1 : Preliminaries. - 1. 1 Markov processes. - 1. 2 Stochastic monotonicity. - 1. 3 Birth-death processes. - 1. 4 Some notation and terminology. - 2 : Natural Birth-Death Processes. - 2. 1 Some basic properties. - 2. 2 The spectral representation. - 2. 3 Exponential ergodicity. - 2. 4 The moment problem and related topics. - 3 : Dual Birth-Death Processes. - 3. 1 Introduction. - 3. 2 Duality relations. - 3. 3 Ergodic properties. - 4 : Stochastic Monotonicity: General Results. - 4. 1 The case ?0 = 0. - 4. 2 The case ?0 > 0. - 4. 3 Properties of E(t). - 5 : Stochastic Monotonicity: Dependence on the Initial State Distribution. - 5. 1 Introduction to the case of a fixed initial state. - 5. 2 The transient and null recurrent process. - 5. 3 The positive recurrent process. - 5. 4 The case of an initial state distribution with finite support. - 6 : The M/M/S Queue Length Process. - 6. 1 Introduction. - 6. 2 The spectral function. - 6. 3 Stochastic monotonicity. - 6. 4 Exponential ergodicity. - 7 : A Queueing Model Where Potential Customers are Discouraged by Queue Length. - 7. 1 Introduction. - 7. 2 The spectral representation. - 7. 3 Stochastic monotonicity and exponential ergodicity. - 8 : Linear Growth Birth-Death Processes. - 8. 1 Introduction. - 8. 2 Stochastic monotonicity. - 9 : The Mean of Birth-Death Processes. - 9. 1 Introduction. - 9. 2 Representations. - 9. 3 Sufficient conditions for finiteness. - 9. 4 Behaviour of the mean in special cases. - 10 : The Truncated Birth-Death Process. - 10. 1 Introduction. - 10. 2 Preliminaries. - 10. 3 The sign structure of P'(t). - 10. 4 Stochastic monotonicity. - Appendix 1 : Proof of the Sign Variation Diminishing Property of Strictly Totally Positive Matrices. - Appendix 2: On Products of Infinite Matrices. - Appendix 3: On the Sign of Certain Quantities. - Appendix 4: Proof of Theorem 10. 2. 8. -References. - Notation Index. - Author Index. Language: English