FUNDAMENTALS OF STOCHASTIC NETWORKS

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5 FUNDAMENTALS OF STOCHASTIC NETWORKS OLIVER C. IBE University of Massachusetts Lowell Massachusetts A JOHN WILEY & SONS INC. PUBLICATION

6 Copyright 11 by John Wiley & Sons Inc. All rights reserve. Publishe by John Wiley & Sons Inc. Hoboken New Jersey. Publishe simultaneously in Canaa. No part of this publication may be reprouce store in a retrieval system or transmitte in any form or by any means electronic mechanical photocopying recoring scanning or otherwise ecept as permitte uner Section 17 or 18 of the 1976 Unite States Copyright Act without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center Inc. Rosewoo Drive Danvers MA 193 (978) fa (978) or on the web at Requests to the Publisher for permission shoul be aresse to the Permissions Department John Wiley & Sons Inc. 111 River Street Hoboken NJ 73 (1) fa (1) or online at Limit of Liability/Disclaimer of Warranty: While the publisher an author have use their best efforts in preparing this book they make no representations or warranties with respect to the accuracy or completeness of the contents of this book an specifically isclaim any implie warranties of merchantability or fitness for a particular purpose. No warranty may be create or etene by sales representatives or written sales materials. The avice an strategies containe herein may not be suitable for your situation. You shoul consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial amages incluing but not limite to special inciental consequential or other amages. For general information on our other proucts an services or for technical support please contact our Customer Care Department within the Unite States at (8) outsie the Unite States at (317) or fa (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley proucts visit our web site at Library of Congress Cataloging-in-Publication Data: Ibe Oliver C. (Oliver Chukwui) 1947 Funamentals of stochastic networks / Oliver C. Ibe. p. cm. Inclues bibliographical references an ine. ISBN (cloth) 1. Queuing theory.. Stochastic analysis. I. Title. QA74.8.I ' c Printe in the Unite States of America obook ISBN: epdf ISBN: epub ISBN: iv

7 CONTENTS Preface Acknowlegments ii i 1 Basic Concepts in Probability Introuction 1 1. Ranom Variables Distribution Functions 1.. Discrete Ranom Variables Continuous Ranom Variables Epectations Moments of Ranom Variables an the Variance Transform Methos The s-transform Moment-Generating Property of the s-transform The z-transform Moment-Generating Property of the z-transform Covariance an Correlation Coefficient Sums of Inepenent Ranom Variables Ranom Sum of Ranom Variables Some Probability Distributions The Bernoulli Distribution The Binomial Distribution The Geometric Distribution The Pascal Distribution The Poisson Distribution The Eponential Distribution The Erlang Distribution The Uniform Distribution The Hypereponential Distribution The Coian Distribution 17 v

8 vi CONTENTS The General Phase-Type Distribution Normal Distribution Limit Theorems Markov Inequality Chebyshev Inequality Law of Large Numbers The Central Limit Theorem 3 Overview of Stochastic Processes 6.1 Introuction 6. Classification of Stochastic Processes 7.3 Stationary Ranom Processes Strict-Sense Stationary Processes 7.3. WSS Processes 8.4 Counting Processes 8.5 Inepenent Increment Processes 9.6 Stationary Increment Process 9.7 Poisson Processes 3.8 Renewal Processes The Renewal Equation The Elementary Renewal Theorem Ranom Incience an Resiual Time 35.9 Markov Processes Discrete-Time Markov Chains State Transition Probability Matri The n-step State Transition Probability State Transition Diagrams Classification of States Limiting State Probabilities Doubly Stochastic Matri Continuous-Time Markov Chains Birth an Death Processes Local Balance Equations 54.1 Gaussian Processes 56 3 Elementary Queueing Theory Introuction Description of a Queueing System The Kenall Notation The Little s Formula The M/M/1 Queueing System Stochastic Balance Total Time an Waiting Time Distributions of the M/M/1 Queueing System 7

9 CONTENTS vii 3.6 Eamples of Other M/M Queueing Systems The M/M/c Queue: The c-server System The M/M/1/K Queue: The Single-Server Finite-Capacity System The M/M/c/c Queue: The c-server Loss System The M/M/1//K Queue: The Single Server Finite Customer Population System M/G/1 Queue Waiting Time Distribution of the M/G/1 Queue The M/E k /1 Queue The M/D/1 Queue The M/M/1 Queue The M/H k /1 Queue 86 4 Avance Queueing Theory Introuction M/G/1 Queue with Priority Nonpreemptive Priority Preemptive-Resume Priority Preemptive-Repeat Priority G/M/1 Queue The E k /M/1 Queue The D/M/1 Queue The H /M/1 Queue The G/G/1 Queue Linley s Integral Equation Laplace Transform of F W (w) Bouns of Mean Waiting Time Special Queueing Systems M/M/1 Vacation Queueing Systems with Eceptional First Vacation M/M/1 Threshol Queueing Systems Queueing Networks Introuction Burke s Output Theorem an Tanem Queues Jackson or Open Queueing Networks Close Queueing Networks BCMP Networks Routing Behavior Service Time Distributions Service Disciplines The BCMP Theorem 134

10 viii CONTENTS 5.6 Algorithms for Prouct-Form Queueing Networks The Convolution Algorithm The MVA Networks with Positive an Negative Customers G-Networks with Signals Etensions of the G-Network Approimations of Queueing Systems an Networks Introuction Flui Approimation Flui Approimation of a G/G/1 Queue Flui Approimation of a Queueing Network Diffusion Approimations Diffusion Approimation of a G/G/1 Queue Brownian Approimation for a G/G/1 Queue Brownian Motion with Drift Reflecte Brownian Motion Scale Brownian Motion Functional Central Limit Theorem Brownian Motion Approimation of the G/G/1 Queue Diffusion Approimation of Open Queueing Networks Diffusion Approimation of Close Queueing Networks Elements of Graph Theory Introuction Basic Concepts Subgraphs an Cliques Ajacency Matri Directe Graphs Weighte Graphs Connecte Graphs Cut Sets Briges an Cut Vertices Euler Graphs Hamiltonian Graphs Trees an Forests Minimum Weight Spanning Trees Bipartite Graphs an Matchings The Hungarian Algorithm Inepenent Set Domination an Covering Complement of a Graph Isomorphic Graphs 188

11 CONTENTS i 7.13 Planar Graphs Euler s Formula for Planar Graphs Graph Coloring Ege Coloring The Four-Color Problem Ranom Graphs Bernoulli Ranom Graphs Phase Transition Geometric Ranom Graphs Markov Ranom Graph Matri Algebra of Graphs Ajacency Matri Connection Matri Path Matri Laplacian Matri Spectral Properties of Graphs Spectral Raius Spectral Gap 7.18 Graph Entropy Directe Acyclic Graphs 1 7. Moral Graphs 7.1 Triangulate Graphs 7. Chain Graphs Factor Graphs 4 8 Bayesian Networks Introuction 9 8. Bayesian Networks Classification of BNs General Conitional Inepenence an -Separation Probabilistic Inference in BNs The Sum-Prouct Algorithm The Junction Tree Algorithm Belief Propagation in a Junction Tree Learning BNs Parameter Learning Maimum Likelihoo Estimation Maimum A Posteriori Estimation Structure Learning Dynamic Bayesian Networks 31 9 Boolean Networks Introuction Introuction to GRNs 36

12 CONTENTS 9.3 Boolean Network Basics Ranom Boolean Networks State Transition Diagram Behavior of Boolean Networks Petri Net Analysis of Boolean Networks Introuction to PNs Behavioral Properties of PNs PN Moel of Boolean Networks Probabilistic Boolean Networks Dynamics of a PBN Avantages an Disavantages of Boolean Networks 5 1 Ranom Networks Introuction Characterization of Comple Networks Degree Distribution Geoesic Distances Centrality Measures Clustering Network Entropy Percolation an the Emergence of Giant Component Moels of Comple Networks The Small-Worl Network Scale-Free Networks Ranom Networks Degree Distribution Emergence of Giant Component Connecteness an Diameter Clustering Coefficient Scale-Free Properties Ranom Regular Networks Consensus over Ranom Networks Consensus over Fie Networks Time to Convergence in a Fie Network Consensus over Ranom Networks Summary 74 References 76 Ine 8

13 PREFACE This book brings into one volume two network moels that can be broaly classifie as queueing network moels an graphical network moels. Queueing networks are systems where customers move among service stations where they receive service. Usually the service times an the orer in which customers visit the service stations are ranom. The orer in which service is receive at the service stations is governe by a probabilistic routing scheule. Queueing networks are popularly use in traffic moeling in computer an telecommunications networks transportation systems an manufacturing networks. Graphical moels are systems that use graphs to moel ifferent types of problems. They inclue Bayesian networks which are also calle irecte graphical moels Boolean networks an ranom networks. Graphical moels are use in statistics ata mining an social networks. The nee for a book of this nature arises from the fact that we live in an era of interisciplinary stuies an research activities when both networks are becoming important in areas that they were not originally use. Thus any person involve in such interisciplinary stuies or research activities nees to have a goo unerstaning of both types of networks. This book is intene to meet this nee. The book is organize into three parts. The first part Chapters 1 an eals with the basic concepts of probability (Chapter 1) an stochastic processes (Chapter ). The secon part Chapters 3 6 eals with queueing systems. Specifically Chapter 3 eals with basic queueing theory particularly a class of queueing systems that we refer to as Markovian queueing systems. Chapter 4 eals with avance queueing systems particularly the non-markovian queueing systems. Chapter 5 eals with queueing networks an Chapter 6 eals with approimations of queueing networks. The thir part Chapters 7 1 eals i

14 ii Preface Traffic Engineering Transportation Networks Manufacturing Networks Epert Systems Statistics Social Sciences Figure 1 Preceence relations of chapters. with graphical moels. Chapter 7 eals with an introuction to graph theory Chapter 8 eals with Bayesian networks Chapter 9 eals with Boolean networks an Chapter 1 eals with ranom networks. The book is self-containe an is written with a view to circumventing the proof theorem format that is traitionally use in stochastic systems moeling books. It is intene to be an introuctory grauate tet on stochastic networks an presents the basic results without much emphasis on proving theorems. Thus it is esigne for science an engineering applications. Stuents who have an interest in traffic engineering transportation an manufacturing networks will nee to cover parts 1 an as well as Chapter 1 in part 3 while stuents with an interest in epert systems statistics an social sciences will nee to cover parts 1 an 3. The preceence relations among the chapters are shown in Figure 1. ACKNOWLEDGMENTS I woul like to epress my sincere gratitue to my wife Christie for bearing with my writing yet another book. She has been my greatest fan when it comes to my writing books. I woul also like to acknowlege the encouraging wors from our chilren Chiinma Ogechi Amanze an Ugonna. I woul like to epress my sincere gratitue to my eitor Susanne Steitz-Filler for her encouragement an for checking on me regularly to make sure that we met the ealines. Finally I woul like to thank the anonymous reviewers for their useful comments an suggestions that helpe to improve the quality of the book. Oliver C. Ibe Lowell Massachusetts January 11

15 1 BASIC CONCEPTS IN PROBABILITY 1.1 INTRODUCTION The concepts of eperiments an events are very important in the stuy of probability. In probability an eperiment is any process of trial an observation. An eperiment whose outcome is uncertain before it is performe is calle a ranom eperiment. When we perform a ranom eperiment the collection of possible elementary outcomes is calle the sample space of the eperiment which is usually enote by Ω. We efine these outcomes as elementary outcomes because eactly one of the outcomes occurs when the eperiment is performe. The elementary outcomes of an eperiment are calle the sample points of the sample space an are enote by w i i = 1... If there are n possible outcomes of an eperiment then the sample space is Ω = {w 1 w... w n }. An event is the occurrence of either a prescribe outcome or any one of a number of possible outcomes of an eperiment. Thus an event is a subset of the sample space. 1. RANDOM VARIABLES Consier a ranom eperiment with sample space Ω. Let w be a sample point in Ω. We are intereste in assigning a real number to each w Ω. A ranom variable (w) is a single-value real function that assigns a real number Funamentals of Stochastic Networks First Eition. Oliver C. Ibe. 11 John Wiley & Sons Inc. Publishe 11 by John Wiley & Sons Inc. 1

16 Basic Concepts in PrOBABILITy calle the value of (w) to each sample point w Ω. That is it is a mapping of the sample space onto the real line. Generally a ranom variable is represente by a single letter instea of the function (w). Therefore in the remainer of the book we use to enote a ranom variable. The sample space Ω is calle the omain of the ranom variable. Also the collection of all numbers that are values of is calle the range of the ranom variable. Let be a ranom variable an a fie real value. Let the event A efine the subset of Ω that consists of all real sample points to which the ranom variable assigns the number. That is A = { w ( w) = } = [ = ]. Since A is an event it will have a probability which we efine as follows: p = P[ A ]. We can efine other types of events in terms of a ranom variable. For fie numbers a an b we can efine the following: [ ] = { w ( w) } [ > ] = { w ( w) > } [ a < < b] = { w a < ( w) < b}. These events have probabilities that are enote by P[ ] is the probability that takes a value less than or equal to. P[ > ] is the probability that takes a value greater than ; this is equal to 1 P[ ]. P[a < < b] is the probability that takes a value that strictly lies between a an b Distribution Functions Let be a ranom variable an be a number. As state earlier we can efine the event [ ] = { (w) }. The istribution function (or the cumulative istribution function [CDF]) of is efine by: [ ] < <. F ( ) = P That is F () enotes the probability that the ranom variable takes on a value that is less than or equal to. Some properties of F () inclue: 1. F () is a nonecreasing function which means that if 1 < then F ( 1 ) F ( ). Thus F () can increase or stay level but it cannot go own.

17 Ranom Variables 3. F () 1 3. F () = 1 4. F ( ) = 5. P[a < b] = F (b) F (a) 6. P[ > a] = 1 P[ a] = 1 F (a) 1.. Discrete Ranom Variables A iscrete ranom variable is a ranom variable that can take on at most a countable number of possible values. For a iscrete ranom variable the probability mass function (PMF) p () is efine as follows: p ( ) = P[ = ]. The PMF is nonzero for at most a countable or countably infinite number of values of. In particular if we assume that can only assume one of the values 1... n then: p ( i ) i = 1 n p ( ) = otherwise. The CDF of can be epresse in terms of p () as follows: F ( ) = p ( k). k The CDF of a iscrete ranom variable is a step function. That is if takes on values where 1 < < 3 < then the value of F () is constant in the interval between i 1 an i an then takes a jump of size p ( i ) at i i = Thus in this case F () represents the sum of all the probability masses we have encountere as we move from to Continuous Ranom Variables Discrete ranom variables have a set of possible values that are either finite or countably infinite. However there eists another group of ranom variables that can assume an uncountable set of possible values. Such ranom variables are calle continuous ranom variables. Thus we efine a ranom variable to be a continuous ranom variable if there eists a nonnegative function f () efine for all real ( ) having the property that for any set A of real numbers P[ A] = f ( ). The function f () is calle the probability ensity function (PDF) of the ranom variable an is efine by: A

18 4 Basic Concepts in PrOBABILITy The properties of f () are as follows: f F ( ) ( ) =. 1. f (). Since must assume some value f ( ) = 1 3. P[ a b] = b a f ( ) which means that P[ = a] = a a f ( ) =. Thus the probability that a continuous ranom variable will assume any fie value is zero. a 4. P[ < a] = P[ a] = F ( a) = f ( ) 1..4 Epectations If is a ranom variable then the epectation (or epecte value or mean) of enote by E[] is efine by: [ ] = E i p ( ) i i f ( ) iscrete continuous Thus the epecte value of is a weighte average of the possible values that can take where each value is weighte by the probability that takes that value. The epecte value of is sometimes enote by Moments of Ranom Variables an the Variance n n The nth moment of the ranom variable enote by E[ ] = is efine by: n n E[ ] = = i p ( ) i n i n f ( ) iscrete continuous for n = The first moment E[] is the epecte value of. We can also efine the central moments (or moments about the mean) of a ranom variable. These are the moments of the ifference between a ranom variable an its epecte value. The nth central moment is efine by ( ) n E = ( ) n = ( ) ( ) n i p i iscrete i n ( ) f ( ) continuous

19 Transform Methos 5 The central moment for the case of n = is very important an carries a special name the variance which is usually enote by σ. Thus σ ( ) = E 1.3 TRANSFORM METHODS ( i ) ( i ) i = ( ) = ( ) f ( ) p iscrete continuous Different types of transforms are use in science an engineering. In this book we consier two types of transforms: the z-transform of PMFs an the s-transform of PDFs of nonnegative ranom variables. These transforms are particularly use when ranom variables take only nonnegative values which is usually the case in many applications iscusse in this book The s-transform Let f () be the PDF of the continuous ranom variable that takes only nonnegative values; that is f () = for <. The s-transform of f () enote by M (s) is efine by: s s M ( s) = E[ e ] = e f ( ). One important property of an s-transform is that when it is evaluate at the point s = its value is equal to 1. That is s= M ( s) = f ( ) = 1. For eample the value of K for which the function A( s) = K ( s + 5 ) is a vali s-transform of a PDF is obtaine by setting A() = 1 which gives: K 5 = 1 K = Moment-Generating Property of the s-transform One of the primary reasons for stuying the transform methos is to use them to erive the moments of the ifferent probability istributions. By efinition: s M ( s) = e f ( ). Taking ifferent erivatives of M (s) an evaluating them at s = we obtain the following results:

20 6 Basic Concepts in PrOBABILITy In general s s M s s e f s s e s ( ) = f ( ) = ( ) s = e f ( ) M ( s) = f ( ) s= ( ) = ( 1) s M s s s= = E[ ] e s M ( s ) = f ( ) = E[ ] The z-transform s s n s M s n E n n ( ) s= = ( 1 ) [ ]. f ( ) = e f ( ) Let p () be the PMF of the iscrete ranom variable. The z-transform of p () enote by G (z) is efine by: G z E z z p. ( ) = [ ] = ( ) Thus the PMF p () is require to take on only nonnegative integers as we state earlier. The sum is guarantee to converge an therefore the z-transform eists when evaluate on or within the unit circle (where z 1). Note that: G ( 1) = p ( ) = 1. This means that a vali z-transform of a PMF reuces to unity when evaluate at z = 1. However this is a necessary but not sufficient conition for a function to the z-transform of a PMF. By efinition G ( z) = z p ( ) 3 = p ( ) + zp ( 1) + z p ( ) + z p ( 3 ) +. This means that P[ = k] = p (k) is the coefficient of z k in the series epansion. Thus given the z-transform of a PMF we can uniquely recover the PMF.

21 Transform Methos 7 The implication of this statement is that not every function of z that has a value of 1 when evaluate at z = 1 is a vali z-transform of a PMF. For eample consier the function A(z) = z 1. Although A(1) = 1 the function contains invali coefficients in the sense that these coefficients either have negative values or positive values that are greater than one. Thus for a function of z to be a vali z-transform of a PMF it must have a value of 1 when evaluate at z = 1 an the coefficients of z must be nonnegative numbers that cannot be greater than 1. The iniviual terms of the PMF can also be etermine as follows: p 1 z G z ( ) = ( ) = 1.! This feature of the z-transform is the reason it is sometimes calle the probability generating function. z= Moment-Generating Property of the z-transform As state earlier one of the major motivations for stuying transform methos is their usefulness in computing the moments of the ifferent ranom variables. Unfortunately the moment-generating capability of the z-transform is not as computationally efficient as that of the s-transform. The moment-generating capability of the z-transform lies in the results obtaine from evaluating the erivatives of the transform at z = 1. For a iscrete ranom variable with PMF p () we have that: G ( z) = z p ( ) z G z z p z z z ( ) = ( ) = p ( 1 1 ) = z p ( ) = z p ( ) ( ) = = ( ) z G z p z 1 = p ( ) = E [ ]. Similarly 1 z G z z p 1 z z z 1 p z ( ) = ( ) = ( ) = ( 1) p ( ) ( ) z= = ( ) ( ) = z G z p p p ( ) ( ) = ( ) p ( ) = E[ ] E[ ] [ ] = ( z ) E z G z G z z= 1 + ( ) z = 1.

22 8 Basic Concepts in PrOBABILITy Thus the variance is obtaine as follows: ( ) σ E E = [ ] [ ] { } z G z z G z = ( ) + ( ) z G ( z ). z= COVARIANCE AND CORRELATION COEFFICIENT Consier two ranom variables an Y with epecte values E[] = μ an E[Y] = μ Y respectively an variances σ an σ Y respectively. The covariance of an Y which is enote by Cov( Y) or σ Y is efine by: [ ] Cov( Y ) = σ Y = E ( µ )( Y µ Y ) = E[ Y µ Y µ Y + µ µ Y ] = E[ Y ] µ µ Y µ µ + µ µ = E[ Y ] µ µ. Y Y Y If an Y are inepenent then E[Y] = μ μ Y an Cov( Y) =. However the converse is not true; that is if the covariance of an Y is zero it oes not mean that an Y are inepenent ranom variables. If the covariance of two ranom variables is zero we efine the two ranom variables to be uncorrelate. We efine the correlation coefficient of an Y enote by ρ( Y) or ρ Y as follows: ρ Y = Cov( Y ) σ Y Var ( ) Var ( Y ) = σ σ Y. The correlation coefficient has the property that: 1 ρ Y SUMS OF INDEPENDENT RANDOM VARIABLES Consier two inepenent continuous ranom variables an Y. We are intereste in computing the CDF an PDF of their sum g( Y) = U = + Y. The ranom variable S can be use to moel the reliability of systems with stan-by connections. In such systems the component A whose time-to-failure is represente by the ranom variable is the primary component an the component B whose time-to-failure is represente by the ranom variable Y is the backup component that is brought into operation when the primary component fails. Thus S represents the time until the system fails which is the sum of the lifetimes of both components.

23 Ranom Sum of Ranom Variables 9 Their CDF can be obtaine as follows: FS ( s) = P[ S s] = P[ + Y s] = fy ( y) y where f Y ( y) is the joint PDF of an Y an D is the set D = {( y) + y s}. Thus F ( s) = f ( y) y = f ( ) f ( y) y S s y Y s y { } ( ) Y y y = f ( ) f = F ( s y) f ( y) y Y. The PDF of S is obtaine by ifferentiating the CDF as follows: f D s y s s F s ( ) = ( ) = F s y f y y s ( ) ( ) = s F ( s y ) f Y ( y ) y = f ( s y ) f Y ( y ) y S S Y where we have assume that we can interchange ifferentiation an integration. The epression on the right-han sie is a well-known result in signal analysis calle the convolution integral. Thus we fin that the PDF of the sum S of two inepenent ranom variables an Y is the convolution of the PDFs of the two ranom variables; that is fs ( s) = f ( s) fy ( s). In general if S is the sum on n mutually inepenent ranom variables 1... n whose PDFs are f i ( ) i = 1... n then we have that: S = n f ( s) = f ( s) f ( s) f ( s). S 1 n Thus the s-transform of the PDF of S is given by: M ( s) = M ( s). S n i= RANDOM SUM OF RANDOM VARIABLES Let be a continuous ranom variable with PDF f () whose s-transform is M (s). We know that if Y is the sum of n inepenent an ientically i Y

24 1 Basic Concepts in PrOBABILITy istribute ranom variables with the PDF f () then from the results in the previous section the s-transform of the PDF of Y is given by: M ( s) = M ( s) Y n [ ]. This result assumes that n is a fie number. However there are certain situations when the number of ranom variables in a sum is itself a ranom variable. For this case let N enote a iscrete ranom variable with PMF p N (n) whose z-transform is G N (z). Our goal is to fin the s-transform of the PDF of Y when the number of ranom variables is itself a ranom variable N. Thus we consier the sum: Y = N where N has a known PMF which in turn has a known z-transform. Now let N = n. Then with N fie at n we have that: Y N = n = n n MY N ( s n) = [ M ( s) ] n MY ( s) = pn ( n) MY N ( s n) = pn ( n) [ M ( s) ] = G ( M ( s) ) n n N. That is the s-transform of the PDF of a ranom sum of inepenent an ientically istribute ranom variables is the z-transform of the PMF of the number of variables evaluate at the s-transform of the PDF of the constituent ranom variables. Now let u = M (s). Then { }{ } s M s s G M s GN ( u) Y ( ) = N ( ( )) = u s M s GN u Y ( ) s = { ( ) }{ u } =. u s = When s = u s= = M () = 1. Thus we obtain: Also s u s s M s GN ( u) u GN ( u) M ( s) Y ( ) s= = = u s s= u u= s E[ Y ] = E[ N] ( E[ ]) = E[ N] E[ ] E Y E N E. { }{ } [ ] = [ ] [ ] s M s GN ( u) Y ( ) = s u { }{ u } s u GN ( u) GN u = { } { } s u 1 s= u GN u = ( ) { s } s { u } + GN ( u) u + ( ) u u s { } u s

25 Some PrOBABILITy DistrIBUTIOns 11 s M s E Y u Y s= s ( ) ( ) = [ ] = { } = [ ] = E N The variance of Y is given by: GN u u { } GN u + ( ) u { E } { E[ N ] E[ N] } + E[ N] E[ ] σ Y E Y ( E Y ) = [ ] [ ] u s [ ]{ E[ ]} + E[ N] E[ ] E[ N] { E[ ]}. = [ ]{ [ ]} + [ ] [ ] [ ] [ ] E N E E N E E N { E } ( E[ N] E[ ]) E N E { E } E E N ( E N ) = [ ]{ [ ] [ ] } + ( [ ]) { [ ] [ ] } = E[ N] σ + ( E[ ]) σ N. If is also a iscrete ranom variable then we obtain: G ( z) = G ( G ( z) ) Y N s= ; u= 1 an the results for E[Y] an σ Y still hol. 1.7 SOME PROBABILITY DISTRIBUTIONS Ranom variables with special probability istributions are encountere in ifferent fiels of science an engineering. In this section we escribe some of these istributions incluing their epecte values variances an s-transforms (or z-transforms as the case may be) The Bernoulli Distribution A Bernoulli trial is an eperiment that results in two outcomes: success an failure. One eample of a Bernoulli trial is the coin-tossing eperiment which results in heas or tails. In a Bernoulli trial we efine the probability of success an probability of failure as follows: [ ] = [ ] = P success p p 1 P failure 1 p Let us associate the events of the Bernoulli trial with a ranom variable such that when the outcome of the trial is a success we efine = 1 an when the outcome is a failure we efine =. The ranom variable is calle a Bernoulli ranom variable an its PMF is given by: 1 p = P ( ) = p = 1

26 1 Basic Concepts in PrOBABILITy An alternative way to efine the PMF of is as follows: 1 p ( ) = p ( 1 p) = 1. The CDF is given by: < F ( ) = 1 p < The epecte value of is given by: [ ] = E ( 1 p) + 1 ( p) = p. Similarly the secon moment of is given by: [ ] = Thus the variance of is given by: E ( 1 p) + 1 ( p) = p. σ E { E } = p p = p( 1 p). = [ ] [ ] The z-transform of the PMF is given by: 1 G ( z) = z p ( ) = z p ( ) = z ( 1 p) + z p = 1 p + zp The Binomial Distribution Suppose we conuct n inepenent Bernoulli trials an we represent the number of successes in those n trials by the ranom variable (n). Then (n) is efine as a binomial ranom variable with parameters (n p). The PMF of a ranom variable (n) with parameters (n p) is given by: p ( n) n p p n ( ) = ( 1 ) = 1 n. n The binomial coefficient represents the number of ways of arranging successes an n failures. The CDF mean an variance of (n) an the z-transform of its PMF are given by:

27 Some PrOBABILITy DistrIBUTIOns 13 F ( ) = P ( n) ( n) [ ] = E ( n) E ( n) [ ] = np n( n 1) p + np n k p k ( 1 p ) k= [ ] = σ n = [ ( )] ( ) G ( n) n k { [ ]} = ( 1 ) E n E n np p ( ) ( z) = ( zp + 1 p) n The Geometric Distribution The geometric ranom variable is use to escribe the number of inepenent Bernoulli trials until the first success occurs. Let be a ranom variable that enotes the number of Bernoulli trials until the first success. If the first success occurs on the th trial then we know that the first 1 trials resulte in failures. Thus the PMF of a geometric ranom variable is given by: 1 p ( ) = p( 1 p) = 1 3. The CDF mean an variance of an the z-transform of its PMF are given by: [ ] = F ( ) = P 1 ( 1 p) E[ ] = 1 p p E[ ] = p 1 σ = E[ ] { E[ ]} = p p zp G ( z) =. 1 z ( 1 p ) The Pascal Distribution The Pascal ranom variable is an etension of the geometric ranom variable. A Pascal ranom variable of orer k escribes the number of trials until the kth success which is why it is sometimes calle the kth-orer interarrival time for a Bernoulli process. The Pascal istribution is also calle the negative binomial istribution. Let k be a kth-orer Pascal ranom variable. Then its PMF is given by: p k n 1 k n k ( n) = p p k n k k k ( 1 ) = 1 ; =

28 14 Basic Concepts in PrOBABILITy The CDF mean an variance of k an the z-transform of its PMF are given by: [ ] = F P k k ( ) = n 1 p k 1 n= k k ( 1 p) E( k ) = k p k + k ( 1 p) E[ k ] = p k ( 1 p) σ = E[ k ] { E[ k ]} = k p G k zp ( z) = 1 z( 1 p) The Poisson Distribution A iscrete ranom variable K is calle a Poisson ranom variable with parameter λ where λ > if its PMF is given by: p K k k λ ( k) = k e λ k = 1.! The CDF mean an variance of K an the z-transform of its PMF are given by: F ( k) = P K k K E K E K [ ] = λ [ ] = λ The Eponential Distribution [ ] = λ r λ r! e r= σk = E[ K ] { E[ K] } = λ λ( z 1) G ( z) = e. K A continuous ranom variable is efine to be an eponential ranom variable (or has an eponential istribution) if for some parameter λ > its PDF is given by: f λe ( ) λ = < The CDF mean an variance of an the s-transform of its PDF are given by: k λ n k

29 Some PrOBABILITy DistrIBUTIOns 15 [ ] = F ( ) = P 1 λ e E[ ] = 1 λ E[ ] = λ σ = E[ ] { E[ ]} = 1 λ λ M ( s) =. s + λ The Erlang Distribution The Erlang istribution is a generalization of the eponential istribution. While the eponential ranom variable escribes the time between ajacent events the Erlang ranom variable escribes the time interval between any event an the kth following event. A ranom variable is referre to as a kthorer Erlang (or Erlang-k) ranom variable with parameter λ if its PDF is given by: f k k 1 λ λ e k = 1 3 ; k ( ) = ( k 1)! < The CDF mean an variance of k an the s-transform of its PDF are given by F P k k ( ) = [ k ] = E [ k ] = E M σ k [ ] = 1 k 1 j= j ( λ) e j! k λ k ( k + 1) λ k E { E } = λ = [ ] [ ] k k k k λ ( s) =. s + λ The Uniform Distribution A continuous ranom variable is sai to have a uniform istribution over the interval [a b] if its PDF is given by: 1 a b f ( ) = b a otherwise The CDF mean an variance of an the s-transform of its PDF are given by: λ

30 16 Basic Concepts in PrOBABILITy < a a F ( ) = P[ ] = a < b b a 1 b E[ ] = b + a b + ab + a E[ ] = 3 ( b a) σ = E[ ] { E[ ]} = 1 as bs e e M ( s) = s( b a) The Hypereponential Distribution The Erlang istribution belongs to a class of istributions that are sai to have a phase-type istribution. This arises from the fact that the Erlang istribution is the sum of inepenent eponential istributions. Thus an Erlang ranom variable can be thought of as the time to go through a sequence of phases or stages each of which requires an eponentially istribute length of time. For eample since an Erlang-k ranom variable k is the sum of k eponentially istribute ranom variables with mean 1/µ an we can visualize k as the time it takes to complete a task that must go through k stages where the time the task spens at each stage is. Thus we can represent the time to complete that task by the series of stages shown in Figure 1.1. The hypereponential istribution is another type of the phase-type istribution. The ranom variable H k is use to moel a process where an item can choose one of k branches. The probability that it chooses branch i is α i i = 1... k. The time it takes the item to traverse branch i is eponentially istribute with a mean of 1/µ i. Thus the PDF of H k is given by: µ i f ( ) = α µ e Hk i i i= 1 k i= 1 i k α = 1. Task Arrives µ µ µ µ Stage 1 Stage Stage i Stage k Task Complete Figure 1.1 Graphical representation of the Erlang-k ranom variable.