{"id":6658,"date":"2018-09-17T16:15:41","date_gmt":"2018-09-17T15:15:41","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6658"},"modified":"2022-12-03T23:02:00","modified_gmt":"2022-12-03T22:02:00","slug":"regime-permanent","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/steady-state\/","title":{"rendered":"Steady state"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"6658\" class=\"elementor elementor-6658\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4a2c5bc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4a2c5bc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8db3d91\" data-id=\"8db3d91\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bfadcf4 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"bfadcf4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Markov process<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8336f23\" data-id=\"8336f23\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a26d29c elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"a26d29c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Home page<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5dba4ab\" data-id=\"5dba4ab\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-362bc56 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"362bc56\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Processus_de_Markov_%C3%A0_temps_continu\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d0d209d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d0d209d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fd2eb6f\" data-id=\"fd2eb6f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1d9f39b elementor-widget elementor-widget-progress\" data-id=\"1d9f39b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"progress.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<span class=\"elementor-title\" id=\"elementor-progress-bar-1d9f39b\">\n\t\t\t\tDifficulty\t\t\t<\/span>\n\t\t\n\t\t<div aria-labelledby=\"elementor-progress-bar-1d9f39b\" class=\"elementor-progress-wrapper\" role=\"progressbar\" aria-valuemin=\"0\" aria-valuemax=\"100\" aria-valuenow=\"50\" aria-valuetext=\"50% (Moyen)\">\n\t\t\t<div class=\"elementor-progress-bar\" data-max=\"50\">\n\t\t\t\t<span class=\"elementor-progress-text\">Average<\/span>\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-progress-percentage\">50%<\/span>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-198b13e4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"198b13e4\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-63d64086\" data-id=\"63d64086\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-53c417cb elementor-widget elementor-widget-text-editor\" data-id=\"53c417cb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/steady-state\/#Regime-permanent\" >Steady state<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/steady-state\/#Exemple\" >Example<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Regime-permanent\"><\/span>Steady state<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">markov chain<\/a> in continuous time (and irreducible in discrete time), the vector of stationary probabilities always exists and is independent of the initial distribution (steady state). This vector \u03c0 is the solution of the following system:<\/p>\n\n<figure class=\"wp-block-image\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6670\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba46.png\" alt=\"steady state\" width=\"849\" height=\"255\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba46.png 849w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba46-300x90.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba46-768x231.png 768w\" sizes=\"(max-width: 849px) 100vw, 849px\" \/><\/figure>\n\n<p>The system is called the balance equations.<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemple\"><\/span>Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Two identical machines run continuously unless they are broken. A repairer available as needed to repair the machines.<\/p>\n\n<p>The repair time follows an exponential distribution with an average of 0.5 day. Once repaired, the uptime of a machine before its next breakage follows an exponential distribution of an average of 1 day. We assume that these distributions are independent. Consider the random process defined in terms of the number of failed machines.<\/p>\n\n<p>Consider the random variable X (t &#039;) describing the number of machines down at time t&#039;. The states of the random variable are {0, 1, 2}. The repair time and the breakage time follow an exponential distribution so we are in the presence of a continuous time Markov chain. The repair time follows an exponential distribution with an average of 0.5 day. The repair rate is the reverse, ie 2 machines per day. Likewise, we deduce that the debris rate is 1 day. When the two machines are working, we have a breakage rate = machine1 + machine2 = 2.<\/p>\n\n<p>The states describe the number of machines that have failed. The two machines cannot break at the same time so q<sub>02<\/sub> = 0. The repairer only repairs one machine at a time so q<sub>20<\/sub> = 0. The repair rate is 2 machines per day. The breakage rate for a machine is 1 machine per day, and 2 per day if both machines are running. Which gives us the following continuous state Markov chain:<\/p>\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" class=\"alignnone wp-image-6660\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba40.png\" alt=\"steady state\" width=\"736\" height=\"148\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba40.png 736w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba40-300x60.png 300w\" sizes=\"(max-width: 736px) 100vw, 736px\" \/><\/figure>\n\n<p>If we take the balance equations, we have the following system:<\/p>\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" class=\"alignnone wp-image-6672\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba47.png\" alt=\"steady state\" width=\"903\" height=\"187\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba47.png 903w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba47-300x62.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba47-768x159.png 768w\" sizes=\"(max-width: 903px) 100vw, 903px\" \/><\/figure>\n\n<p>Which gives for solution the vector (0.4, 0.4, 0.2). If one seeks to calculate the average number of broken machines, it suffices to calculate the mathematical expectation since the states represent the number of broken machines: 0 * 0.4 + 1 * 0.4 + 2 * 0.2 = 0.8.<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Markov Process Wiki Home Difficulty Medium 50% Steady State In a continuous-time (and irreducible in discrete-time) Markov chain, the \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":5007,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-6658","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6658","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=6658"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6658\/revisions"}],"predecessor-version":[{"id":18673,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6658\/revisions\/18673"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=6658"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}