Multi-threshold Control of the BMAP/SM/1/K Queue with Group Services

We consider a finite capacity queue in which arrivals occur according to a batch Markovian arrival process (BMAP). The customers are served in groups of varying sizes. The services are governed by a controlled semi-Markovian process according to a multithreshold strategy. We perform the steady-state analysis of this model by computing (a) the queue length distributions at departure and arbitrary epochs, (b) the Laplace-Stieltjes transform of the sojourn time distribution of an admitted customer, and (c) some selected system performance measures. An optimization problem of interest is presented and some numerical examples are illustrated

Dynamic resource allocation for energy management in data centers

In this dissertation we study the problem of allocating computational resources and managing applications in a data center to serve incoming requests in such a way that the energy usage, reliability and quality of service considerations are balanced. The problem is motivated by the growing energy consumption by data centers in the world and their overall inefficiency. This work is focused on designing flexible and robust strategies to manage the resources in such a way that the system is able to meet the service agreements even when the load conditions change. As a first step, we study the control of a Markovian queueing system with controllable number of servers and service rates (M=Mt=kt ) to minimize effort and holding costs. We present structural properties of the optimal policy and suggest an algorithm to find good performance policies even for large cases. Then we present a reactive/proactive approach, and a tailor-made wavelet-based forecasting procedure to determine the resource allocation in a single application setting; the method is tested by simulation with real web traces. The main feature of this method is its robustness and flexibility to meet QoS goals even when the traffic behavior changes. The system was tested by simulating a system with a time service factor QoS agreement. Finally, we consider the multi-application setting and develop a novel load consolidation strategy (of combining applications that are traditionally hosted on different servers) to reduce the server-load variability and the number of booting cycles in order to obtain a better capacity allocation

Analysis of some batch arrival queueing systems with balking, reneging, random breakdowns, fluctuating modes of service and Bernoulli schedulled server vacations.

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonThe purpose of this research is to investigate and analyse some batch arrival queueing systems with Bernoulli scheduled vacation process and single server providing service. The study aims to explore and extend the work done on vacation and unreliable queues with a combination of assumptions like balking and re-service, reneging during vacations, time homogeneous random breakdowns and fluctuating modes of service. We study the steady state properties, and also transient behaviour of such queueing systems. Due to vacations the arriving units already in the system may abandon the system without receiving any service (reneging). Customers may decide not to join the queue when the server is in either working or vacation state (balking). We study this phenomenon in the framework of two models; a single server with two types of parallel services and two stages of service. The model is further extended with re-service offered instantaneously. Units which join the queue but leave without service upon the absence of the server; especially due to vacation is quite a natural phenomenon. We study this reneging behaviour in a queueing process with a single server in the context of Markovian and non-Markovian service time distribution. Arrivals are in batches while each customer can take the decision to renege independently. The non-Markovian model is further extended considering service time to follow a Gamma distribution and arrivals are due to Geometric distribution. The closed-form solutions are derived in all the cases. Among other causes of service interruptions, one prime cause is breakdowns. We consider breakdowns to occur both in idle and working state of the server. In this queueing system the transient and steady state analysis are both investigated. Applying the supplementary variable technique, we obtain the probability generating function of queue size at random epoch for the different states of the system and also derive some performance measures like probability of server‟s idle time, utilization factor, mean queue length and mean waiting time. The effect of the parameters on some of the main performance measures is illustrated by numerical examples to validate the analytical results obtained in the study. The Mathematica 10 software has been used to provide the numerical results and presentation of the effects of some performance measures through plots and graphs

A study of some M[x]/G/1 type queues with random breakdowns and bernouilli schedule server vacations based on a single vacation policy

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Queueing systems arise in modelling of many practical applications related to computer sciences, telecommunication networks, manufacturing and production, human computer interaction, and so on. The classical queueing system, even vacation queues or queues subject to breakdown, might not be sufficiently realistic. The purpose of this research is to extend the work done on vacation queues and on unreliable queues by studying queueing systems which take into consideration both phenomena. We study the behavior of a batch arrival queueing system with a single server, where the system is subject to random breakdowns which require a repair process, and on the other hand, the server is allowed to take a vacation after finishing a service. The breakdowns are assumed to occur while serving a customer, and when the system breaks down, it enters a repair process immediately while the customer whose service is interrupted comes back to the head of the queue waiting for the service to resume. Server vacations are assumed to follow a Bernoulli schedule under single vacation policy. We consider the above assumptions for different queueing models: queues with generalized service time, queues with two-stages of heterogeneous service, queues with a second optional service, and queues with two types of service. For all the models mentioned above, it is assumed that the service times, vacation times, and repair times all have general arbitrary distributions. Applying the supplementary variable technique, we obtain probability generating functions of queue size at a random epoch for different states of the system, and some performance measures such as the mean queue length, mean waiting time in the queue, proportion of server's idle time, and the utilization factor. The results obtained in this research, show the effect of vacation and breakdown parameters upon main performance measures of interest. These effects are also illustrated using some numerical examples and graphs.This work is funded by the Ministry of Education, Kingdom of Bahrain

Controlling single server queues

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From car traffic to production flows:a guided tour through solvable stochastic transport processes

The purpose of this thesis is to show on explicit examples how various theoretical concepts, ranging from statistical mechanics to stochastic control and from traffic theory to queuing systems, can be transferred to transport processes, encountered in particular in manufacturing systems, with benefic implications for their dynamical understanding, optimization and control. The thesis collects several articles where such implications are exposed [38]-[43]. We start with the observation that car traffic and production flows share several common dynamical properties (chapter 3). The main reason for the similarities are the presence of non-linear interactions in both settings. In traffic theory the interactions are between competing cars and originate from a trade off between safe and fast driving. They directly influence the speed of the cars. In production flow engineering the interactions are between cooperating work-cells forming the manufacturing system. They govern the production policy and hence the throughput of the manufacturing system. We exploit this analogy in case of a serial production line where the influence on the production rate of a work-cell is determined by the contents of its adjacent buffers (fig. 0.1) and derive a dictionary between the two fields. As a first result, this analogy allows the recognition of free-flow and jamming-flow regimes —well studied in traffic theory — in the context of production lines. Fig. 0.1. Above: Sketch of a serial production line composed of N machines Mi with production rates vi and N -1 buffers Bi with buffer content yi. Below: Sketch of a one-lane traffic system composed of N cars with velocities vi and headways xi. Dynamical similarities between cars and work-cells: the production rates and the car velocities, depend both on their environment e.g., the content of the next nearest buffers vi = vi(yi-1, yi) resp. the distances to the next nearest cars vi = vi(xi-1, xi). Applying a linear stability analysis to a given stationary flow regime, we draw a flow diagram which defines the boundary between the free and the jammed regime as a function of the control parameters. The relevant conclusions include the introduction of a dimensionless performance parameter, an enlightening connection between transient and stationary performance measures for production lines, a discussion of both the bull-whip effect and the stabilizing effect of pull production controls in serial production lines. The traffic models used in the analogy with serial production lines are socalled optimal-velocity car following models which assume that the velocity of a car is adapted to a distance dependent optimal velocity which reflects the safety requirements of two neighboring cars. This optimal velocity is chosen in an ad hoc fashion by traffic engineers and is not related to a cost functional which defines "optimality" via a minimization procedure. Here we calculate in the context of serial production lines the "optimal velocity" (i.e., the optimal production control) based on a specific cost functional. We solve in chapter 4 an optimal control problem for the production rates where the cost structure penalizes the entrance of the buffer content into a boundary state. We show that the optimal control is of four thresholds type and give the optimal position of the thresholds. The optimal control problem, explicitly discussed for a serial two-stage production line, can not be solved analytically for longer lines. This forces us to look in chapter 5 for other ways to describe relations between the throughput and the work in process of production flows. The analogous quantities in traffic theory — flow of cars and car density — are related in the so-called fundamental diagram (fig. 0.2). It encodes in a single graph the functional relation between the flow of cars and the car-density. Inspired by the micro-macro paradigm of mechanical statistics, we derive from a mesoscopic level the fundamental diagram introduced by Greenshields in 1931. The study is based on the Boltzmann equations introduced by Ruijgrok and Wu, which we derive from a space discrete interacting particle system. The fundamental kinetic features of the microscopic model are migration, reaction and collisions of particles. Performing the hydrodynamic limit of the model, we have that the macroscopic density distribution ρ is governed by the Burgers equation and that the macroscopic flow J is proportional to the logistic equation. Fig. 0.2. Generic form of the density-flow relation in one-lane car traffic. Another property of production flows shared with cars in traffic is the simple fact that the circulating items have spatial extensions. This is of foremost importance especially when multiplexing structures are present in the production line and/or the traffic network. The distribution of items flowing out of a merge structure into a single collecting flow definitely depends on the physical size of the circulating items. In chapter 6 we will study a discrete materials flow merge system connected to a downstream station (fig. 0.3). The outflow process from the merge as a function of the the items extensions is given. Fig. 0.3. Merging of N streams of items into a buffer B. A conveyor transports the items from B to M. The spatial extensions of the items are crucial for the outflow. The mentioned discrete velocities Boltzmann equations of Ruijgrok and Wu are related to random evolutions. They are particularly well adapted to model the dynamics of failure prone machines switching between their states (e.g., between "on" and "off"). For the inhomogeneous two-states case (i.e., when the switching rates depend on the environment), we show in chapter 7 that the probability density and the associated probability current are in a supersymmetric relation — a algebraic structure well known in quantum mechanics. The quest to optimize throughput in stochastic manufacturing systems and vehicles flow in traffic systems can be unified through the following question: Given the initial distribution of items (of workload or cars) how do I have to influence the noisy dynamics in order to efficiently transport the items involved (workpieces or cars) to a given final distribution? This point of view seems natural to us and is directly related to a problem addressed by E. Schrödinger in 1931. He asks for a Markov diffusion process satisfying given initial and final conditions and which minimizes some energy functional. Based on this, we propose in chapter 8 an efficiency measure relevant for a large class of diffusion-mediated transport processes

Frequency regulation in electric power systems using deferrable loads

Incluye bibliografía y anexosCon el advenimiento del paradigma de la red inteligente (Smart Grid) y las energías renovables, se hace necesario estudiar el almacenamiento de energía generada que no se consume al momento. En esta tesis, se indaga en el papel de un “load agregator” que administra un conjunto de cargas eléctricas y aprovecha la flexibilidad de las mismas para regular la frecuencia de una red. Se estudia el problema desde un punto de vista macroscópico, sin entrar en detalles de cargas individuales. Se propone un set de modelos ODE para predecir la evolución de la potencia consumida por el cluster de cargas y se diseñan controladores para estos modelos, con el fin de poder seguir las referencias de potencia externa. Finalmente, se sugieren algunos algoritmos posibles para implementar el control a cargas individuales. Las simulaciones muestran que este sistema podría proporcionar valiosos servicios a las redes eléctricas, si existiese suficiente infraestructura de comunicaciones.ANII - POS_NAC_2013_1_11675