{"id":28694,"date":"2025-06-20T14:36:14","date_gmt":"2025-06-20T14:36:14","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=28694"},"modified":"2025-06-20T14:36:15","modified_gmt":"2025-06-20T14:36:15","slug":"what-is-the-probability-that-the-service-facility-will-be-idle","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/what-is-the-probability-that-the-service-facility-will-be-idle\/","title":{"rendered":"What is the probability that the service facility will be idle"},"content":{"rendered":"\n<p>Number of Servers Arrival Rate 100 Service Rate 10.00 P(0), probability that there are no customers in the system 30% Lq: average length of the queue W: average time in the system 0.33 average number of customers in the system 2.33 Wq: average time in the queue 0.23 Utilization factor of the system 70% Table What is the probability that the service facility will be idle? Refer to the Table A) 0.23 B) 0.70 C) 0.33 D) 0.30 E) 0.233<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"390\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-98.png\" alt=\"\" class=\"wp-image-28695\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-98.png 700w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-98-300x167.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-light-green-cyan-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The correct answer is <strong>D) 0.30<\/strong>.<\/p>\n\n\n\n<p>In queuing theory, the probability that a service facility will be idle corresponds to the probability that there are <strong>zero customers in the system<\/strong>, denoted as <strong>P(0)<\/strong>. This value indicates the likelihood that the system is empty and the server is not occupied with any tasks. From the table, <strong>P(0) is given as 30%<\/strong>, or <strong>0.30<\/strong>. This means that 30% of the time, there are no customers either waiting or being served, and the service facility is entirely idle.<\/p>\n\n\n\n<p>To understand this more thoroughly, consider the structure of a single-server queuing system, where entities arrive at a certain rate and are served one at a time. The probability of the system being empty depends on the <strong>arrival rate (\u03bb)<\/strong> and the <strong>service rate (\u03bc)<\/strong>. In this case, although the arrival rate is shown as 100, that appears inconsistent with other values, suggesting that this may either be an error or placeholder. However, since <strong>P(0) is explicitly listed as 0.30<\/strong>, we take it as a given.<\/p>\n\n\n\n<p>The <strong>utilization factor (\u03c1)<\/strong> is another important concept, measuring the proportion of time the server is busy. In this table, utilization is given as 70%, meaning the server is occupied 70% of the time. The remaining 30% of the time, it is idle. This is precisely aligned with <strong>P(0) = 0.30<\/strong>, confirming that when the server is not utilized, it is in an idle state.<\/p>\n\n\n\n<p>Options like 0.23 or 0.233 reflect average queue lengths or waiting times, and should not be confused with the probability of idleness. Hence, <strong>option D, 0.30, is the correct and most appropriate choice based on the table&#8217;s data<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-241.jpeg\" alt=\"\" class=\"wp-image-28696\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-241.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-241-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-241-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Number of Servers Arrival Rate 100 Service Rate 10.00 P(0), probability that there are no customers in the system 30% Lq: average length of the queue W: average time in the system 0.33 average number of customers in the system 2.33 Wq: average time in the queue 0.23 Utilization factor of the system 70% Table [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-28694","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28694","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=28694"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28694\/revisions"}],"predecessor-version":[{"id":28697,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28694\/revisions\/28697"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=28694"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=28694"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=28694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}