UID:
almafu_9960119133802883
Format:
1 online resource (169 pages) :
,
digital, PDF file(s)
ISBN:
0-511-98402-2
Content:
Markov chains are an important idea, related to random walks, which crops up widely in applied stochastic analysis. They are used, for example, in performance modelling and evaluation of computer networks, queuing networks, and telecommunication systems. The main point of the present book is to provide methods, based on the construction of Lyapunov functions, of determining when a Markov chain is ergodic, null recurrent, or transient. These methods can also be extended to the study of questions of stability. Of particular concern are reflected random walks and reflected Brownian motion. The authors provide not only a self-contained introduction to the theory but also details of how the required Lyapunov functions are constructed in various situations.
Note:
Bibliographic Level Mode of Issuance: Monograph
,
Cover -- Frontmatter -- Contents -- Errata -- Introduction and history -- Preliminaries -- 1.1 Irreducibility and aperiodicity -- 1.2 Classification -- 1.3 Continuous time -- 1.4 Classical examples -- General criteria -- 2.1 Criteria involving semi-martingales -- 2.2 Criteria for countable Markov chains -- Explicit construction of Lyapounov functions -- 3.1 Markov chains in a half-strip -- 3.2 Random walks in main definitions and interpretation -- 3.3 Classification of random walks in -- 3.4 Zero drifts -- 3.5 Jackson networks -- 3.6 Asymptotically small drifts -- 3.7 Stability and invariance principle -- Ideology of induced chains -- 4.1 Second vector field -- 4.2 Classification of paths -- 4.3 Gluing Lyapounov functions together -- 4.4 Classification in -- Random walks in two-dimensional complexes -- 5.1 Introduction and preliminary results -- 5.2 Random walks on hedgehogs -- 5.3 Formulation of the main result -- 5.4 Quasi-deterministic process -- 5.5 Proof of the ergodicity in theorem 5.3.4 -- 5.6 Proof of the transience -- 5.7 Proof of the recurrence -- 5.8 Proof of the non-ergodicity -- 5.9 Queueing applications -- 5.10 Remarks and problems -- Stability -- 6.1 A necessary and sufficient condition for continuity -- 6.2 Continuity of stationary probabilities -- 6.3 Continuity of random walks in -- Exponential convergence and analyticity -- 7.1 Analytic Lyapounov families -- 7.2 Proof of the exponential convergence -- 7.3 General analyticity theorem -- 7.4 Proof of analyticity completed -- 7.5 Examples of analyticity -- Bibliography -- Index.
,
English
Additional Edition:
ISBN 0-521-06447-3
Additional Edition:
ISBN 0-521-46197-9
Language:
English
URL:
https://doi.org/10.1017/CBO9780511984020
