Advances in Combinatorial Methods and Applications to Probability and Statistics

Transcription

1 Advances in Combinatorial Methods and Applications to Probability and Statistics N. Balakrishnan Editor 1997 Birkhäuser Boston Basel Berlin

2 Contents Preface Sri Gopal Mohanty Life and Works List of Contributors List of Tables List of Figures xvii xix xxvii xxxi xxxiii PART I LATTICE PATHS AND COMBINATORIAL METHODS 1 Lattice Paths and Faber Polynomials Ira M. Gessel and Sangwook Ree Introduction, Faber Polynomials, Counting Paths, A Positivity Result, Examples, 11 References, 13 2 Lattice Path Enumeration and Umbral Calculus Heinrich Niederhausen Introduction, Notation, Initial Value Problems, Theroleof e x, Piecewise affine boundaries, Applications: Bounded paths, Systems of Operator Equations, Applications: Lattice paths with several step directions, Symmetrie Sheffer Sequences, Applications: Weighted left turns, Paths inside a band, 23

3 2.5 Geometrie Sheffer Sequences, Applications: Crossings, 25 References, 26 3 The Enumeration of Lattice Paths With Respect to Their Number of Turns C. Krattenthaler 3.1 Introduction, Notation, Motivating Examples, Turn Enumeration of (Single) Lattice Paths, Applications, Nonintersecting Lattice Paths and Turns, 47 References, 55 4 Lattice Path Counting, Simple Random Walk Statistics, and Randomizations: An Analytic Approach Wolfgang Panny and Walter Katzenbeisser 4.1 Introduction, Lattice Paths, Simple Random Walks, Randomized Random Walks, 70 References, 74 5 Combinatorial Identities: A Generalization of Dougall's Identity Erik Sparre Andersen and Mogens Esrom, Larsen 5.1 Introduction, The Generalized Pfaff-Saalschütz Formula, A Modified Pfaff-Saalschütz Sum of Type JJ(4,4,1)7V, A Well-Balanced 77(5, 5,1)7V Identity, A Generalization of Dougall's Well-Balanced 11(7, 7,1)7V Identity, 85 References, 87 6 A Comparison of Two Methods for Random Labelling of Balls by Vectors of Integers Doron Zeilberger 6.1 First Way, Second Way, Variance and Standard Deviation, 91

4 Contents ix 6.4 Analysis of the Second Way, 92 References, 93 PART II APPLICATIONS TO PROBABILITY PROBLEMS 7 On the Ballot Theorems Lajos Takdcs Introduction, The Classical Ballot Theorem, The Original Proofs of Theorem 7.2.1, Historical Background, The General Ballot Theorem, Some Combinatorial Identities, Another Extension of The Classical Ballot Theorem, 109 References, Some Results for Two-Dimensional Random Walk Endre Csdki Introduction, Identities and Distributions, Pairs of LRW Paths, 120 References, Random Walks on SL(2, F2) and Jacobi Symbols of Quadratic Residues Toshihiro Watanabe Introduction, Preliminaries, A Calculation of the Character xi a M,m,) and Its Relation, 129 References, Rank Order Statistics Related to a Generalized Random Walk Jagdish Saran and Sarita Rani Introduction, Some Auxiliary Results, The Technique, Defmitions of Rank Order Statistics, Distributions of JV+*(a) and Ä+*(a), Distributions of A+ (a) and Ä/+ n (a), Distributions of JV*' n (a) and R* ß ' v {a), 148 References, 151

5 X Contents 11 On a Subset Sum Algorithm and Its Probabilistic and Other Applications V. G. Voinov and M. S. Nikulin Introduction, A Derivation of the Algorithm, A Class of Discrete Probability Distributions, A Remark on a Summation Procedure When Constructing Partitions, 160 References, I and J Polynomials in a Potpourri of Probability Problems Milton Sobel Introduction, Guide to the Problems of this Paper, Triangulär Network with Common Failure Probability q for Each Unit, Duality Levels in a Square with Diagonals That Do Not Intersect: Problem 12.5, 177 References, Stirling Numbers and Records N. Baiakrishnan and V. B. Nevzorov Stirling Numbers, Generalized Stirling Numbers, Stirling Numbers and Records, Generalized Stirling Numbers and Records in the.f a -scheme, Record Values from Discrete Distributions and Generalized Stirling Numbers, 197 References, 198 PART III APPLICATIONS TO URN MODELS 14 Advances in Urn Models During The Past Two Decades Samuel Kotz and N. Baiakrishnan Introduction, Pölya-Eggenberger Urns and Their Generalizations and Modifications, Generalizations of the Classical Occupancy Model, Ehrenfest Urn Model, 219

6 Contents XI 14.5 Pölya Urn Model with a Continuum of Colors, Stopping Problems in Urns, Limit Theorems for Urns with Random Drawings, Limit Theorems for Sequential Occupancy, Limit Theorems for Infinite Urn Models, Urn Models with Indistinguishable Balls (Böse-Einstein Statistics), Ewens Sampling Formula and Coalescent Urn Models, Reinforcement-Depletion (Compartmental) Urn Models, Urn Models for Interpretation of Mathematical and Probabilistic Concepts and Engineering and Statistical Applications, 243 References, A Unified Derivation of Occupancy and Sequential Occupancy Distributions Ch. A. Charalambides Introduction, Occupancy Distributions, Sequential Occupancy Distributions, 267 References, Moments, Binomial Moments and Combinatorics Janos Galambos Basic Relations, Linear Inequalities in Sk, p r and q r, A Statistical Paradox and an Urn Model with Applications, Quadratic Inequalities, 281 References, 283 PART IV APPLICATIONS TO QUEUEING THEORY 17 Nonintersecting Paths and Applications to Queueing Theory Walter Böhm, Introduction, Dissimilar Bernoulli Processes, The r-node Series Jackson Network, The Dummy Path Lemma for Poisson Processes, A Special Variant of D/M/l Queues, 297

7 Xll Contents References, Transient Busy Period Analysis of Initially Non-Empty M/G/l Queues Lattice Path Approach Kanwar Sen and Manju Agarwal Introduction, Lattice Path Approach, Discretized M/C 2 /l Model, Transition probabilities, Counting of lattice paths, Busy period probability, Continuous M/C 2 /l Model, Particular Cases, 313 References, Single Server Queueing System with Poisson Input: A Review of Some Recent Developments J. Medhi Introduction, Exceptional Service for the First Unit in Each Busy Period, M/G/l With Random Setup Time S, M/G/l System Under iv-policy, M/G/l Under iv-policy and With Setup Time, Queues With Vacation: M/G/l Queueing System With Vacation, M/G/l - V m System, M/G/l - V m With Exceptional First Vacation, M/G/l - V s System, M/G/l System With Vacation and Under iv-policy (With Threshold N), M x /G/l System With Batch Arrival, M x /G/l Under iv-policy, M x /G/l - V m and M x /G/l - V s, M x /G/l Vacation Queues Under iv-policy, Concluding Remarks, 335 References, Recent Advances in the Analysis of Polling Systems Diwakar Gupta and Yavuz Günalay Introduction, Notations and Preliminaries, Main Results, 346

8 Contents Xlll 20.4 Some Related Models, Customer routing, Stopping only at a preferred Station, Gated or mixed Service policy, State-dependent Setups, Periodic monitoring during idle period, Insights, Future Directions, 357 References, 357 s PART V APPLICATIONS TO WAITING TIME PROBLEMS 21 Waiting Times and Number of Appearances of Events in a Sequence of Discrete Random Variables Markos V. Kontras Introduction, Definitions and Notations, General Results, Waiting Times and Number of Occurrences of Delayed Recurrent Events, Distribution of the Number of Success Runs in a Two-State Markov Chain, Non-overlapping success runs, Success runs of length at least k, Overlapping success runs, Number of non-overlapping Windows of length at most k containing exactly 2 successes, Conclusions, 380 References, On Sooner and Later Problems Between Success and Failure Runs Sigeo Aki Introduction, Number of Ocurrences of the Sooner Event Until the Later Waiting Time, Joint Distribution of Numbers of Runs, 397 References, 399

9 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 Introduction, Numbers of Success-Runs in Higher Order Markov Chain, Case l < m, 408 References, 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 Introduction, Independent Trials, Generalized Sequence of Order k, 420 References, A Multivariate Negative Binomial Distribution of Order k Arising When Success Runs are Allowed to Overlap Gregory A. Tripsiannis and Andreas N. Philippou Introduction, Multivariate Negative Binomial Distribution of Order k, Type III, Characteristics and Distributional Properties of MNB kjii {r; qi,..., q m ), 431 References, 436 PART VI APPLICATIONS TO DISTRIBUTION THEORY 26 The Joint Energy Distributions of the Bose-Einstein and of the Fermi-Dirac Particles /. Vincze and R. Törös Introduction, Derivation of the Joint Distribution and of the Joint Entropy, On the method, Joint distribution of the number of particles in energy intervals, Determination of the Limit Distributions, Discussion, 448

10 References, On Modified g-bessel Functions and Some Statistical Applications A. W. Kemp Introduction, Notation, The Distribution of the Difference of Two Euler Random Variables, The Distribution of the Difference of Two Heine Random Variables, Comments on the Distribution of the Difference of Two Generalized Euler Random Variables, 460 References, A g-logarithmic Distribution C. David Kem,p Introduction, A g-logarithmic Distribution, A Group Size Model for the Distribution, 469 References, Bernoulli Learning Models: Uppuluri Numbers K. G. Janardan Introduction, The General Model, Special cases of the general probabilistic model, Waiting Time Learning Models, Special cases of waiting time learning modeis, 478 References, 480 PART VII APPLICATIONS TO NONPARAMETRIC STATISTICS 30 Linear Nonparametric Tests Against Restricted Alternatives: The Simple-Tree Order and The Simple Order S. Chakraborti and W. Schaafsm,a Introduction, Background, Objectives, Exploration and Reformulation, Test for the Simple-Tree Problem, Some particular cases, Derivation of the MSSMP test, 493

11 XVI Contents 30.6 Test for the Simple Order Problem, Derivation of the (A)MSSMP test, Power comparisons, Extending the Class of SMP Tests, 501 Appendix, 503 References, Nonparametric Estimation of the Ratio of Variance Components M. Mahibbur Rahm.an and Z. Govindarajulu Introduction, Proposed Estimation Procedure, Monte Carlo Comparison, Ad.justment for Bias, 514 References, Limit Theorems for M-Processes Via Rank Statistics Processes M. Huskovd Introduction, Case 8 1 = = 0 n, Change Point Alternatives, 526 References, 533 Author Index 535 Subject Index 545