Advances in Combinatorial Methods and Applications to Probability and Statistics

N. Balakrishnan Editor

1997 Birkhäuser Boston • Basel • Berlin

Contents

Preface xvii

Sri Gopal Mohanty—Life and Works xix

List of Contributors xxvii

List of Tables xxxi

List of Figures xxxiii

PART I — L A T T I C E PATHS AND COMBINATORIAL M E T H O D S

1 Lattice Paths and Faber Polynomials Ira M. Gessel and Sangwook Ree 3

1.1 Introduction, 3 1.2 Faber Polynomials, 6 1.3 Counting Paths, 7 1.4 A Positivity Result, 10 1.5 Examples, 11

References, 13

2 Lattice Path Enumeration and Umbral Calculus Heinrich Niederhausen 15

2.1 Introduction, 15 2.1.1 Notation, 16

2.2 Initial Value Problems, 16 2.2.1 Theroleof ex, 18 2.2.2 Piecewise affine boundaries, 18 2.2.3 Applications: Bounded paths, 19

2.3 Systems of Operator Equations, 20 2.3.1 Applications: Lattice paths with

several step directions, 21 2.4 Symmetrie Sheffer Sequences, 21

2.4.1 Applications: Weighted left turns, 22 2.4.2 Paths inside a band, 23

2.5 Geometrie Sheffer Sequences, 24 2.5.1 Applications: Crossings, 25 References, 26

3 T h e E n u m e r a t i o n of Lat t ice P a t h s W i t h R e s p e c t t o The ir N u m b e r of Turns C. Krattenthaler

3.1 Introduction, 29 3.2 Notation, 31 3.3 Motivating Examples, 31 3.4 Turn Enumeration of (Single) Lattice Paths, 36 3.5 Applications, 44 3.6 Nonintersecting Lattice Paths and Turns, 47

References, 55

4 Lat t i ce P a t h Count ing , S imple R a n d o m Walk Stat i s t i c s , and R a n d o m i z a t i o n s : A n Ana ly t i c A p p r o a c h Wolfgang Panny and Walter Katzenbeisser

4.1 Introduction, 59 4.2 Lattice Paths , 60 4.3 Simple Random Walks, 64 4.4 Randomized Random Walks, 70

References, 74

5 Combinator ia l Ident i t ies: A Genera l izat ion of Dougal l ' s Ident i ty Erik Sparre Andersen and Mogens Esrom, Larsen

5.1 Introduction, 77 5.2 The Generalized Pfaff-Saalschütz Formula, 80 5.3 A Modified Pfaff-Saalschütz Sum of Type

JJ(4,4,1)7V, 82 5.4 A Well-Balanced 77(5, 5,1)7V Identity, 83 5.5 A Generalization of Dougall's Well-Balanced

11(7, 7,1)7V Identity, 85 References, 87

6 A C o m p a r i s o n of T w o M e t h o d s for R a n d o m Label l ing of Bal ls by Vectors of Integers Doron Zeilberger

6.1 First Way, 89 6.2 Second Way, 89 6.3 Variance and Standard Deviation, 91

Contents ix

6.4 Analysis of the Second Way, 92 References, 93

P A R T I I — A P P L I C A T I O N S T O P R O B A B I L I T Y P R O B L E M S

7 O n t h e Ba l lo t T h e o r e m s Lajos Takdcs 97

7.1 Introduction, 97 7.2 The Classical Ballot Theorem, 97 7.3 The Original Proofs of Theorem 7.2.1, 100 7.4 Historical Background, 102 7.5 The General Ballot Theorem, 104 7.6 Some Combinatorial Identities, 107 7.7 Another Extension of The Classical Ballot

Theorem, 109 References, 111

8 S o m e R e s u l t s for T w o - D i m e n s i o n a l R a n d o m Walk Endre Csdki 115

8.1 Introduction, 115 8.2 Identities and Distributions, 118 8.3 Pairs of LRW Paths, 120

References, 123

9 R a n d o m Walks on SL(2, F2) and Jacobi S y m b o l s of Quadrat ic R e s i d u e s Toshihiro Watanabe 125

9.1 Introduction, 125 9.2 Preliminaries, 126 9.3 A Calculation of the Character xiaM,m,) and

Its Relation, 129 References, 133

10 R a n k Order Stat i s t ics R e l a t e d to a General ized R a n d o m Walk Jagdish Saran and Sarita Rani 135

10.1 Introduction, 135 10.2 Some Auxiliary Results, 136 10.3 The Technique, 138 10.4 Defmitions of Rank Order Statistics, 139 10.5 Distributions of JV+*(a) and Ä+*(a) , 140 10.6 Distributions of A+„(a) and Ä / + n ( a ) , 144 10.7 Distributions of JV*'n(a) and R*ß'v{a), 148

References, 151

X Contents

11 O n a Subse t S u m A l g o r i t h m a n d Its Probabi l i s t i c a n d O t h e r Appl i ca t ions V. G. Voinov and M. S. Nikulin 153

11.1 Introduction, 153 11.2 A Derivation of the Algorithm, 154 11.3 A Class of Discrete Probability Distributions, 159 11.4 A Remark on a Summation Procedure When

Constructing Partit ions, 160 References, 162

12 I a n d J P o l y n o m i a l s in a P o t p o u r r i of Probabi l i ty P r o b l e m s Milton Sobel 165

12.1 Introduction, 165 12.2 Guide to the Problems of this Paper, 166 12.3 Triangulär Network with Common Failure

Probability q for Each Unit, 171 12.4 Duality Levels in a Square with Diagonals

Tha t Do Not Intersect: Problem 12.5, 177 References, 183

13 St ir l ing N u m b e r s and R e c o r d s N. Baiakrishnan and V. B. Nevzorov 189

13.1 Stirling Numbers, 189 13.2 Generalized Stirling Numbers, 190 13.3 Stirling Numbers and Records, 193 13.4 Generalized Stirling Numbers and Records

in the .Fa-scheme, 195 13.5 Record Values from Discrete Distributions

and Generalized Stirling Numbers, 197 References, 198

P A R T I I I — A P P L I C A T I O N S T O U R N M O D E L S

14 A d v a n c e s in U r n M o d e l s D u r i n g T h e P a s t T w o D e c a d e s Samuel Kotz and N. Baiakrishnan 2 0 3

14.1 Introduction, 203 14.2 Pölya-Eggenberger Urns and Their Generalizations

and Modifications, 206 14.3 Generalizations of the Classical Occupancy

Model, 216 14.4 Ehrenfest Urn Model, 219

Contents XI

14.5 Pölya Urn Model with a Continuum of Colors, 225 14.6 Stopping Problems in Urns, 226 14.7 Limit Theorems for Urns with Random

Drawings, 227 14.8 Limit Theorems for Sequential Occupancy, 228 14.9 Limit Theorems for Infinite Urn Models, 230 14.10 Urn Models with Indistinguishable Balls

(Böse-Einstein Statistics), 231 14.11 Ewens Sampling Formula and Coalescent

Urn Models, 233 14.12 Reinforcement-Depletion (Compartmental)

Urn Models, 237 14.13 Urn Models for Interpretation of Mathematical

and Probabilistic Concepts and Engineering and Statistical Applications, 243 References, 247

15 A Unified Derivation of Occupancy and Sequential Occupancy Distributions Ch. A. Charalambides 259

15.1 Introduction, 259 15.2 Occupancy Distributions, 260 15.3 Sequential Occupancy Distributions, 267

References, 273

16 Moments, Binomial Moments and Combinatorics Janos Galambos 275

16.1 Basic Relations, 275 16.2 Linear Inequalities in Sk, pr and qr, 277 16.3 A Statistical Paradox and an Urn Model

with Applications, 280 16.4 Quadratic Inequalities, 281

References, 283

P A R T I V — A P P L I C A T I O N S T O Q U E U E I N G T H E O R Y

17 Nonintersecting Paths and Applications to Queueing Theory Walter Böhm, 287

17.1 Introduction, 287 17.2 Dissimilar Bernoulli Processes, 288 17.3 The r-Node Series Jackson Network, 291 17.4 The Dummy Path Lemma for Poisson Processes, 295 17.5 A Special Variant of D/M/l Queues, 297

X l l Contents

References, 299

18 Transient Busy Period Analysis of Initially Non-Empty M / G / l Queues—Lattice Path Approach Kanwar Sen and Manju Agarwal 301

18.1 Introduction, 301 18.2 Lattice Path Approach, 304 18.3 Discretized M/C 2 / l Model, 304

18.3.1 Transition probabilities, 304 18.3.2 Counting of lattice paths, 307 18.3.3 Busy period probability, 310

18.4 Continuous M/C 2 / l Model, 312 18.5 Particular Cases, 313

References, 313

19 Single Server Queueing System with Poisson Input: A Review of Some Recent Developments J. Medhi 317

19.1 Introduction, 317 19.2 Exceptional Service for the First Unit in

Each Busy Period, 319 19.3 M/G/l With Random Setup Time S, 320 19.4 M/G/l System Under iV-Policy, 322 19.5 M/G/l Under iV-Policy and With Setup Time, 323 19.6 Queues With Vacation: M/G/l Queueing System

With Vacation, 324 19.7 M/G/l - Vm System, 325 19.8 M/G/l - Vm With Exceptional First Vacation, 326 19.9 M/G/l - Vs System, 327 19.10 M/G/l System With Vacation and Under iV-Policy

(With Threshold N), 328 19.11 Mx/G/l System With Batch Arrival, 332 19.12 Mx/G/l Under iV-Policy, 332 19.13 Mx/G/l - Vm and Mx/G/l - Vs, 334 19.14 Mx/G/l Vacation Queues Under iV-Policy, 334 19.15 Concluding Remarks, 335

References, 336

20 Recent Advances in the Analysis of Polling Systems Diwakar Gupta and Yavuz Günalay 339

20.1 Introduction, 339 20.2 Notations and Preliminaries, 342 20.3 Main Results, 346

Contents X l l l

20.4 Some Related Models, 350 20.4.1 Customer routing, 350 20.4.2 Stopping only at a preferred Station, 351 20.4.3 Gated or mixed Service policy, 351 20.4.4 State-dependent Setups, 352 20.4.5 Periodic monitoring during idle period, 354

20.5 Insights, 355 20.6 Future Directions, 357

References, 357 s

P A R T V — A P P L I C A T I O N S T O W A I T I N G T I M E P R O B L E M S

21 Waiting Times and Number of Appearances of Events in a Sequence of Discrete Random Variables Markos V. Kontras 363

21.1 Introduction, 363 21.2 Definitions and Notations, 365 21.3 General Results, 366 21.4 Waiting Times and Number of Occurrences of

Delayed Recurrent Events, 370 21.5 Distribution of the Number of Success Runs in

a Two-State Markov Chain, 373 21.5.1 Non-overlapping success runs, 374 21.5.2 Success runs of length at least k, 376 21.5.3 Overlapping success runs, 378 21.5.4 Number of non-overlapping Windows of length at

most k containing exactly 2 successes, 378 21.6 Conclusions, 380

References, 380

22 On Sooner and Later Problems Between Success and Failure Runs Sigeo Aki 385

22.1 Introduction, 385 22.2 Number of Ocurrences of the Sooner Event Until

the Later Waiting Time, 387 22.3 Joint Distribution of Numbers of Runs, 397

References, 399

XIV

23 Distributions of Numbers of Success-Runs Until the First Consecutive k Successes in Higher Order Markov Dependent Trials Katuomi Hirano, Sigeo Aki and Masayuki Uchida 401

23.1 Introduction, 401 23.2 Numbers of Success-Runs in Higher Order

Markov Chain, 403 23.3 Case l < m, 408

References, 409

24 On Multivariate Distributions of Various Orders Obtained by Waiting for the r-th Success Run of Length k in Trials With Multiple Outcomes Dem,etrios L. Antzoulakos and Andreas N. Philippou 411

24.1 Introduction, 412 24.2 Independent Trials, 413 24.3 Generalized Sequence of Order k, 420

References, 423

25 A Multivariate Negative Binomial Distribution of Order k Arising When Success Runs are Allowed to Overlap Gregory A. Tripsiannis and Andreas N. Philippou 427

25.1 Introduction, 427 25.2 Multivariate Negative Binomial Distribution of

Order k, Type III, 429 25.3 Characteristics and Distributional Properties

of MNBkJII{r; qi, . . . , qm), 431 References, 436

P A R T V I — A P P L I C A T I O N S T O D I S T R I B U T I O N T H E O R Y

26 The Joint Energy Distributions of the Bose-Einstein and of the Fermi-Dirac Particles /. Vincze and R. Törös 441

26.1 Introduction, 441 26.2 Derivation of the Joint Distribution and of the

Joint Entropy, 442 26.2.1 On the method, 442 26.2.2 Joint distribution of the number of particles in

energy intervals, 444 26.3 Determination of the Limit Distributions, 447 26.4 Discussion, 448

References, 449

27 On Modified g-Bessel Functions and Some Stat ist ical Applications A. W. Kemp 451

27.1 Introduction, 451 27.2 Notation, 453 27.3 The Distribution of the Difference of Two Euler

Random Variables, 456 27.4 The Distribution of the Difference of Two Heine

Random Variables, 458 27.5 Comments on the Distribution of the Difference of

Two Generalized Euler Random Variables, 460 References, 462

28 A g-Logarithmic Dis t r ibut ion C. David Kem,p 465

28.1 Introduction, 465 28.2 A g-Logarithmic Distribution, 467 28.3 A Group Size Model for the Distribution, 469

References, 470

29 Bernoull i Learning Models: Uppulur i Number s K. G. Janardan 471

29.1 Introduction, 471 29.2 The General Model, 472

29.2.1 Special cases of the general probabilistic model, 474 29.3 Waiting Time Learning Models, 476

29.3.1 Special cases of waiting time learning modeis, 478 References, 480

PART V I I — A P P L I C A T I O N S T O N O N P A R A M E T R I C STATISTICS

30 Linear Nonparamet r i c Tests Against Res t r ic ted Alternat ives: The Simple-Tree Order and The Simple Order S. Chakraborti and W. Schaafsm,a 483

30.1 Introduction, 484 30.2 Background, 485 30.3 Objectives, 486 30.4 Exploration and Reformulation, 487 30.5 Test for the Simple-Tree Problem, 487

30.5.1 Some particular cases, 490 30.5.2 Derivation of the MSSMP test, 493

XVI Contents

30.6 Test for the Simple Order Problem, 496 30.6.1 Derivation of the (A)MSSMP test, 497 30.6.2 Power comparisons, 499

30.7 Extending the Class of SMP Tests, 501 Appendix, 503 References, 505

31 N o n p a r a m e t r i c E s t i m a t i o n of t h e R a t i o of Variance C o m p o n e n t s M. Mahibbur Rahm.an and Z. Govindarajulu 507

31.1 Introduction, 507 31.2 Proposed Estimation Procedure, 510 31.3 Monte Carlo Comparison, 512 31.4 Ad.justment for Bias, 514

References, 515

32 Limit T h e o r e m s for M - P r o c e s s e s V i a R a n k Stat i s t i c s P r o c e s s e s M. Huskovd 521

32.1 Introduction, 521 32.2 Case 81 = • • • = 0n, 522 32.3 Change Point Alternatives, 526

References, 533

A u t h o r I n d e x 535

S u b j e c t I n d e x 545