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<title>Comparing the Response Time of Three Simple Queueing Systems</title>
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</aside>
<hgroup class="auto-fadein">
<h1>Comparing the Response Time of Three Simple Queueing Systems</h1>
<h2>A shiny queue calculator</h2>
<p>Tim Wise<br/></p>
</hgroup>
<article></article>
</slide>
<!-- SLIDES -->
<slide class="" id="slide-1" style="background:;">
<hgroup>
<h2>A Quick Introduction to Queueing Theory</h2>
</hgroup>
<article data-timings="">
<p><a href="http://en.wikipedia.org/wiki/Queueing_theory">Queueing theory</a> is the mathematical
study of <em>customers</em> waiting in lines for <em>services</em>.
A queueing system can be as simple as a single server,
like waiting to get cash from an ATM machine.
Or it can be a complex network of servers,
like components flowing through a factory assembly line.</p>
<div style='float:left;width:48%;' class='centered'>
<p>The simplest queue is a single server with a waiting line and called
an <a href="http://en.wikipedia.org/wiki/M/M/1_queue">M/M/1 queue</a>. Customers
arrive in the queue at rate \(\lambda\) and
the server handles them at rate \(\mu\).</p>
<p>A generalization is an <a href="http://en.wikipedia.org/wiki/M/M/c_queue">M/M/c queue</a>
that has a single line feeding
C servers. The next person in line is served by the next available
server. It's like a load balancer.</p>
<p>We're interested in comparing the response times of the two queueing systems
for different values of \(C\).</p>
</div>
<div style='float:right;width:48%;'>
<p><img class=center src=./assets/img/MM1-MMc.png></p>
</div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-2" style="background:;">
<hgroup>
<h2>Metrics of M/M/1 Queue and an Example</h2>
</hgroup>
<article data-timings="">
<div style='float:left;width:48%;' class='centered'>
<p>Typical configuration metrics of an M/M/1 queue are: </p>
<ul>
<li>\(\lambda\), arrival rate of customers</li>
<li>\(S\), service time for a single customer</li>
</ul>
<p>from those we can compute:</p>
<ul>
<li>max server rate, or capacity, \(\mu = 1 / S\) </li>
<li>server utilization, \(\rho = \lambda / \mu\)</li>
<li>customer response time, \(R = 1 / (\mu - \lambda)\)</li>
</ul>
<p>A queueing system is in steady-state when the arrival rate is less than
the capacity of the server. For an M/M/1 queue, when
\(\lambda < \mu \equiv \rho < 1\).</p>
</div>
<div style='float:right;width:48%;'>
<pre><code class="r">lambda <- 50 # tx per sec
S <- 0.010 # sec per tx
mu <- 1 / S; mu # max tx per sec
</code></pre>
<pre><code>## [1] 100
</code></pre>
<pre><code class="r"># server utilization
rho <- lambda / mu; percent(rho)
</code></pre>
<pre><code>## Error in eval(expr, envir, enclos): could not find function "percent"
</code></pre>
<pre><code class="r"># tx response time sec
R <- 1 / (mu - lambda); R
</code></pre>
<pre><code>## [1] 0.02
</code></pre>
</div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-3" style="background:;">
<hgroup>
<h2>Consider Three Equivalent Queueing Systems</h2>
</hgroup>
<article data-timings="">
<p>From <a href="http://www.cmg.org/publications/conference-proceedings/conference-proceedings2014/">Developing Our Intuition About Queuing Network Models</a>:</p>
<p><img class=center src=./assets/img/3qnetworks.png></p>
<table><thead>
<tr>
<th>Grocery Store</th>
<th>Bank Teller</th>
<th>Super Server</th>
</tr>
</thead><tbody>
<tr>
<td>A set of six parallel M/M/1 queues</td>
<td>An M/M/c queue with 6 servers</td>
<td>An M/M/1 queue that is \(6x\) faster</td>
</tr>
<tr>
<td>each with an arrival rate \(\lambda\)</td>
<td>with an arrival rate of \(6\lambda\)</td>
<td>with an arrival rate of \(6\lambda\)</td>
</tr>
<tr>
<td>and each server works at rate \(\mu\)</td>
<td>and each server works at rate \(\mu\)</td>
<td>and the server works at rate \(6\mu\)</td>
</tr>
</tbody></table>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-4" style="background:;">
<hgroup>
<h2>Which Queue Provides the Fastest Response Time?</h2>
</hgroup>
<article data-timings="">
<div class="quiz quiz-single well ">
<h3>Question 1</h3>
<p>We've seen that response time of a queue is a function of the
customer arrival rate and the service time. Which of the 3 queueing systems do
you think provides the fastest response time under light load,
i.e., slow customer arrival rate?</p>
<ol>
<li>Grocery Store</li>
<li>Bank Teller</li>
<li><em>Super Server</em></li>
</ol>
<button class="quiz-submit btn btn-primary">Submit</button>
<button class="quiz-toggle-hint btn btn-info">Show Hint</button>
<button class="quiz-show-answer btn btn-success">Show Answer</button>
<button class="quiz-clear btn btn-danger">Clear</button>
<div class="quiz-hint">
<p>Hint: Under light load there is no waiting time. Response time is just the
service time. Which system has the fastest server?</p>
</div>
<div class="quiz-explanation">
<p>The Super Server is 6 times as fast as the other servers.
So its service time \(1/6\)-th that of the other systems.
Under low customer arrival rate, there is no wait time and the response time
is just the service time. So the Super Server has the fastest response time
under low load.</p>
</div>
<div class="quiz-">
<h3>Question 2</h3>
<p>Now, which system do you think provides better response time under very
high load? To find out, go use our simple shiny app
<a href="http://timwise.shinyapps.io/queue-calculator/">queue calculator</a>.</p>
</div>
</div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="backdrop"></slide>
</slides>
<div class="pagination pagination-small" id='io2012-ptoc' style="display:none;">
<ul>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=1 title='A Quick Introduction to Queueing Theory'>
1
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=2 title='Metrics of M/M/1 Queue and an Example'>
2
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=3 title='Consider Three Equivalent Queueing Systems'>
3
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=4 title='Which Queue Provides the Fastest Response Time?'>
4
</a>
</li>
</ul>
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