advances in combinatorial methods and applications to
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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
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