{"id":6694,"date":"2018-09-21T10:40:38","date_gmt":"2018-09-21T09:40:38","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6694"},"modified":"2022-12-03T23:02:02","modified_gmt":"2022-12-03T22:02:02","slug":"la-file-m-m-1","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/the-file-mm-1\/","title":{"rendered":"The M \/ M \/ 1 queue"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"6694\" class=\"elementor elementor-6694\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d96fc12 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d96fc12\" 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-74aee66\" data-id=\"74aee66\" 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-8d24995 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"8d24995\" 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-b630415\" data-id=\"b630415\" 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-f1efdfa elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"f1efdfa\" 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-2161422\" data-id=\"2161422\" 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-bda1c9c elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"bda1c9c\" 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\/File_M\/M\/1\" 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-8ca27fc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8ca27fc\" 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-afd0ef5\" data-id=\"afd0ef5\" 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-586139f elementor-widget elementor-widget-progress\" data-id=\"586139f\" 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-586139f\">\n\t\t\t\tDifficulty\t\t\t<\/span>\n\t\t\n\t\t<div aria-labelledby=\"elementor-progress-bar-586139f\" class=\"elementor-progress-wrapper\" role=\"progressbar\" aria-valuemin=\"0\" aria-valuemax=\"100\" aria-valuenow=\"25\" aria-valuetext=\"25% (Facile)\">\n\t\t\t<div class=\"elementor-progress-bar\" data-max=\"25\">\n\t\t\t\t<span class=\"elementor-progress-text\">Easy<\/span>\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-progress-percentage\">25%<\/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-4334defa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4334defa\" 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-1d935574\" data-id=\"1d935574\" 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-3ecf86b4 elementor-widget elementor-widget-text-editor\" data-id=\"3ecf86b4\" 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\/the-file-mm-1\/#File-MM1\" >Queue M \/ M \/ 1<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"File-MM1\"><\/span>Queue M \/ M \/ 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>An M \/ M \/ 1 queue follows an exponential law for the arrival and service of customers. An M \/ M \/ 1 queue is represented as follows:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6697\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba56.png\" alt=\"Queue M \/ M \/ 1\" width=\"363\" height=\"194\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba56.png 363w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba56-300x160.png 300w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/figure>\n<\/div>\n\n<p>In the majority of cases, the customer in a service is included in the number of customers in the <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/queues\/\">waiting line<\/a>.<\/p>\n\n<p>The number of customers in the queue is modeled by the <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">markov chain<\/a> following continuous time:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6698\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba57.png\" alt=\"Queue M \/ M \/ 1\" width=\"459\" height=\"115\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba57.png 459w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba57-300x75.png 300w\" sizes=\"(max-width: 459px) 100vw, 459px\" \/><\/figure>\n<\/div>\n\n<p>Stationary probabilities exist because the chain is irreducible. Denote by p (n) the probability that the number of clients in the queue N (t) = n as t tends to infinity. The equilibrium equations give the following system:<\/p>\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" class=\"alignnone wp-image-6699\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba58.png\" alt=\"Queue M \/ M \/ 1\" width=\"268\" height=\"141\" title=\"\"><\/figure>\n\n<p>If we set \u03c1 = \u03bb \/ \u03bc then we find p (n) = \u03c1<sup>not<\/sup>p (0), which implies:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6700\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba59.png\" alt=\"Queue M \/ M \/ 1\" width=\"252\" height=\"94\" title=\"\"><\/figure>\n\n<p>We deduce that the queue is stable if \u03c1 &lt;1. That is to say that the average customer processing time is strictly less than the average arrival time of a customer (i.e. the average time between 2 customer arrivals). The queue is unstable if \u03c1\u22651, in this case the clients accumulate ad infinitum in the queue.<\/p>\n\n<p>All the performance parameters are calculated in steady state if the queue is stable. If we apply Little&#039;s law and the performance measures to M \/ M \/ 1 queues (and more generally to M \/ M \/ S queues), with \u03c1 = A:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6702\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba60.png\" alt=\"File M \/ M \/ S\" width=\"870\" height=\"430\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba60.png 870w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba60-300x148.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba60-768x380.png 768w\" sizes=\"(max-width: 870px) 100vw, 870px\" \/><\/figure>\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 Easy 25% M\/M\/1 Queue An M\/M\/1 queue follows an exponential law for customer arrival and service. \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-6694","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6694","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=6694"}],"version-history":[{"count":6,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6694\/revisions"}],"predecessor-version":[{"id":18686,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6694\/revisions\/18686"}],"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=6694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}