exercise:25488e32c5: Difference between revisions
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(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> Consider a queueing process such that in each minute either 1 or 0 customers arrive with probabilities <math>p</math> or <math>q = 1 - p</math>, respectively. (The number <math>p</math> is called the ''arrival rate''.) When a customer starts s...") |
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Consider a queueing process such that in each minute either 1 or 0 customers arrive with probabilities <math>p</math> or | |||
<math>q = 1 - p</math>, respectively. (The number <math>p</math> is called the ''arrival rate''.) | <math>q = 1 - p</math>, respectively. (The number <math>p</math> is called the ''arrival rate''.) | ||
When a customer starts service she finishes in the next minute | When a customer starts service she finishes in the next minute | ||
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Thus when a customer begins being served she will finish | Thus when a customer begins being served she will finish | ||
being served in <math>j</math> minutes with probability <math>(1 - r)^{j -1}r</math>, for <math>j = 1</math>, 2, | being served in <math>j</math> minutes with probability <math>(1 - r)^{j -1}r</math>, for <math>j = 1</math>, 2, | ||
3, | 3, .... | ||
<ul><li> Find the generating function <math>f(z)</math> for the number of customers who | <ul style="list-style-type:lower-alpha"><li> Find the generating function <math>f(z)</math> for the number of customers who | ||
arrive in one minute and the generating function <math>g(z)</math> for the length of time | arrive in one minute and the generating function <math>g(z)</math> for the length of time | ||
that a person spends in service once she begins service. | that a person spends in service once she begins service. | ||
Revision as of 23:50, 14 June 2024
Consider a queueing process such that in each minute either 1 or 0 customers arrive with probabilities [math]p[/math] or [math]q = 1 - p[/math], respectively. (The number [math]p[/math] is called the arrival rate.) When a customer starts service she finishes in the next minute with probability [math]r[/math]. The number [math]r[/math] is called the service rate.) Thus when a customer begins being served she will finish being served in [math]j[/math] minutes with probability [math](1 - r)^{j -1}r[/math], for [math]j = 1[/math], 2, 3, ....
- Find the generating function [math]f(z)[/math] for the number of customers who arrive in one minute and the generating function [math]g(z)[/math] for the length of time that a person spends in service once she begins service.