{"id":216837,"date":"2025-05-20T13:26:14","date_gmt":"2025-05-20T13:26:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=216837"},"modified":"2025-05-20T13:26:15","modified_gmt":"2025-05-20T13:26:15","slug":"consider-a-taxi-station-where-taxis-looking-for-passengers-and-passengers-looking-for-taxis-arrive-according-to-poisson-processes-with-mean-rates-per-minute-of-1-and-1-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/20\/consider-a-taxi-station-where-taxis-looking-for-passengers-and-passengers-looking-for-taxis-arrive-according-to-poisson-processes-with-mean-rates-per-minute-of-1-and-1-5\/","title":{"rendered":"Consider a taxi station where taxis looking for passengers and passengers looking for taxis arrive according to Poisson processes, with mean rates per minute of 1 and 1.5"},"content":{"rendered":"\n<p>Consider a taxi station where taxis looking for passengers and passengers looking for taxis arrive according to Poisson processes, with mean rates per minute of 1 and 1.5. A taxi will wait no matter how many other taxis are in line, but an arriving passenger waits only if the number of passengers already waiting for taxis is two or less.<\/p>\n\n\n\n<p>(a) Define the states, draw the transition diagram, and mark the rates. For clear understanding draw on a paper<br>(b) Write the balance equations and calculate the steady-state probabilities<br>(c) Find the mean number of taxis waiting for passengers.<br>(d) Find the mean number of passengers waiting for taxis.<br>(e) Find the mean number of passengers who in the course of an hour do not join the waiting line because at least three passengers were already waiting.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let\u2019s work through this <strong>queueing system<\/strong> scenario step-by-step. The key idea is to model the number of people (taxis or passengers) waiting. This is a <strong>birth-death process<\/strong>, where the state of the system represents the <em>net number of people waiting<\/em> (positive for passengers, negative for taxis, zero for a perfect match).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(a) <strong>Define States and Transition Diagram<\/strong><\/h3>\n\n\n\n<p>Let state <code>n<\/code> represent the number of people waiting:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If <code>n = 0<\/code>, no one is waiting \u2014 each arrival (passenger or taxi) instantly pairs with the other.<\/li>\n\n\n\n<li>If <code>n > 0<\/code>, then <code>n<\/code> passengers are waiting for taxis.<\/li>\n\n\n\n<li>If <code>n &lt; 0<\/code>, then <code>|n|<\/code> taxis are waiting for passengers.<\/li>\n<\/ul>\n\n\n\n<p><strong>Allowed states<\/strong>:<br><code>n \u2208 {\u2026, -3, -2, -1, 0, 1, 2, 3}<\/code><br>(Note: Passengers are allowed to wait only if \u2264 2 are in queue, so maximum is 3 passengers \u2014 the arriving one plus two already waiting.)<\/p>\n\n\n\n<p><strong>Transitions<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Passenger arrival (rate \u03bb = 1.5): causes state to move <strong>+1<\/strong>, unless already in state 3.<\/li>\n\n\n\n<li>Taxi arrival (rate \u03bc = 1): causes state to move <strong>\u20131<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Transition rates<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From state <code>n<\/code> to <code>n+1<\/code>: rate 1.5 if <code>n &lt; 3<\/code><\/li>\n\n\n\n<li>From state <code>n<\/code> to <code>n\u20131<\/code>: rate 1 always (taxis always arrive and wait)<\/li>\n<\/ul>\n\n\n\n<p>Diagram: States from \u2013\u221e to 3, with arrows between each pair of adjacent states and labeled rates.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(b) <strong>Balance Equations and Steady-State Probabilities<\/strong><\/h3>\n\n\n\n<p>Let <code>\u03c0_n<\/code> be the steady-state probability of being in state <code>n<\/code>.<\/p>\n\n\n\n<p>We can model this as a <strong>birth-death process<\/strong>, and define the balance equations recursively:<\/p>\n\n\n\n<p>Let\u2019s assume the system is stable and truncates at state 3 (since no more passengers can join when 3 are waiting). For states <code>n \u2264 2<\/code>:<\/p>\n\n\n\n<p>$$<br>\\pi_{n+1} = \\left(\\frac{\\lambda}{\\mu}\\right) \\pi_n = 1.5 \\cdot \\pi_n<br>$$<\/p>\n\n\n\n<p>Starting from <code>\u03c0_0<\/code>, we get:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03c0\u2081 = 1.5 \u03c0\u2080<\/li>\n\n\n\n<li>\u03c0\u2082 = (1.5)^2 \u03c0\u2080<\/li>\n\n\n\n<li>\u03c0\u2083 = (1.5)^3 \u03c0\u2080<\/li>\n<\/ul>\n\n\n\n<p>For taxis waiting (<code>n &lt; 0<\/code>):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03c0\u208b\u2081 = (1 \/ 1.5) \u03c0\u2080 = (2\/3) \u03c0\u2080<\/li>\n\n\n\n<li>\u03c0\u208b\u2082 = (2\/3)^2 \u03c0\u2080<\/li>\n\n\n\n<li>\u03c0\u208b\u2083 = (2\/3)^3 \u03c0\u2080<\/li>\n\n\n\n<li>and so on\u2026<\/li>\n<\/ul>\n\n\n\n<p><strong>Normalize<\/strong>: Total probability = 1<\/p>\n\n\n\n<p>$$<br>\\pi_0 \\left[ \\sum_{n=0}^{3} (1.5)^n + \\sum_{n=1}^{\\infty} \\left(\\frac{2}{3}\\right)^n \\right] = 1<br>$$<\/p>\n\n\n\n<p>Compute the sums:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Forward (n=0 to 3): $\\sum_{n=0}^{3} (1.5)^n = 1 + 1.5 + 2.25 + 3.375 = 8.125$<\/li>\n\n\n\n<li>Backward (n=1 to \u221e): geometric series $$<br>\\sum_{n=1}^{\\infty} \\left(\\frac{2}{3}\\right)^n = \\frac{\\frac{2}{3}}{1 &#8211; \\frac{2}{3}} = 2<br>$$<\/li>\n<\/ul>\n\n\n\n<p>So total = \u03c0\u2080 \u00d7 (8.125 + 2) = \u03c0\u2080 \u00d7 10.125 \u2192 \u03c0\u2080 = 1 \/ 10.125 \u2248 0.09877<\/p>\n\n\n\n<p>Now calculate other probabilities:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03c0\u2081 \u2248 1.5 \u00d7 0.09877 \u2248 0.14815<\/li>\n\n\n\n<li>\u03c0\u2082 \u2248 (1.5)^2 \u00d7 0.09877 \u2248 0.22222<\/li>\n\n\n\n<li>\u03c0\u2083 \u2248 (1.5)^3 \u00d7 0.09877 \u2248 0.33333<\/li>\n\n\n\n<li>\u03c0\u208b\u2081 \u2248 (2\/3) \u00d7 0.09877 \u2248 0.06585<\/li>\n\n\n\n<li>\u03c0\u208b\u2082 \u2248 (2\/3)^2 \u00d7 0.09877 \u2248 0.04390<\/li>\n\n\n\n<li>\u03c0\u208b\u2083 \u2248 (2\/3)^3 \u00d7 0.09877 \u2248 0.02927<\/li>\n\n\n\n<li>And so on\u2026<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(c) <strong>Mean Number of Taxis Waiting<\/strong><\/h3>\n\n\n\n<p>Taxis wait in states: <code>n = -1, -2, -3, ...<\/code><\/p>\n\n\n\n<p>Expected number:<\/p>\n\n\n\n<p>$$<br>E[T] = \\sum_{n=1}^{\\infty} n \\cdot \\pi_{-n} = \\sum_{n=1}^{\\infty} n \\cdot \\left(\\frac{2}{3}\\right)^n \\cdot \\pi_0<br>$$<\/p>\n\n\n\n<p>Use formula for expected value of geometric series:<\/p>\n\n\n\n<p>$$<br>\\sum_{n=1}^{\\infty} n r^n = \\frac{r}{(1 &#8211; r)^2}, \\text{ with } r = \\frac{2}{3}<br>$$<\/p>\n\n\n\n<p>$$<br>E[T] = \\pi_0 \\cdot \\frac{\\frac{2}{3}}{(1 &#8211; \\frac{2}{3})^2} = \\pi_0 \\cdot \\frac{\\frac{2}{3}}{(\\frac{1}{3})^2} = \\pi_0 \\cdot \\frac{2}{3} \\cdot 9 = 6 \\cdot \\pi_0 \u2248 6 \u00d7 0.09877 \u2248 0.5926<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(d) <strong>Mean Number of Passengers Waiting<\/strong><\/h3>\n\n\n\n<p>Passengers wait in states <code>n = 1, 2, 3<\/code>.<\/p>\n\n\n\n<p>$$<br>E[P] = \\sum_{n=1}^{3} n \\cdot \\pi_n = 1 \u00d7 \u03c0\u2081 + 2 \u00d7 \u03c0\u2082 + 3 \u00d7 \u03c0\u2083<br>= 0.14815 + 2 \u00d7 0.22222 + 3 \u00d7 0.33333<br>= 0.14815 + 0.44444 + 1.0 = 1.5926<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(e) <strong>Passengers Who Do Not Join Queue per Hour<\/strong><\/h3>\n\n\n\n<p>Passengers are turned away when system is in state <code>n = 3<\/code>.<\/p>\n\n\n\n<p>Arrival rate is 1.5\/min, and fraction of time in state 3 is \u03c0\u2083 \u2248 0.33333. So:<\/p>\n\n\n\n<p>$$<br>\\text{Blocked rate} = 1.5 \u00d7 \u03c0\u2083 = 1.5 \u00d7 0.33333 = 0.5 \\text{ passengers\/min}<br>$$<\/p>\n\n\n\n<p>In 1 hour:<\/p>\n\n\n\n<p>$$<br>0.5 \u00d7 60 = \\boxed{30 \\text{ passengers}}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>(c)<\/strong> Mean taxis waiting: \u2248 <strong>0.5926<\/strong><\/li>\n\n\n\n<li><strong>(d)<\/strong> Mean passengers waiting: \u2248 <strong>1.5926<\/strong><\/li>\n\n\n\n<li><strong>(e)<\/strong> Passengers turned away per hour: <strong>30<\/strong><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Consider a taxi station where taxis looking for passengers and passengers looking for taxis arrive according to Poisson processes, with mean rates per minute of 1 and 1.5. A taxi will wait no matter how many other taxis are in line, but an arriving passenger waits only if the number of passengers already waiting for [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-216837","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216837","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=216837"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216837\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216837"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216837"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216837"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}