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Probability (including stochastic processes) is now being applied to virtually every academic discipline, especially to the sciences. An area of substantial application is that known as operations research or industrial engineering, which incorporates subjects such as queueing theory, optimization, and network flow. This book provides a compact introduction to that field for students with minimal preparation, knowing mainly calculus and having "mathe matical maturity." Beginning with the basics of probability, the develop ment is self-contained but not abstract, that is, without measure theory and its probabilistic counterpart. Although the text is reasonably short, a course based on this book will normally occupy two semesters or three quarters. There are many points in the discussions and problems which require the assistance of an instructor for completeness and clarity. The book is designed to give equal emphasis to those applications which motivate the subject and to appropriatemathematical techniques. Thus, the student who has successfully completed the course is ready to turn in either of two directions: towards direct study of research papers in operations research, or towards a course in abstract probability, for which this text provides the intuitive background. Frank A. Haight Pennsylvania State University vii Contents 1. Discrete Probability .................................................. 1 1.1. Applied Probability. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Sample Spaces ......................................................... 3 1.3. Probability Distributions and Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4. The Connection between Distributions and Sample Points: Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . . .
1. Discrete Probability.- 1.1. Applied Probability.- 1.2. Sample Spaces.- 1.3. Probability Distributions and Parameters.- 1.4. The Connection between Distributions and Sample Points: Random Variables.- 1.5. Events and Indicators.- 1.6. Mean and Variance.- 1.7. Calculation of the Mean and Variance.- 1.8. The Distribution Function.- 1.9. The Gamma Function and the Beta Function.- 1.10. The Negative Binomial Distribution.- 1.11. The Probability Generating Function.- 1.12. The Catalan Distribution.- 1.13. More about the p.g.f.; The Equation s = ?(s).- 1.14. Problems.- 2. Conditional Probability.- 2.1. Introduction. An Example.- 2.2. Conditional Probability and Bayes' Theorem.- 2.3. Conditioning.- 2.4. Independence and Bernoulli Trials.- 2.5. Moments, Distribution Functions, and Generating Functions.- 2.6. Convolutions and Sums of Random Variables.- 2.7. Computing Convolutions: Examples.- 2.8. Diagonal Distributions.- 2.9. Problems.- 3. Markov Chains.- 3.1. Introduction: Random Walk.- 3.2.Definitions.- 3.3. Matrix and Vector.- 3.4. The Transition Matrix and Initial Vector.- 3.5. The Higher-Order Transition Matrix: Regularity.- 3.6. Reducible Chains.- 3.7. Periodic Chains.- 3.8. Classification of States. Ergodic Chains.- 3.9. Finding Equilibrium Distributions-The Random Walk Revisited.- 3.10. A Queueing Model.- 3.11. The Ehrenfest Chain.- 3.12. Branching Chains.- 3.13. Probability of Extinction.- 3.14. The Gambler's Ruin.- 3.15. Probability of Ruin as Probability of Extinction.- 3.16. First-Passage Times.- 3.17. Problems.- 4. Continuous Probability Distributions.- 4.1. Examples.- 4.2. Probability Density Functions.- 4.3. Change of Variables.- 4.4. Convolutions of Density Functions.- 4.5. The Incomplete Gamma Function.- 4.6. The Beta Distribution and the Incomplete Beta Function.- 4.7. Parameter Mixing.- 4.8. Distribution Functions.- 4.9. Stieltjes Integration.- 4.10. The Laplace Transform.- 4.11. Properties of the Laplace Transform.- 4.12. Laplace Inversion.- 4.13. Random Sums.- 4.14. Problems.- 5. Continuous Time Processes.- 5.1. Introduction and Notation.- 5.2. Renewal Processes.- 5.3. The Poisson Process.- 5.4. Two-State Processes.- 5.5. Markov Processes.- 5.6. Equilibrium.- 5.7. The Method of Marks.- 5.8. The Markov Infinitesimal Matrix.- 5.9. The Renewal Function.- 5.10. The Gap Surrounding an Arbitrary Point.- 5.11. Counting Distributions.- 5.12. The Erlang Process.- 5.13. Displaced Gaps.- 5.14. Divergent Birth Processes.- 5.15. Problems.- 6. The Theory of Queues.- 6.1. Introduction and Classification.- 6.2. The M? / M? / 1 Queue: General Solution.- 6.3. The M? / M? / 1 Queue: Oversaturation.- 6.4. The M? / M? / 1 Queue: Equilibrium.- 6.5. The M? / M? / n Queue in Equilibrium: Loss Formula.- 6.6. The M? / G? / 1 Queue and the Imbedded Markov Chain.- 6.7. The Pollaczek-Khintchine Formula.- 6.8. Waiting Time.- 6.9. Virtual Queueing Time.- 6.10. The Equation y = xe?x.- 6.11. Busy Period: Borel's Method.- 6.12. The Busy Period Treated as a BranchingProcess: The M / G /1 Queue.- 6.13. The Continuous Busy Period and the M / G /1 Queue.- 6.14. Generalized Busy Periods.- 6.15. The G / M /1 Queue.- 616 Balking.- 6.17. Priority Service.- 6.18. Reverse-Order Service (LIFO).- 6.19. Problems.
