## ＜電子ブック＞Stochastic Monotonicity and Queueing Applications of Birth-Death Processes

責任表示 by E. A. Doorn Doorn, E. A SpringerLink (Online service) English (英語) Springer New York 1981- New York, NY, United States シリーズ Lecture Notes in Statistics ; 4 A stochastic process {X(t): 0 S t < =} with discrete state space S c ~ is said to be stochastically increasing (decreasing) on an interval T if the probabilities Pr{X(t) > i}, i E S, are increasing (d...ecreasing) with t on T. Stochastic monotonicity is a basic structural property for process behaviour. It gives rise to meaningful bounds for various quantities such as the moments of the process, and provides the mathematical groundwork for approximation algorithms. Obviously, stochastic monotonicity becomes a more tractable subject for analysis if the processes under consideration are such that stochastic mono tonicity on an inter val 0 < t < E implies stochastic monotonicity on the entire time axis. DALEY (1968) was the first to discuss a similar property in the context of discrete time Markov chains. Unfortunately, he called this property "stochastic monotonicity", it is more appropriate, however, to speak of processes with monotone transition operators. KEILSON and KESTER (1977) have demonstrated the prevalence of this phenomenon in discrete and continuous time Markov processes. They (and others) have also given a necessary and sufficient condition for a (temporally homogeneous) Markov process to have monotone transition operators. Whether or not such processes will be stochas tically monotone as defined above, now depends on the initial state distribution. Conditions on this distribution for stochastic mono tonicity on the entire time axis to prevail were given too by KEILSON and KESTER (1977).続きを見る 1 : Preliminaries1.1 Markov processes1.2 Stochastic monotonicity1.3 Birth-death processes1.4 Some notation and terminology2 : Natural Birth-Death Processes2.1 Some basic properties2.2 The spectral representation2.3 Exponential ergodicity2.4 The moment problem and related topics3 : Dual Birth-Death Processes3.1 Introduction3.2 Duality relations3.3 Ergodic properties4 : Stochastic Monotonicity: General Results4.1 The case ?0 = 04.2 The case ?0 > 04.3 Properties of E(t)5 : Stochastic Monotonicity: Dependence on the Initial State Distribution5.1 Introduction to the case of a fixed initial state5.2 The transient and null recurrent process5.3 The positive recurrent process5.4 The case of an initial state distribution with finite support6 : The M/M/S Queue Length Process6.1 Introduction6.2 The spectral function6.3 Stochastic monotonicity6.4 Exponential ergodicity7 : A Queueing Model Where Potential Customers are Discouraged by Queue Length7.1 Introduction7.2 The spectral representation7.3 Stochastic monotonicity and exponential ergodicity8 : Linear Growth Birth-Death Processes8.1 Introduction8.2 Stochastic monotonicity9 : The Mean of Birth-Death Processes9.1 Introduction9.2 Representations9.3 Sufficient conditions for finiteness9.4 Behaviour of the mean in special cases10 : The Truncated Birth-Death Process10.1 Introduction10.2 Preliminaries10.3 The sign structure of P'(t)10.4 Stochastic monotonicityAppendix 1 : Proof of the Sign Variation Diminishing Property of Strictly Totally Positive MatricesAppendix 2: On Products of Infinite MatricesAppendix 3: On the Sign of Certain QuantitiesAppendix 4: Proof of Theorem 10.2.8ReferencesNotation IndexAuthor Index.続きを見る http://hdl.handle.net/2324/1000036392 Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

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レコードID 3646470 QA273.A1-274.9 QA274-274.9 519.2 Mathematics. Distribution (Probability theory). Mathematics. Probability Theory and Stochastic Processes. ssj0001298605 9780387905471(print) 9781461258834 2020.06.27 2020.06.28