A Course Material on Probability and Queueing Theory By Mrs. V.Sumathi ASSISTANT PROFESSOR DEPARTMENT OF SCIENCE AND HUMANITIES SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM – 638 056 QUALITY CERTIFICATE This is to certify that the e-course material Subject Code : MA6453 Subject : Probability and Queueing Theory Class : II Year CSE being prepared by me and it meets the knowledge requirement of the university curriculum. Signature of the Author Name: V. Sumathi Designation: AP This is to certify that the course material being prepared by Mss. V.Sumathi is of adequate quality. He has referred more than five books amont them minimum one is from aborad author. Signature of HD Name: Mrs. P. Murugapriya SEAL S.NO CONTENTS Page.NO UNIT I RANDOM VARIABLES 1 Introduction 1 2 Discrete Random Variables 2 3 Continuous Random Variables 5 4 Moments 14 5 Moment generating functions 14 6 Binomial distribution 18 7 Poisson distribution 21 8 Geometric distribution 25 9 Uniform distribution 27 10 Exponential distribution 29 11 Gamma distribution 31 UNIT II TWO –DIMENSIONAL RANDOM VARIABLES 11 Introduction 36 12 Joint distribution 36 13 Marginal and Conditional Distribution 38 14 Covariance 42 15 Correlation Coefficient 42 16 Problems 43 17 Linear Regression 40 18 Transformation of random variables 45 19 Problems 46 UNIT III RANDOM PROCESSES 20 Introduction 49 21 Classification 50 22 stationary processes 51 23 Markov processes 55 24 Poisson processes 61 25 Discrete parameter Markov chains 62 26 Chapman Kolmogorov Equation 63 27 Limiting distribution 64 UNIT IV QUEUEING MODELS 28 Introduction 67 29 Markovian Queues 68 30 Birth & Death processes 70 31 Single and Multiple server queueing models 74 32 Little’s formula 69 33 Queues with finite waiting rooms 81 34 Queues with impatient customers 82 35 Balking and reneging 83 UNIT V ADVANCED QUEUEING MODELS 36 Introduction 89 37 (M/G/I):(GD) queue 89 38 Pollaczek-Khintchine formula 90 39 M/D/1 and M/EK/1 as special cases 91 40 Problems 92 41 Series queues 93 42 Open Jackson Networks 94 43 Problems 95 MA 6453 PROBABILITY AND QUEUEING THEORY L T P C 3 1 0 4 OBJECTIVE: To provide the required mathematical support in real life problems and develop probabilistic models which can be used in several areas of science and engineering. UNIT I RANDOM VARIABLES 9+3 Discrete and continuous random variables – Moments – Moment generating functions – Binomia l ,Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions. UNIT II TWO - DIMENSIONAL RANDOM VARIABLES 9+3 Joint distributions – Marginal and conditional distributions – Covariance – Correlation and Linear regression – Transformation of random variables. UNIT III RANDOM PROCESSES 9+3 Classification – Stationary process – Markov process - Poisson process – Discrete parameter Markov chain – Chapman Kolmogorov equations – Limiting distributions . UNIT IV QUEUEING MODELS 9+3 Markovian queues – Birth and Death processes – Single and multiple server queueing models – Little’s formula - Queues with finite waiting rooms – Queues with impatient customers: Balking and reneging. UNIT V ADVANCED QUEUEING MODELS 9+3 Finite source models - M/G/1 queue – Pollaczek Khinchin formula - M/D/1 and M/EK/1 as special cases – Series queues – Open Jackson networks. TOTAL (L:45+T:15): 60 PERIODS OUTCOMES: 1. The students will have a fundamental knowledge of the probability concepts. 2. Acquire skills in analyzing queueing models. 3. It also helps to understand and characterize phenomenon which evolve with respect to time in a probabilistic manner. TEXT BOOKS: 1. Ibe. O.C., "Fundamentals of Applied Probability and Random Processes", Elsevier, 1st Indian Reprint, 2007. 2. Gross. D. and Harris. C.M., "Fundamentals of Queueing Theory", Wiley Student edition, 2004. REFERENCES: 1. Robertazzi, "Computer Networks and Systems: Queueing Theory and performance evaluation", Springer, 3rd Edition, 2006. 2. Taha. H.A., "Operations Research", Pearson Education, Asia, 8th Edition, 2007. 3. Trivedi.K.S., "Probability and Statistics with Reliability, Queueing and Computer Science Applications", John Wiley and Sons, 2nd Edition, 2002. 4. Hwei Hsu, "Schaum’s Outline of Theory and Problems of Probability, Random Variables and Random Processes", Tata McGraw Hill Edition, New Delhi, 2004. 5. Yates. R.D. and Goodman. D. J., "Probability and Stochastic Processes", Wiley India Pvt. Ltd.,Bangalore, 2nd Edition, 2012. MA6453 PROBABILITY AND QUEUEING THEORY UNIT - I RANDOM VARIABLES Introduction Consider an experiment of throwing a coin twice. The outcomes {HH, HT, TH, TT} consider the sample space. Each of these outcome can be associated with a number by specifying a rule of association with a number by specifying a rule of association (eg. The number of heads). Such a rule of association is called a random variable. We denote a random variable by the capital letter (X, Y, etc) and any particular value of the random variable by x and y. Thus a random variable X can be considered as a function that maps all elements in the sample space S into points on the real line. The notation X(S)=x means that x is the value associated with the outcomes S by the Random variable X. 1.1 SAMPLE SPACE Consider an experiment of throwing a coin twice. The outcomes S = {HH, HT, TH, TT} constitute the sample space. 1.2 RANDOM VARIABLE In this sample space each of these outcomes can be associated with a number by specifying a rule of association. Such a rule of association is called a random variables. Eg : Number of heads We denote random variable by the letter (X, Y, etc) and any particular value of the random variable by x or y. S = {HH, HT, TH, TT} X(S) = {2, 1, 1, 0} Thus a random X can be the considered as a fun. That maps all elements in the sample space S into points on the real line. The notation X(S) = x means that x is the value associated with outcome s by the R.V.X. Example 1.1 In the experiment of throwing a coin twice the sample space S is S = {HH,HT,TH,TT}. Let X be a random variable chosen such that X(S) = x (the number of heads). SCE 1 Department of CSE MA6453 PROBABILITY AND QUEUEING THEORY Note Any random variable whose only possible values are 0 and 1 is called a Bernoulli random variable. 1.1.1 DISCRETE RANDOM VARIABLE Definition : A discrete random variable is a R.V.X whose possible values consitute finite set of values or countably infinite set of values. Example 1.1 All the R.V.’s from Example : 1 are discrete R.V’s Remark The meaning of P(X ≤a). P(X ≤a) is simply the probability of the set of outcomes ‘S’ in the sample space for which X(s) ≤ a. OrP(X≤a) = P{S : X(S) ≤ a} In the above example : 1 we should write 3 P(X ≤ 1) = P(HH, HT, TH) = 4 3 Here P(X≤1) = means the probability of the R.V.X (the number of heads) is less than 4 3 or equal to 1 is . 4 Distribution function of the random variable X or cumulative distribution of the random variable X Def : The distribution function of a random variable X defined in (-∞, ∞) is given by F(x) = P(X ≤ x) = P{s : X(s) ≤ x} Note Let the random variable X takes values x1, x2, ….., xn with probabilities P1, P2, ….., Pn and let x1< x2< ….. <xn SCE 2 Department of CSE MA6453 PROBABILITY AND QUEUEING THEORY Then we have F(x) = P(X < x1) = 0, -∞ < x < x, F(x) = P(X < x1) = 0, P(X < x1) + P(X = x1) = 0 + p1 = p1 F(x) = P(X < x2) = 0, P(X < x1) + P(X = x1) + P(X = x2) = p1 + p2 F(x) = P(X < xn) = P(X < x1) + P(X = x1) + ….. + P(X = xn) = p1 + p2+ ………. + pn = 1 1.1.2 PROPERTIES OF DISTRIBUTION FUNCTIONS Property : 1 P(a < X ≤ b) = F(b) – F(a), where F(x) = P(X ≤ x) Property : 2 P(a ≤ X ≤ b) = P(X = a) + F(b) – F(a) Property : 3 P(a < X < b) = P(a < X ≤ b) - P(X = b) = F(b) – F(a) – P(X = b) by prob (1) 1.1.3 PROBABILITY MASS FUNCTION (OR) PROBABILITY FUNCTION Let X be a one dimenstional discrete R.V. which takes the values x1, x2, …… To each possible outcome ‘xi’ we can associate a number pi. i.e., P(X = xi) = P(xi) = pi called the probability of xi. The number pi = P(xi) satisfies the following conditions. ∞ (i) p( xii) ≥∀ 0, (ii) ∑ p(xi )= 1 i1= The function p(x) satisfying the above two conditions is called the probability mass function (or) probability distribution of the R.V.X. The probability distribution {xi, pi} can be displayed in the form of table as shown below. X = xi x1 x2 ……. xi P(X = p1 p2 ……. pi xi) = pi Notation Let ‘S’ be a sample space. The set of all outcomes ‘S’ in S such that X(S) = x is denoted by writing X = x. SCE 3 Department of CSE MA6453 PROBABILITY AND QUEUEING THEORY P(X = x) = P{S : X(s) = x} |||ly P(x ≤ a) = P{S : X() ∈ (-∞, a)} and P(a < x ≤ b) = P{s : X(s) ∈ (a, b)} P(X = a or X = b) = P{(X = a) ∪ (X = b)} P(X = a and X = b) = P{(X = a) ∩ (X = b)} and so on. Theorem 1 If X1 and X2 are random variable and K is a constant then KX1, X1 + X2, X1X2, K1X1 + K2X2, X1-X2 are also random variables. Theorem 2 If ‘X’ is a random variable and f(•) is a continuous function, then f(X) is a random variable.