Queueing Theory Help M/M/3
$begingroup$
Comtex plc employs three people in its mail room to sort and despatch mail going through its internal mail system. Letters arrive at an average rate of 150 an hour and each employee can deal with 60 letters an hour. Assuming you can treat this as an M/M/3/∞/FIFO system, calculate:
- The probability that there are no letters waiting to be sorted
- The proportion of the time each sorter is busy
- The probability that there are more than 4 letters in the sorting office
I have attempted these questions... please can someone tell me if I am doing it right
service rate is now S$mu$ so traffic intensity is now $lambda$/$mu$S where S is 3 since there are 3 people. So $rho$ = 150/180 = 0.833333333 and 1-$rho$=0.16666 = $p_0$
the proportion of the time each sorter is busy
$rho$= 0.833333333 so 83.3333% of the time? not sure if this is rightsince question is asking if there are more than 4 letters in the sorting office now $n>S$ so
$p_4$ = $2.5^4$/4! * 0.166666 = 0.27126...
Is this all correct? thanks
queueing-theory operations-research
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
$endgroup$
add a comment |
$begingroup$
Comtex plc employs three people in its mail room to sort and despatch mail going through its internal mail system. Letters arrive at an average rate of 150 an hour and each employee can deal with 60 letters an hour. Assuming you can treat this as an M/M/3/∞/FIFO system, calculate:
- The probability that there are no letters waiting to be sorted
- The proportion of the time each sorter is busy
- The probability that there are more than 4 letters in the sorting office
I have attempted these questions... please can someone tell me if I am doing it right
service rate is now S$mu$ so traffic intensity is now $lambda$/$mu$S where S is 3 since there are 3 people. So $rho$ = 150/180 = 0.833333333 and 1-$rho$=0.16666 = $p_0$
the proportion of the time each sorter is busy
$rho$= 0.833333333 so 83.3333% of the time? not sure if this is rightsince question is asking if there are more than 4 letters in the sorting office now $n>S$ so
$p_4$ = $2.5^4$/4! * 0.166666 = 0.27126...
Is this all correct? thanks
queueing-theory operations-research
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
$endgroup$
$begingroup$
@callculus not sure if you can help?
$endgroup$
– Charlotte Sacks
Dec 6 '18 at 23:01
add a comment |
$begingroup$
Comtex plc employs three people in its mail room to sort and despatch mail going through its internal mail system. Letters arrive at an average rate of 150 an hour and each employee can deal with 60 letters an hour. Assuming you can treat this as an M/M/3/∞/FIFO system, calculate:
- The probability that there are no letters waiting to be sorted
- The proportion of the time each sorter is busy
- The probability that there are more than 4 letters in the sorting office
I have attempted these questions... please can someone tell me if I am doing it right
service rate is now S$mu$ so traffic intensity is now $lambda$/$mu$S where S is 3 since there are 3 people. So $rho$ = 150/180 = 0.833333333 and 1-$rho$=0.16666 = $p_0$
the proportion of the time each sorter is busy
$rho$= 0.833333333 so 83.3333% of the time? not sure if this is rightsince question is asking if there are more than 4 letters in the sorting office now $n>S$ so
$p_4$ = $2.5^4$/4! * 0.166666 = 0.27126...
Is this all correct? thanks
queueing-theory operations-research
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
$endgroup$
Comtex plc employs three people in its mail room to sort and despatch mail going through its internal mail system. Letters arrive at an average rate of 150 an hour and each employee can deal with 60 letters an hour. Assuming you can treat this as an M/M/3/∞/FIFO system, calculate:
- The probability that there are no letters waiting to be sorted
- The proportion of the time each sorter is busy
- The probability that there are more than 4 letters in the sorting office
I have attempted these questions... please can someone tell me if I am doing it right
service rate is now S$mu$ so traffic intensity is now $lambda$/$mu$S where S is 3 since there are 3 people. So $rho$ = 150/180 = 0.833333333 and 1-$rho$=0.16666 = $p_0$
the proportion of the time each sorter is busy
$rho$= 0.833333333 so 83.3333% of the time? not sure if this is rightsince question is asking if there are more than 4 letters in the sorting office now $n>S$ so
$p_4$ = $2.5^4$/4! * 0.166666 = 0.27126...
Is this all correct? thanks
queueing-theory operations-research
queueing-theory operations-research
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
asked Dec 6 '18 at 22:45
Charlotte SacksCharlotte Sacks
132
132
$begingroup$
@callculus not sure if you can help?
$endgroup$
– Charlotte Sacks
Dec 6 '18 at 23:01
add a comment |
$begingroup$
@callculus not sure if you can help?
$endgroup$
– Charlotte Sacks
Dec 6 '18 at 23:01
$begingroup$
@callculus not sure if you can help?
$endgroup$
– Charlotte Sacks
Dec 6 '18 at 23:01
$begingroup$
@callculus not sure if you can help?
$endgroup$
– Charlotte Sacks
Dec 6 '18 at 23:01
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029163%2fqueueing-theory-help-m-m-3%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029163%2fqueueing-theory-help-m-m-3%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
@callculus not sure if you can help?
$endgroup$
– Charlotte Sacks
Dec 6 '18 at 23:01