Document Zbl 1361.60062

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Algebraic, stochastic and analysis structures for networks, data classification and optimization. Cham: Springer (ISBN 978-3-319-42104-9/hbk; 978-3-319-42105-6/ebook). Springer Proceedings in Mathematics & Statistics 179, 151-222 (2016). </div> <div class="abstract"><div class="pre">Summary: New algorithms for computing asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1365.74007">Zbl 1365.74007</a>].</div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6703848">Cited in <strong>11</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60J22" title="MSC2020">60J22</a> </td> <td class="space"> Computational methods in Markov chains </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60K15" title="MSC2020">60K15</a> </td> <td class="space"> Markov renewal processes, semi-Markov processes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60J10" title="MSC2020">60J10</a> </td> <td class="space"> Markov chains (discrete-time Markov processes on discrete state spaces) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60J80" title="MSC2020">60J80</a> </td> <td class="space"> Branching processes (Galton-Watson, birth-and-death, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C40" title="MSC2020">65C40</a> </td> <td class="space"> Numerical analysis or methods applied to Markov chains </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asemi-Markov+process">semi-Markov process</a>; <a href="/?q=ut%3Abirth-death-type+process">birth-death-type process</a>; <a href="/?q=ut%3Astationary+distribution">stationary distribution</a>; <a href="/?q=ut%3ALaurent+asymptotic+expansion">Laurent asymptotic expansion</a>; <a href="/?q=ut%3Aalgorithms">algorithms</a>; <a href="/?q=ut%3Ahitting+time">hitting time</a>; <a href="/?q=ut%3Anonlinear+perturbation">nonlinear perturbation</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1361.60062" data-ciurl="/ci/06703848" data-biburl="/bibtex/06703848.bib" data-amsurl="/amsrefs/06703848.bib" data-xmlurl="/xml/06703848.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06703848.pdf" title="Zbl 1361.60062 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/978-3-319-42105-6_10" aria-label="DOI for “Asymptotic expansions for stationary distributions of perturbed semi-Markov processes”" title="10.1007/978-3-319-42105-6_10">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1603.03891" title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abadov, Z.A.: Asymptotical expansions with explicit estimation of constants for exponential moments of sums of random variables defined on a markov chain and their applications to limit theorems for first hitting times. Candidate of Science dissertation, Kiev State University (1984)</td> </tr><tr> <td>[2]</td> <td class="space">Abbad, M., Filar, J.A.: Algorithms for singularly perturbed Markov control problems: a survey. In: Leondes, C.T. (ed.) Techniques in Discrete-Time Stochastic Control Systems. Control and Dynamic Systems, vol. 73, pp. 257–289. Academic Press, New York (1995) · <a href="/0843.93094" class="nowrap">Zbl 0843.93094</a> · <a href="https://doi.org/10.1016/S0090-5267(05)80010-6" class="nowrap">doi:10.1016/S0090-5267(05)80010-6</a></td> </tr><tr> <td>[3]</td> <td class="space">Albeverio, S., Koroliuk, V.S., Samoilenko, I.V.: Asymptotic expansion of semi-Markov random evolutions. Stochastics 81(5), 477–502 (2009) · <a href="/1179.60058" class="nowrap">Zbl 1179.60058</a></td> </tr><tr> <td>[4]</td> <td class="space">Albrecht, A.R., Howlett, P.G., Pearce, C.E.M.: Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series. J. Math. Anal. Appl. 383(1), 95–110 (2011) · <a href="/1223.47014" class="nowrap">Zbl 1223.47014</a> · <a href="https://doi.org/10.1016/j.jmaa.2011.05.007" class="nowrap">doi:10.1016/j.jmaa.2011.05.007</a></td> </tr><tr> <td>[5]</td> <td class="space">Albrecht, A.R., Howlett, P.G., Pearce, C.E.M.: The fundamental equations for inversion of operator pencils on Banach space. J. Math. Anal. Appl. 413(1), 411–421 (2014) · <a href="/1308.47013" class="nowrap">Zbl 1308.47013</a> · <a href="https://doi.org/10.1016/j.jmaa.2013.11.060" class="nowrap">doi:10.1016/j.jmaa.2013.11.060</a></td> </tr><tr> <td>[6]</td> <td class="space">Alimov, D., Shurenkov, V.M.: Markov renewal theorems in triangular array model. Ukr. Mat. Zh. 42, 1443–1448 (1990) (English translation in Ukr. Math. J. 42, 1283–1288) · <a href="/0719.60070" class="nowrap">Zbl 0719.60070</a></td> </tr><tr> <td>[7]</td> <td class="space">Alimov, D., Shurenkov, V.M.: Asymptotic behavior of terminating Markov processes that are close to ergodic. Ukr. Mat. Zh. 42, 1701–1703 (1990) (English translation in Ukr. Math. J. 42 1535–1538) · <a href="/0732.60098" class="nowrap">Zbl 0732.60098</a></td> </tr><tr> <td>[8]</td> <td class="space">Allen, B., Anderssen, R.S., Seneta, E.: Computation of stationary measures for infinite Markov chains. In: Neuts, M.F. (ed.) Algorithmic Methods in Probability. Studies in the Management Sciences, vol. 7, pp. 13–23. North-Holland, Amsterdam (1977) · <a href="/0386.60048" class="nowrap">Zbl 0386.60048</a></td> </tr><tr> <td>[9]</td> <td class="space">Altman, E., Avrachenkov, K.E., Núñez-Queija, R.: Perturbation analysis for denumerable Markov chains with application to queueing models. Adv. Appl. Probab. 36(3), 839–853 (2004) · <a href="/1062.60066" class="nowrap">Zbl 1062.60066</a> · <a href="https://doi.org/10.1017/S0001867800013148" class="nowrap">doi:10.1017/S0001867800013148</a></td> </tr><tr> <td>[10]</td> <td class="space">Andersson, F., Silvestrov, S.: The mathematics of Internet search engines. Acta Appl. Math. 104, 211–242 (2008) · <a href="/1184.68174" class="nowrap">Zbl 1184.68174</a> · <a href="https://doi.org/10.1007/s10440-008-9254-y" class="nowrap">doi:10.1007/s10440-008-9254-y</a></td> </tr><tr> <td>[11]</td> <td class="space">Anisimov, V.V. Limit theorems for semi-Markov processes. I, II. Teor. Veroyatn. Mat. 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