exercise:D094f2f7d6: Difference between revisions
(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> A discrete time queueing system of capacity <math>n</math> consists of the person being served and those waiting to be served. The queue length <math>x</math> is observed each second. If <math>0 < x < n</math>, then with probability <math>p<...") |
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\newcommand{\mathds}{\mathbb}</math></div> A discrete time queueing system of capacity <math>n</math> | \newcommand{\mathds}{\mathbb}</math></div> A discrete time queueing system of capacity <math>n</math> consists of the person being served and those waiting to be served. The queue length <math>x</math> is observed each second. If <math>0 < x < n</math>, then with probability <math>p</math>, the queue size is increased by one by an arrival and, inependently, with probability <math>r</math>, it is decreased by one because the person being served finishes service. If <math>x = 0</math>, only an arrival (with probability <math>p</math>) is possible. If <math>x= n</math>, an arrival will depart without waiting for service, and so only the departure (with probability <math>r</math>) of the person being served is possible. Form a Markov chain with states given by the number of customers in the queue. Modify the program ''' FixedVector''' so that you can input <math>n</math>, <math>p</math>, and <math>r</math>, and the program will construct the transition matrix and compute the fixed vector. The quantity <math>s = p/r</math> is called the ''traffic intensity.'' Describe the differences in the fixed vectors according as <math>s < 1</math>, <math>s = 1</math>, or <math>s > 1</math>. | ||
consists of the person being served and those waiting to be served. The queue | |||
length <math>x</math> is observed each second. If <math>0 < x < n</math>, then with probability <math>p</math>, | |||
the queue size is increased by one by an arrival and, inependently, with | |||
probability <math>r</math>, it is decreased by one because the person being served | |||
finishes | |||
service. If <math>x = 0</math>, only an arrival (with probability <math>p</math>) is possible. If | |||
<math>x | |||
= n</math>, an arrival will depart without waiting for service, and so only the | |||
departure (with probability <math>r</math>) of the person being served is possible. Form | |||
a | |||
Markov chain with states given by the number of customers in the queue. Modify | |||
the program ''' FixedVector''' so that you can input <math>n</math>, <math>p</math>, and <math>r</math>, and the | |||
program will construct the transition matrix and compute the fixed vector. The | |||
quantity <math>s = p/r</math> is called the ''traffic intensity.'' Describe the | |||
differences in the fixed vectors according as <math>s < 1</math>, <math>s = 1</math>, or <math>s > 1</math>. | |||
Latest revision as of 22:59, 17 June 2024
A discrete time queueing system of capacity [math]n[/math] consists of the person being served and those waiting to be served. The queue length [math]x[/math] is observed each second. If [math]0 \lt x \lt n[/math], then with probability [math]p[/math], the queue size is increased by one by an arrival and, inependently, with probability [math]r[/math], it is decreased by one because the person being served finishes service. If [math]x = 0[/math], only an arrival (with probability [math]p[/math]) is possible. If [math]x= n[/math], an arrival will depart without waiting for service, and so only the departure (with probability [math]r[/math]) of the person being served is possible. Form a Markov chain with states given by the number of customers in the queue. Modify the program FixedVector so that you can input [math]n[/math], [math]p[/math], and [math]r[/math], and the program will construct the transition matrix and compute the fixed vector. The quantity [math]s = p/r[/math] is called the traffic intensity. Describe the differences in the fixed vectors according as [math]s \lt 1[/math], [math]s = 1[/math], or [math]s \gt 1[/math].