for studying queueing systems with shared service came from a particular process in inventory management called kitting. A kitting process collects the necessary parts for an end product in a box prior to sending it to the assembly area. The parts and their inventories being the customers and queues, we get ``shared service'' as kitting cannot proceed if some parts are absent. Still in the area of inventory management, the decoupling inventory of a hybrid make-to-stock/make-to-order system exhibits shared service. The production process prior to the decoupling inventory is make-to-stock and driven by demand forecasts. In contrast, the production process after the decoupling inventory is make-to-order and driven by actual demand as items from the decoupling inventory are customised according to customer specifications. At the decoupling point, the decoupling inventory is complemented with a queue of outstanding orders. As customisation only starts when the decoupling inventory is nonempty and there is at least one order, there is again shared service. Moving to applications in telecommunications, shared service applies to energy harvesting sensor nodes. Such a sensor node scavenges energy from its environment to meet its energy expenditure or to prolong its lifetime. A rechargeable battery operates very much like a queue, customers being discretised as chunks of energy. As a sensor node requires both sensed data and energy for transmission, shared service can again be identified. In the Markovian framework, "solving" a queueing system corresponds to finding the steady-state solution of the Markov process that describes the queueing system at hand. Indeed, most performance measures of interest of the queueing system can be expressed in terms of the steady-state solution of the underlying Markov process. For a finite ergodic Markov process, the steady-state solution is the unique solution of $N-1$ balance equations complemented with the normalisation condition, $N$ being the size of the state space. For the queueing systems with shared service, the size of the state space of the Markov processes grows exponentially with the number of queues involved. Hence, even if only a moderate number of queues are considered, the size of the state space is huge. This is the state-space explosion problem. As direct solution methods for such Markov processes are computationally infeasible, this dissertation aims at exploiting structural properties of the Markov processes, as to speed up computation of the steady-state solution. The first property that can be exploited is sparsity of the generator matrix of the Markov process. Indeed, the number of events that can occur in any state --- or equivalently, the number of transitions to other states --- is far smaller than the size of the state space. This means that the generator matrix of the Markov process is mainly filled with zeroes. Iterative methods for sparse linear systems --- in particular the Krylov subspace solver GMRES --- were found to be computationally efficient for studying kitting processes only if the number of queues is limited. For more queues (or a larger state space), the methods cannot calculate the steady-state performance measures sufficiently fast. The applications related to the decoupling inventory and the energy harvesting sensor node involve only two queues. In this case, the generator matrix exhibits a homogene block-tridiagonal structure. Such Markov processes can be solved efficiently by means of matrix-geometric methods, both in the case that the process has finite size and --- even more efficiently --- in the case that it has an infinite size and a finite block size. Neither of the former exact solution methods allows for investigating systems with many queues. Therefore we developed an approximate numerical solution method, based on Maclaurin series expansions. Rather than focussing on structural properties of the Markov process for any parameter setting, the series expansion technique exploits structural properties of the Markov process when some parameter is sent to zero. For the queues with shared exponential service and the service rate sent to zero, the resulting process has a single absorbing state and the states can be ordered such that the generator matrix is upper-diagonal. In this case, the solution at zero is trivial and the calculation of the higher order terms in the series expansion around zero has a computational complexity proportional to the size of the state space. This is a case of regular perturbation of the parameter and contrasts to singular perturbation which is applied when the service times of the kitting process are phase-type distributed. For singular perturbation, the Markov process has no unique steady-state solution when the parameter is sent to zero. However, similar techniques still apply, albeit at a higher computational cost. Finally we note that the numerical series expansion technique is not limited to evaluating queues with shared service. Resembling shared queueing systems in that a Markov process with multidimensional state space is considered, it is shown that the regular series expansion technique can be applied on an epidemic model for opinion propagation in a social network. Interestingly, we find that the series expansion technique complements the usual fluid approach of the epidemic literature

The power-series algorithm:A numerical approach to Markov processes

Abstract: The development of computer and communication networks and flexible manufacturing systems has led to new and interesting multidimensional queueing models. The Power-Series Algorithm is a numerical method to analyze and optimize the performance of such models. In this thesis, the applicability of the algorithm is extended. This is illustrated by introducing and analyzing a wide class of queueing networks with very general dependencies between the different queues. The theoretical basis of the algorithm is strengthened by proving analyticity of the steady-state distribution in light traffic and finding remedies for previous imperfections of the method. Applying similar ideas to the transient distribution renders new analyticity results. Various aspects of Markov processes, analytic functions and extrapolation methods are reviewed, necessary for a thorough understanding and efficient implementation of the Power-Series Algorithm.

Introduction to Queueing Theory and Stochastic Teletraffic Models

The aim of this textbook is to provide students with basic knowledge of stochastic models that may apply to telecommunications research areas, such as traffic modelling, resource provisioning and traffic management. These study areas are often collectively called teletraffic. This book assumes prior knowledge of a programming language, mathematics, probability and stochastic processes normally taught in an electrical engineering course. For students who have some but not sufficiently strong background in probability and stochastic processes, we provide, in the first few chapters, background on the relevant concepts in these areas.Comment: 298 page

A Robust Queueing Network Analyzer Based on Indices of Dispersion

In post-industrial economies, modern service systems are dramatically changing the daily lives of many people. Such systems are often complicated by uncertainty: service providers usually cannot predict when a customer will arrive and how long the service will be. Fortunately, useful guidance can often be provided by exploiting stochastic models such as queueing networks. In iterating the design of service systems, decision makers usually favor analytical analysis of the models over simulation methods, due to the prohibitive computation time required to obtain optimal solutions for service operation problems involving multidimensional stochastic networks. However, queueing networks that can be solved analytically require strong assumptions that are rarely satisfied, whereas realistic models that exhibit complicated dependence structure are prohibitively hard to analyze exactly. In this thesis, we continue the effort to develop useful analytical performance approximations for the single-class open queueing network with Markovian routing, unlimited waiting space and the first-come first-served service discipline. We focus on open queueing networks where the external arrival processes are not Poisson and the service times are not exponential. We develop a new non-parametric robust queueing algorithm for the performance approximation in single-server queues. With robust optimization techniques, the underlying stochastic processes are replaced by samples from suitably defined uncertainty sets and the worst-case scenario is analyzed. We show that this worst-case characterization of the performance measure is asymptotically exact for approximating the mean steady-state workload in G/G/1 models in both the light-traffic and heavy-traffic limits, under mild regularity conditions. In our non-parametric Robust Queueing formulation, we focus on the customer flows, defined as the continuous-time processes counting customers in or out of the network, or flowing from one queue to another. Each flow is partially characterized by a continuous function that measures the change of stochastic variability over time. This function is called the index of dispersion for counts. The Robust Queueing algorithm converts the index of dispersion for counts into approximations of the performance measures. We show the advantage of using index of dispersion for counts in queueing approximation by a renewal process characterization theorem and the ordering of the mean steady-state workload in GI/M/1 models. To develop generalized algorithm for open queueing networks, we first establish the heavy-traffic limit theorem for the stationary departure flows from a GI/GI/1 model. We show that the index of dispersion for counts function of the stationary departure flow can be approximately characterized as the convex combination of the arrival index of dispersion for counts and service index of dispersion for counts with a time-dependent weight function, revealing the non-trivial impact of the traffic intensity on the departure processes. This heavy-traffic limit theorem is further generalized into a joint heavy-traffic limit for the stationary customer flows in generalized Jackson networks, where the external arrival are characterized by independent renewal processes and the service times are independent and identically distributed random variables, independent of the external arrival processes. We show how these limiting theorems can be exploited to establish a set of linear equations, whose solution serves as approximations of the index of dispersion for counts of the flows in an open queueing network. We prove that this set of equations is asymptotically exact in approximating the index of dispersion for counts of the stationary flows. With the index of dispersion for counts available, the network is decomposed into single-server queues and the Robust Queueing algorithm can be applied to obtain performance approximation. This algorithm is referred to as the Robust Queueing Network Analyzer. We perform extensive simulation study to validate the effectiveness of our algorithm. We show that our algorithm can be applied not only to models with non-exponential distirbutions but also to models with more complex arrival processes than renewal processes, including those with Markovian arrival processes

Towards large deviations in stochastic systems with memory

supported by QMUL Research-IT and funded by EPSRC Grant No. EP/K000128/1.The theory of large deviations can help to shed light on systems in non-equilibrium statistical mechanics and, more generically, on non-reversible stochastic processes. For this purpose, we target trajectories in space time rather than static con figurations and study time-extensive observables. This suggests that the details of the evolution law such as the presence of time correlations take on a major role. In this thesis, we investigate selected models with stochastic dynamics that incorporate memory by means of diff erent mechanisms, devise a numerical approach for such models, and quantify to what extent the memory aff ects the large deviation functionals. The results are relevant for real-world situations, where simpli ed memoryless (Markovian) models may not always be appropriate. After an original introduction to the mathematics of stochastic processes, we explore, analytically and numerically, an open-boundary zero-range process which incorporates memory by means of hidden variables that a ect particle congestion. We derive the exact solution for the steady state of the one-site system, as well as a mean- eld approximation for larger one-dimensional lattices. Then, we focus on the large deviation properties of the particle current in such a system. This reveals that the time correlations can be apparently absorbed in a memoryless description for the steady state and the small uctuation regime. However, they can dramatically alter the probability of rare currents. Di erent regimes are separated by dynamical phase transitions. Subsequently, we address systems in which the memory cannot be encoded in hidden variables or the waiting-time distributions depend on the whole trajectory. Here, the di culty in obtaining exact analytical results is exacerbated. To tackle these systems, we have proposed a version of the so-called \cloning" algorithm for the evaluation of large deviations that can be applied consistently for both Markovian and non-Markovian dynamics. The e cacy of this approach is con rmed by numerical results for some of the rare non-Markovian models whose large deviation functions can be obtained exactly. We nally adapt this machinery to a technological problem, speci cally the performance evaluation of communication systems, where temporal correlations and large deviations are important.EPSRC Grant No. EP/K000128/1

Traffic and resource management in content-centric networks (design and evaluation)

Dans les dernières années, l utilisation d Internet a sensiblement changé en passant d un modèle de communication centré sur les machines á un centré sur les contenus. La plus part de services utilisés par les clients d Internet aujourd hui sont déjà centré sur les contenus même et pas sur leurs emplacement. Dans ce contexte, beaucoup de projets de recherche proposent un changement de l architecture de l Internet, en mettent des contenu identifié par leur nom au centre du réseau. Ce group de proposition est identifiés sous le nom de Information Centric Networking (ICN). Cette thèse se focalise sur la proposition Content-Centric Network (CCN). Dans une premier temps, nous analysons les performance du modèle de communication CCN en se concentrent sur le partage de la bande passante et de la mémoire et en proposant des formules pour la caractérisation du temps de transfert. Deuxièmement, nous proposons un protocole de contrôle de congestion et des mécanismes de forwarding pour CCN. En particulier on présent un premier mécanisme de contrôle de congestion, Interest Control Protocol (ICP), qui utilise une fenêtre contrôlé avec le mécanisme Additive Increase Multiplicative Decrease au récepteur. En complément avec ça, nous présentons un mécanisme distribué (hop-by-hop) pour obtenir une détection/réaction à la congestion plus rapide. Nous proposons aussi une modification d'ICP en implémentant le mécanisme Remote Adaptive Active Queue Management pour exploiter efficacement le multi-chemin. En fin, nous présentons un mécanisme de forwarding distribué qui base ses décisions sur des mesure de qualité d interface par chaque préfixe disponible dans les tableaux de routage.The advent of the World Wide Web has radically changed Internet usage from host-to-host to service access and data retrieval. The majority of services used by Internet s clients are content-centric (e.g. web). However, the original Internet revolves around host-to-host communication for which it was conceived. Even if Internet has been able to address the challenges offered by new applications, there is an evident mismatch between the architecture and its current usage. Many projects in national research agencies propose to redesign the Internet architecture around named data. Such research efforts are identified under the name of Information Centric Networking. This thesis focuses on the Content-Centric Networking (CCN) proposition. We first analyze the CCN communication model with particular focus on the bandwidth and storage sharing performance, We compute closed formulas for data delivery time, that we use in the second part of the thesis as guideline for network protocol design. Second, we propose some CCN congestion control and forwarding mechanisms. We present a first window based receiver driven flow control protocol, Interest Control Protocol (ICP). We also introduce a hop-by-hop congestion control mechanism to obtain early congestion detection and reaction. We then extend the original ICP congestion control protocol implementing a Remote Adaptive Active Queue Management mechanism in order to efficiently exploit heterogeneous (joint/disjoint) network paths. Finally, we introduce a distributed forwarding mechanism that bases its decisions on per prefix and per interface quality measurement without impacting the system scalability.PARIS-Télécom ParisTech (751132302) / SudocSudocFranceF

Advanced data management system analysis techniques study

The state of the art of system analysis is reviewed, emphasizing data management. Analytic, hardware, and software techniques are described

Распознавание точек привязки трехмерных объектов по стереоизображению в системах технического зрения

В процессе реконструкции формы трехмерного тела по стереоизображению возникает задача распознавания точек привязки на поверхности объекта. Эти точки используются в алгоритме триангуляции, когда формируется описание модели объекта. Один из способов уточнения координат точек привязки – анализ карты диспаритета для выделенного сегмента стереоизображения. Предлагаемый метод распознавания основан на использовании марковской сети для описания карты диспаритета, преобразовании графа этой сети путём анализа распределения яркостей пикселей изображения. Возможность выбора приоритетного направления смещения при обходе графа карты диспаритета обеспечивает эффективность работы алгоритма, основанного на описанном метод

Stationary waiting time distribution in G/M/n/r with random renovation policy

Consideration is given to G/M/n/r FCFS queue with random renovation policy. This policy implies that a customer upon service completion with probability q(l) removes l customers from the queue and then leaves the system. The choice of customers to be removed from the queue is done in a random fashion. Customer from the queue are served in FCFS order. For this system in a series of papers there were obtained stationary characteristics related to the number of customers in the system. Here we obtain several expressions for the stationary time-related characteristics and specifically dwell on the stationary waiting time distribution, which is obtained in terms of the transforms