10.2 Markov Chains and Queueing Networks

10.2 Markov Chains and Queueing Networks - CS 797...

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CS 797 Independent Study Report on Markov Chains and Queueing Networks By: Vibhu Saujanya Sharma (Roll No. Y211165, CSE, IIT Kanpur) Under the supervision of: Prof. S. K. Iyer (Dept. Of Mathematics, IIT Kanpur)
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Abstract Quantitative analysis of computer systems has become very important as they are being used to handle various mission critical and performance intensive applications. In this independent study I studied methods to model computer systems for analyzing them for reliability, availability, and performance metrics. The techniques that were covered include modeling using Discrete Time Markov Chains (DTMC), Continuous Time Markov Chains (CTMC) and Queueing Networks. Each of the topics, which were studied, are very extensive and have many books devoted solely to each one of them. Thus, in this report I present a summary of the important definitions, assumptions, and results pertaining to each one of these as well as provide some handle on using them practically. 1. Introduction Modeling of computer systems has become very necessary with quantitative analysis gaining importance these days. Almost in every related field the emphasis is now on concrete numbers rather than intuitive arguments in support of the system being built or sold. Ascertaining a number of desirable properties of computer systems such as reliability, availability, and performance metrics such as throughput, response times, etc. mandate the need for accurately modeling these systems. Queueing networks and Markov chains offer fairly simple yet effective and practical modeling solutions to such problems. The aim of this independent study was to study these topics in detail and also learn about their practical applications. The text Probability and Statistics with Reliability, Queuing, and Computer Science Applications by Dr. Kishor S. Trivedi [1] was used as the guiding reference for this study along with some reading from [2] and [3]. Discrete Time Markov Chains, Continuous Time Markov Chains, Stochastic Petri Nets, and Network of Queues, were studied as a part of this course along with some other paradigms and basics required for stochastic modeling. I also solved many problems given in the text as a part of this course. This report contains a gist of some of the important and basic topics, which were studied. I have tried to present the important results and techniques along with some intuition behind them. I have also included some examples showing the use of the techniques. The reader is suggested to refer to [1], for detailed derivations as well as other examples. 2 Discrete Time Markov Chains 2.1 Introduction A Markov Chain is a stochastic process whose state space I is discrete (finite or countably infinite) and the probability distributions for its future development depend only on the present state and not on the path (consisting of past states), that was followed to reach this state. If we further assume that the parameter space T is also discrete, then we have a
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10.2 Markov Chains and Queueing Networks - CS 797...

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