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Probability and Statistics withReliability, Queuing, andComputer Science ApplicationsKishor Shridharbhai TrivediDuke UniversityDurham, North Carolina

A Wiley-Interscience PublicationJOHN WILEY & SONS, INC.

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Contents

Preface to the Second Edition ixPreface to the First Ed ition xiAcronyms xiii

1 I n t r o d u c t i o n 11.1 M otiv atio n 11.2 Pro bab ility Mod els 21.3 Sam ple Space 31.4 Ev en ts 61.5 Alg ebra of Ev ents 71.6 Gra phica l M etho ds of Rep resenting Eve nts 111.7 Pr ob ab ility Axiom s 131.8 Co mb inatorial Prob lems 191.9 Con ditional Prob ability 231.10 Ind epe nd enc e of Ev ents 251.11 Ba ye s' Ru le 381.12 Be rno ulli Tria ls 45

2 D i s c r e t e R a n d o m V a r i a b l e s 6 12.1 Intro du ction 612.2 Ra nd om Variables and The ir Event Spaces 622.3 Th e Pro bab ility Mass Function 642.4 Dis tributio n Fun ctions 662.5 Special Discrete Distrib utions 682.6 Analysis of Pro gram MA X 922.7 Th e Probab il i ty Gen erat ing Function 962.8 Discrete Ran dom Vectors 992.9 Inde pen den t Ran dom Variables 104

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3 C o n t i n u o u s R a n d o m V a r i a b le s 1 1 53.1 Intro du ction 1153.2 Th e Ex po nen tial D istribution 1193.3 T he Reliability and Failure R ate 1243.4 Some Im po rtan t Distr ibut ions 1293.5 Fun ctions of a Ra ndo m Variable 1483.6 Join tly D istribu ted Ran dom Variables 1533.7 O rder Statis tics 1573.8 D istribu tion of Sum s 1673.9 Fu nction s of No rmal Ran dom Variables 182

4 E x p e c t a t i o n 1 934.1 Introd uction 1934.2 M om ents 1974.3 Ex pecta t ion Based on M ult iple Rando m Variables 2004.4 Transform M ethod s 2084.5 M om ents and Transforms of Some D istributio ns 2174.6 Co m pu tatio n of Mean Tim e to Failure 2284.7 Inequali t ies and Limit Theo rems 237

5 C o n d i t i o n a l D i s t r i b u t i o n a n d E x p e c t a t i o n 2 4 75.1 Intro du ctio n 2475.2 M ixtu re D istribu tions 2555.3 Con ditional Ex pec tation 2625.4 Impefect Fa ult Coverage an d Reliability 2685.5 Ran dom Sums 279

6 S t o c h a s t i c P r o c e s s e s 2 8 96.1 Introd uction 2896.2 Classification of Stoc has tic Proce sses 2946.3 T he Bernoulli Proces s 3006.4 T he Poisson Process 3046.5 Renew al Processes 3146.6 Av ailability An alysis 3196.7 Ra nd om Incidence 3286.8 Renew al Model of Pro gram Behavior 332

7 D i s c r e t e - T i m e M a r k o v C h a i n s 3 3 77.1 Intro du ction 3377.2 Co m pu tatio n of n-s tep Tran sition Prob abilit ies 3417.3 Sta te Classification and Limiting Prob abilit ies 3477.4 D istribu tion of Tim es Between Sta te Chan ges 3567.5 Markov M od ulate d Bernoulli Process 3587.6 Irreducible Fin ite Ch ains with Ap eriodic Sta tes 3617.7 * T he M/Q/l Qu euing System 377

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7.8 Discrete-Time B irth -D ea th Processes 3857.9 Fin ite Markov Cha ins with Absorbing States 3928 C o n t i n u o u s - T i m e M a r k o v C h a i n s 4 0 5

8.1 Intro du ction 4058.2 Th e Bi r th- D eat h Process 4128.3 O the r Special Cases of the Bi rth -D ea th Model 4468.4 N on-B irth-D eath Processes 4548.5 Markov Cha ins with Absorbing Sta tes 4968.6 Solu tion Tech niques 5208.7 Au tom ated Generat ion 530

9 N e t w o r k s o f Q u e u e s 5 5 59.1 Introd uction 5559.2 Op en Qu euing Networks 5609.3 Closed Qu euing Networks . 5689.4 Ge neral Service Distribu tion and M ultiple Jo b Ty pes 5969.5 No n-produc t-form Netw orks 6049.6 Co mp uting Response Time Distr ibut ion 6179.7 Sum ma ry 630

10 Sta t i s t i ca l In feren ce 63710.1 Intro du ction 63710.2 Par am eter Est im ation 63910.3 Hy pothesis Testing 692

1 1 R e g r e s s i o n a n d A n a l y s i s o f V a r i a n c e 7 2 711.1 Intro duc tion 72711.2 Lea st-square s Cu rve Fit ting 73211.3 T he Coefficients of D eter m inat ion 73511.4 Confidence Inter vals in Linear Regression 73811.5 Tren d Detection and Slope Es tima tion 74211.6 Corre lation Analysis 74511.7 Simp le No nlinear Regression 74811.8 Higher-dimensional Least-squ ares Fi t 74911.9 An alysis of Varian ce 751

A B i b l i o g r a p h y 7 6 5A .l Theo ry 765A.2 Ap plications 770

B P r o p e r t i e s o f D i s t r i b u t i o n s 7 7 7C S t a t i s t i c a l T a b l e s 7 8 0D L a p l a c e T r a n s f o r m s 8 0 1

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E Pr og ra m Per fo rm ance Ana lysi s 808A uth or Inde x 811Su bjec t Ind ex 819