M.Tech (Computer Science and Engineering) Random Variables, Stochastic Processes and Queuing Theory Subject Code: MCNE113 Credits: 4:0:0 (4) Course Objectives: The students will 1. Learn the concept of discrete and continuous random variable 2. Understand the Probability mass and density function and probability distributions 3. Learn the theory of Random process and special classes of random variable 4. Learn the concept of continuous time Markov Chains, pure birth , pure death, birth and death process 5. Learn the concept of different queuing models such as M/M/1, M/G/1. Unit I: Discrete Random Variables Introduction, Random variables and their event spaces, Probability Mass function. Distribution functions, Special Distribution functions, Independent Variables Unit II: Continuous Random Variables Introduction, Exponential Distribution, Functions of a random variable, Jointly distributed random variables, Functions of normal random variables Unit III: Random Processes Introduction, Binomial Process, Poisson Process, Ergodic Process, Special Classes of Random Process Unit IV: Markov Process Markov chain and Transition Probabilities, Continuous Parameter Markov chain, Pure birth and pure Death Process, Birth and Death Process, Renewal Process Unit-V: Introduction to Queuing theory and applications Single server with infinite system capacity, queuing Modes (M/M/1):( /FIFO),(M/M/1):(k/FIFO),(M/M/s):( /FIFO), (M/M/s):(k/FIFO),M/G/1 Queuing system characteristics, Case Studies Text Books 1. Sheldon M Ross – Introduction to Probability Models, 10th Edition, Elsevier, 2010 2. T Veera Rajan – Probability, Statistics and Random Process, 3rd Edition, Tata Mc Graw Hill, 2008 Reference Books 1. Kishore S Trivedi – Probability & Statistics with Reliability, Queuing and Computer Science Applications, 2nd edition, John Wiley & Sons, 2008. 2. Athanasios Papoulis. S, Unni Krishna Pillai – Probability, Random Variables and Stochastic Processes, 4th edition, Tata Mc. Graw Hill, 2002. Course Outcomes: The students will be able to 1. Recognize the random variable governing the problem 2. Discuss the different features of the probability distribution which is used in statistical models. 3. Uses of knowledge random processes, evaluate various measures of the system effectiveness such as response time, reliability, etc., 4. Recognize the types of queues and discusses its queuing system characteristics. M.Tech (Information Science) ADVANCED MATHEMATICS Course Code: MSWE11 Credits: 4:1:0 Prerequisites: Nil Contact Hours : 56L + 28T Course objectives: The Students will Learn to solve algebraic and transcendental equations Numerically. Understand the concepts of Vector spaces and its applications to Difference equations and Markov chains. Learn the concepts of Diagonalization by finding The Eigen values and Eigen vectors. Learn the concepts of Random Process,Poisson Process. Understand the different Queuing Models. Course Contents: Unit I Numerical Methods: Errors, Different types of errors, Fixed Point Iteration method, Aitken’s 2 Process method, Newton-Raphson method for a system of two simultaneous equations, Horner’s method, Birje-Vieta method. Introduction to Linear Algebra,Consistency,Gauss Jordan method. Unit II Vector Spaces: Vector spaces and Subspaces, Null Spaces, Column Spaces and Linear Transformations, Linearly Independent Sets, Basis, Coordinate Systems, The Dimension of Vector Space, Rank, Change of Basis, Applications to Difference Equations. Unit III Linear Transformations: Introduction to Linear Transformations, The Matrix of a Linear Transformation, Rank-nullity theorem, Algebra of Linear Transformations. Eigenvalues and Eigenvectors: Characteristic equation, Diagonalization, eigenvectors and linear transformation. Unit IV Probability:Introduction to the theory of Probability,Random variables,Binomial and Normal distribution,Stochatic process and Markov chains. Markov Process: Markov Process, Poisson Process, Pure Birth and Pure Death Process, Birth and Death Process. Unit V Introduction to Queuing theory and Applications: Single server with infinite system capacity, queuing Modes (M/M/1): ( /FIFO),(M/M/1):(k/FIFO),(M/M/s): ( /FIFO), (M/M/s):(k/FIFO),M/G/1 Queuing system characteristics, Case Studies. Tutorials: 2 1. Examples on Aitken’s Process method and Newton-Raphson method. 2. Problems on Horner’s method and Birje-Vieta method. 3. Examples on consistency and Gauss Jordan method. 4. Problems on Vector spaces and Subspaces. 5. Problems on Basis and Change of Basis. 6. Application to Difference Equations and Markov chains. 7. Problems on Linear Transformations. 8. Examples on Eigen values and Eigen vectors. 9. Problems on Random variables and distributions. 10. Problems on Markov and Poisson Process 11. Problems on M/M/1 Queing system. 12. Problems on M/G/1 Queing system. References: 1. M.K.Jain, S.R.K. Iyengar and R.K. Jain- ‘Numerical Methods for Scientific and Engineering Computation’, 5th Edition, New Age International Publishers, 2007. 2. David C. Lay-‘ Linear Algebra and its Applications’, 3rd Edition, Pearson, 2011. 3. T. Veerarajan-‘Probability,Statistics and Random Processes’ 3rd Edition ,Tata McGraw-Hill, 2008. 4. Kishore S.Trivedi- ‘Probability and Statistics with Reliability, queuing and Computer Science Applications’,2nd Edition, John Wiley & Sons,2008. Course outcomes: Students will be able to CO1: Solve the problems of algebraic and transcendental equations using numerical methods. CO2: Apply Vector spaces to solve Difference equations and problems arising in Markov chains. CO3: Diagonalize the Matrix by finding the Eigen values and Eigen vectors. CO4: Apply the concept of Random process to discuss Poisson process,Birth and Death process, Pure Birth and Pure Death process. CO5: Study the system characteristics in analyzing Queuing Models.