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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist"></a><a class="link" href="hyperexponential_dist.html" title="Hyperexponential Distribution">Hyperexponential
Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">hyperexponential</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">hyperexponential_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">hyperexponential</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
<span class="comment">// Constructors:</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">();</span> <span class="comment">// Default.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RateIterT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateIterT2</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span> <span class="comment">// Default equal probabilities.</span>
<span class="identifier">RateIterT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_first</span><span class="special">,</span>
<span class="identifier">RateIterT2</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_last</span><span class="special">);</span> <span class="comment">// Rates using Iterators.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">ProbIterT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateIterT</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbIterT</span> <span class="identifier">prob_first</span><span class="special">,</span> <span class="identifier">ProbIterT</span> <span class="identifier">prob_last</span><span class="special">,</span>
<span class="identifier">RateIterT</span> <span class="identifier">rate_first</span><span class="special">,</span> <span class="identifier">RateIterT</span> <span class="identifier">rate_last</span><span class="special">);</span> <span class="comment">// Iterators.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">ProbRangeT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateRangeT</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbRangeT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">prob_range</span><span class="special">,</span>
<span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_range</span><span class="special">);</span> <span class="comment">// Ranges.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RateRangeT</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_range</span><span class="special">);</span>
<span class="preprocessor">#if</span> <span class="special">!</span><span class="identifier">defined</span><span class="special">(</span><span class="identifier">BOOST_NO_CXX11_HDR_INITIALIZER_LIST</span><span class="special">)</span> <span class="comment">// C++11 initializer lists supported.</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">l1</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">l2</span><span class="special">);</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">l1</span><span class="special">);</span>
<span class="preprocessor">#endif</span>
<span class="comment">// Accessors:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">num_phases</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">probabilities</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">rates</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
An implementation-defined mechanism is provided to avoid ambiguity between
constructors accepting ranges, iterators and constants as parameters.
This should be transparent to the user. See below and the header file
hyperexponential.hpp for details and explanatory comments.
</p></td></tr>
</table></div>
<p>
The class type <code class="computeroutput"><span class="identifier">hyperexponential_distribution</span></code>
represents a <a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">hyperexponential
distribution</a>.
</p>
<p>
A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is a <a href="http://en.wikipedia.org/wiki/Continuous_probability_distribution" target="_top">continuous
probability distribution</a> obtained as a mixture of <span class="emphasis"><em>k</em></span>
<a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>s.
It is also referred to as <span class="emphasis"><em>mixed exponential distribution</em></span>
or parallel <span class="emphasis"><em>k-phase exponential distribution</em></span>.
</p>
<p>
A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is characterized
by two parameters, namely a <span class="emphasis"><em>phase probability vector</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(α<sub>1</sub>,...,α<sub>k</sub>)</em></span> and a
<span class="emphasis"><em>rate vector</em></span> <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(λ<sub>1</sub>,...,λ<sub>k</sub>)</em></span>.
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
density function</a> for random variate <span class="emphasis"><em>x</em></span> in a
hyperexponential distribution is given by:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span>
</p></blockquote></div>
<p>
The following graph illustrates the PDF of the hyperexponential distribution
with five different parameters, namely:
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(1.0)</em></span> (which degenerates to a simple
exponential distribution),
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.1, 0.9)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.9, 0.1)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.2, 0.3, 0.5)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.0, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.5, 0.3, 0.2)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.0, 1.5)</em></span>.
</li>
</ol></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf.svg" align="middle"></span>
</p></blockquote></div>
<p>
Also, the following graph illustrates the PDF of the hyperexponential distribution
(solid lines) where only the <span class="emphasis"><em>phase probability vector</em></span>
changes together with the PDF of the two limiting exponential distributions
(dashed lines):
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.1, 0.9)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.6, 0.4)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.9, 0.1)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
Exponential distribution with parameter <span class="emphasis"><em>λ=0.5</em></span>,
</li>
<li class="listitem">
Exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>.
</li>
</ol></div>
<p>
As expected, as the first element <span class="emphasis"><em>α<sub>1</sub></em></span> of the <span class="emphasis"><em>phase
probability vector</em></span> approaches to <span class="emphasis"><em>1</em></span> (or,
equivalently, <span class="emphasis"><em>α<sub>2</sub></em></span> approaches to <span class="emphasis"><em>0</em></span>),
the resulting hyperexponential distribution nears the exponential distribution
with parameter <span class="emphasis"><em>λ=0.5</em></span>. Conversely, as the first element
<span class="emphasis"><em>α<sub>2</sub></em></span> of the <span class="emphasis"><em>phase probability vector</em></span>
approaches to <span class="emphasis"><em>1</em></span> (or, equivalently, <span class="emphasis"><em>α<sub>1</sub></em></span>
approaches to <span class="emphasis"><em>0</em></span>), the resulting hyperexponential distribution
nears the exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.svg" align="middle"></span>
</p></blockquote></div>
<p>
Finally, the following graph compares the PDF of the hyperexponential distribution
with different number of phases but with the same mean value equal to
<span class="emphasis"><em>2</em></span>:
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(2.0)</em></span> (which degenerates to a simple
exponential distribution),
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.5, 0.5)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.3, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0/3.0, 1.0/3.0, 1.0/3.0)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.2, 1.5, 3.0)</em></span>,
</li>
</ol></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samemean.svg" align="middle"></span>
</p></blockquote></div>
<p>
As can be noted, even if the three distributions have the same mean value,
the two hyperexponential distributions have a <span class="emphasis"><em>longer</em></span>
tail with respect to the one of the exponential distribution. Indeed, the
hyperexponential distribution has a larger variability than the exponential
distribution, thus resulting in a <a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" target="_top">Coefficient
of Variation</a> greater than <span class="emphasis"><em>1</em></span> (as opposed to
the one of the exponential distribution which is exactly <span class="emphasis"><em>1</em></span>).
</p>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.applications"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.applications">Applications</a>
</h4>
<p>
A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is frequently
used in <a href="http://en.wikipedia.org/wiki/Queueing_theory" target="_top">queueing
theory</a> to model the distribution of the superposition of <span class="emphasis"><em>k</em></span>
independent events, like, for instance, the service time distribution of
a queueing station with <span class="emphasis"><em>k</em></span> servers in parallel where
the <span class="emphasis"><em>i</em></span>-th server is chosen with probability <span class="emphasis"><em>α<sub>i</sub></em></span>
and its service time distribution is an exponential distribution with rate
<span class="emphasis"><em>λ<sub>i</sub></em></span> (Allen,1990; Papadopolous et al.,1993; Trivedi,2002).
</p>
<p>
For instance, CPUs service-time distribution in a computing system has
often been observed to possess such a distribution (Rosin,1965). Also,
the arrival of different types of customer to a single queueing station
is often modeled as a hyperexponential distribution (Papadopolous et al.,1993).
Similarly, if a product manufactured in several parallel assembly lines
and the outputs are merged, the failure density of the overall product
is likely to be hyperexponential (Trivedi,2002).
</p>
<p>
Finally, since the hyperexponential distribution exhibits a high Coefficient
of Variation (CoV), that is a CoV &gt; 1, it is especially suited to fit
empirical data with large CoV (Feitelson,2014; Wolski et al.,2013) and
to approximate <a href="http://en.wikipedia.org/wiki/Long_tail" target="_top">long-tail
probability distributions</a> (Feldmann et al.,1998).
</p>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.related_distributions"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.related_distributions">Related
distributions</a>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
When the number of phases <span class="emphasis"><em>k</em></span> is equal to <code class="computeroutput"><span class="number">1</span></code>, the hyperexponential distribution
is simply an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
Distribution</a>.
</li>
<li class="listitem">
When the <span class="emphasis"><em>k</em></span> rates are all equal to <span class="emphasis"><em>λ</em></span>,
the hyperexponential distribution is simple an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
Distribution</a> with rate <span class="emphasis"><em>λ</em></span>.
</li>
</ul></div>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.examples"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.examples">Examples</a>
</h4>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.lifetime_of_appliances"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.lifetime_of_appliances">Lifetime
of Appliances</a>
</h5>
<p>
Suppose a customer is buying an appliance and is choosing at random between
an appliance with average lifetime of 10 years and an appliance with average
lifetime of 12 years. Assuming the lifetime of this appliance follows an
exponential distribution, the lifetime distribution of the purchased appliance
can be modeled as a hyperexponential distribution with phase probability
vector <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1/2,1/2)</em></span>
and rate vector <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(1/10,1/12)</em></span>
(Wolfram,2014).
</p>
<p>
In the rest of this section, we provide an example C++ implementation for
computing the average lifetime and the probability that the appliance will
work for more than 15 years.
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">hyperexponential</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10.0</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12.0</span> <span class="special">};</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Average lifetime: "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">he</span><span class="special">)</span>
<span class="special">&lt;&lt;</span> <span class="string">" years"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that the appliance will work for more than 15 years: "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">he</span><span class="special">,</span> <span class="number">15.0</span><span class="special">))</span>
<span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
The resulting output is:
</p>
<pre class="programlisting"><span class="identifier">Average</span> <span class="identifier">lifetime</span><span class="special">:</span> <span class="number">11</span> <span class="identifier">years</span>
<span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">the</span> <span class="identifier">appliance</span> <span class="identifier">will</span> <span class="identifier">work</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">15</span> <span class="identifier">years</span><span class="special">:</span> <span class="number">0.254817</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.workloads_of_private_cloud_compu"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.workloads_of_private_cloud_compu">Workloads
of Private Cloud Computing Systems</a>
</h5>
<p>
<a href="http://en.wikipedia.org/wiki/Cloud_computing" target="_top">Cloud computing</a>
has become a popular metaphor for dynamic and secure self-service access
to computational and storage capabilities. In (Wolski et al.,2013), the
authors analyze and model workloads gathered from enterprise-operated commercial
<a href="http://en.wikipedia.org/wiki/Cloud_computing#Private_cloud" target="_top">private
clouds</a> and show that 3-phase hyperexponential distributions (fitted
using the <a href="http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm" target="_top">Expectation
Maximization algorithm</a>) capture workload attributes accurately.
</p>
<p>
In this type of computing system, user requests consist in demanding the
provisioning of one or more <a href="http://en.wikipedia.org/wiki/Virtual_machine" target="_top">Virtual
Machines</a> (VMs). In particular, in (Wolski et al.,2013) the workload
experienced by each cloud system is a function of four distributions, one
for each of the following workload attributes:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<span class="emphasis"><em>Request Interarrival Time</em></span>: the amount of time
until the next request,
</li>
<li class="listitem">
<span class="emphasis"><em>VM Lifetime</em></span>: the time duration over which a VM
is provisioned to a physical machine,
</li>
<li class="listitem">
<span class="emphasis"><em>Request Size</em></span>: the number of VMs in the request,
and
</li>
<li class="listitem">
<span class="emphasis"><em>Core Count</em></span>: the CPU core count requested for each
VM.
</li>
</ul></div>
<p>
The authors assume that all VMs in a request have the same core count,
but request sizes and core counts can vary from request to request. Moreover,
all VMs within a request are assumed to have the same lifetime. Given these
assumptions, the authors build a statistical model for the request interarrival
time and VM lifetime attributes by fitting their respective data to a 3-phase
hyperexponential distribution.
</p>
<p>
In the following table, we show the sample mean and standard deviation
(SD), in seconds, of the request interarrival time and of the VM lifetime
distributions of the three datasets collected by authors:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Dataset
</p>
</th>
<th>
<p>
Mean Request Interarrival Time (SD)
</p>
</th>
<th>
<p>
Mean Multi-core VM Lifetime (SD)
</p>
</th>
<th>
<p>
Mean Single-core VM Lifetime (SD)
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
DS1
</p>
</td>
<td>
<p>
2202.1 (2.2e+04)
</p>
</td>
<td>
<p>
257173 (4.6e+05)
</p>
</td>
<td>
<p>
28754.4 (1.6e+05)
</p>
</td>
</tr>
<tr>
<td>
<p>
DS2
</p>
</td>
<td>
<p>
41285.7 (1.1e+05)
</p>
</td>
<td>
<p>
144669.0 (7.9e+05)
</p>
</td>
<td>
<p>
599815.0 (1.7e+06)
</p>
</td>
</tr>
<tr>
<td>
<p>
DS3
</p>
</td>
<td>
<p>
11238.8 (3.0e+04)
</p>
</td>
<td>
<p>
30739.2 (1.6e+05)
</p>
</td>
<td>
<p>
44447.8 (2.2e+05)
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
Whereas in the following table we show the hyperexponential distribution
parameters resulting from the fit:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Dataset
</p>
</th>
<th>
<p>
Request Interarrival Time
</p>
</th>
<th>
<p>
Multi-core VM Lifetime
</p>
</th>
<th>
<p>
Single-core VM Lifetime
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
DS1
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.34561,0.08648,0.56791),
<span class="bold"><strong>λ</strong></span>=(0.008,0.00005,0.02894)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.24667,0.37948,0.37385),
<span class="bold"><strong>λ</strong></span>=(0.00004,0.000002,0.00059)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.09325,0.22251,0.68424),
<span class="bold"><strong>λ</strong></span>=(0.000003,0.00109,0.00109)</em></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
DS2
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.38881,0.18227,0.42892),
<span class="bold"><strong>λ</strong></span>=(0.000006,0.05228,0.00081)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.42093,0.43960,0.13947),
<span class="bold"><strong>λ</strong></span>=(0.00186,0.00008,0.0000008)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.44885,0.30675,0.2444),
<span class="bold"><strong>λ</strong></span>=(0.00143,0.00005,0.0000004)</em></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
DS3
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.39442,0.24644,0.35914),
<span class="bold"><strong>λ</strong></span>=(0.00030,0.00003,0.00257)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.37621,0.14838,0.47541),
<span class="bold"><strong>λ</strong></span>=(0.00498,0.000005,0.00022)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.34131,0.12544,0.53325),
<span class="bold"><strong>λ</strong></span>=(0.000297,0.000003,0.00410)</em></span>
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
In the rest of this section, we provide an example C++ implementation for
computing some statistical properties of the fitted distributions for each
of the analyzed dataset.
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">string</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">ds_info</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">name</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">iat_sample_mean</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">iat_sample_sd</span><span class="special">;</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">iat_he</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">multi_lt_sample_mean</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">multi_lt_sample_sd</span><span class="special">;</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">multi_lt_he</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">single_lt_sample_mean</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">single_lt_sample_sd</span><span class="special">;</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">single_lt_he</span><span class="special">;</span>
<span class="special">};</span>
<span class="comment">// DS1 dataset</span>
<span class="identifier">ds_info</span> <span class="identifier">make_ds1</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">=</span> <span class="string">"DS1"</span><span class="special">;</span>
<span class="comment">// VM interarrival time distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.34561</span><span class="special">,</span><span class="number">0.08648</span><span class="special">,</span><span class="number">0.56791</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.0008</span><span class="special">,</span><span class="number">0.00005</span><span class="special">,</span><span class="number">0.02894</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_mean</span> <span class="special">=</span> <span class="number">2202.1</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_sd</span> <span class="special">=</span> <span class="number">2.2e+4</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">iat_fit_probs</span><span class="special">,</span> <span class="identifier">iat_fit_rates</span><span class="special">);</span>
<span class="comment">// Multi-core VM lifetime distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.24667</span><span class="special">,</span><span class="number">0.37948</span><span class="special">,</span><span class="number">0.37385</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.00004</span><span class="special">,</span><span class="number">0.000002</span><span class="special">,</span><span class="number">0.00059</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_mean</span> <span class="special">=</span> <span class="number">257173</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_sd</span> <span class="special">=</span> <span class="number">4.6e+5</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">multi_lt_fit_probs</span><span class="special">,</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">);</span>
<span class="comment">// Single-core VM lifetime distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.09325</span><span class="special">,</span><span class="number">0.22251</span><span class="special">,</span><span class="number">0.68424</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.000003</span><span class="special">,</span><span class="number">0.00109</span><span class="special">,</span><span class="number">0.00109</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_mean</span> <span class="special">=</span> <span class="number">28754.4</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_sd</span> <span class="special">=</span> <span class="number">1.6e+5</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">single_lt_fit_probs</span><span class="special">,</span> <span class="identifier">single_lt_fit_rates</span><span class="special">);</span>
<span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
<span class="special">}</span>
<span class="comment">// DS2 dataset</span>
<span class="identifier">ds_info</span> <span class="identifier">make_ds2</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">=</span> <span class="string">"DS2"</span><span class="special">;</span>
<span class="comment">// VM interarrival time distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.38881</span><span class="special">,</span><span class="number">0.18227</span><span class="special">,</span><span class="number">0.42892</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.000006</span><span class="special">,</span><span class="number">0.05228</span><span class="special">,</span><span class="number">0.00081</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_mean</span> <span class="special">=</span> <span class="number">41285.7</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_sd</span> <span class="special">=</span> <span class="number">1.1e+05</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">iat_fit_probs</span><span class="special">,</span> <span class="identifier">iat_fit_rates</span><span class="special">);</span>
<span class="comment">// Multi-core VM lifetime distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.42093</span><span class="special">,</span><span class="number">0.43960</span><span class="special">,</span><span class="number">0.13947</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.00186</span><span class="special">,</span><span class="number">0.00008</span><span class="special">,</span><span class="number">0.0000008</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_mean</span> <span class="special">=</span> <span class="number">144669.0</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_sd</span> <span class="special">=</span> <span class="number">7.9e+05</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">multi_lt_fit_probs</span><span class="special">,</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">);</span>
<span class="comment">// Single-core VM lifetime distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.44885</span><span class="special">,</span><span class="number">0.30675</span><span class="special">,</span><span class="number">0.2444</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.00143</span><span class="special">,</span><span class="number">0.00005</span><span class="special">,</span><span class="number">0.0000004</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_mean</span> <span class="special">=</span> <span class="number">599815.0</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_sd</span> <span class="special">=</span> <span class="number">1.7e+06</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">single_lt_fit_probs</span><span class="special">,</span> <span class="identifier">single_lt_fit_rates</span><span class="special">);</span>
<span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
<span class="special">}</span>
<span class="comment">// DS3 dataset</span>
<span class="identifier">ds_info</span> <span class="identifier">make_ds3</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">=</span> <span class="string">"DS3"</span><span class="special">;</span>
<span class="comment">// VM interarrival time distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.39442</span><span class="special">,</span><span class="number">0.24644</span><span class="special">,</span><span class="number">0.35914</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.00030</span><span class="special">,</span><span class="number">0.00003</span><span class="special">,</span><span class="number">0.00257</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_mean</span> <span class="special">=</span> <span class="number">11238.8</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_sd</span> <span class="special">=</span> <span class="number">3.0e+04</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">iat_fit_probs</span><span class="special">,</span> <span class="identifier">iat_fit_rates</span><span class="special">);</span>
<span class="comment">// Multi-core VM lifetime distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.37621</span><span class="special">,</span><span class="number">0.14838</span><span class="special">,</span><span class="number">0.47541</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.00498</span><span class="special">,</span><span class="number">0.000005</span><span class="special">,</span><span class="number">0.00022</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_mean</span> <span class="special">=</span> <span class="number">30739.2</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_sd</span> <span class="special">=</span> <span class="number">1.6e+05</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">multi_lt_fit_probs</span><span class="special">,</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">);</span>
<span class="comment">// Single-core VM lifetime distribution</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.34131</span><span class="special">,</span><span class="number">0.12544</span><span class="special">,</span><span class="number">0.53325</span> <span class="special">};</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.000297</span><span class="special">,</span><span class="number">0.000003</span><span class="special">,</span><span class="number">0.00410</span> <span class="special">};</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_mean</span> <span class="special">=</span> <span class="number">44447.8</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_sd</span> <span class="special">=</span> <span class="number">2.2e+05</span><span class="special">;</span>
<span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">single_lt_fit_probs</span><span class="special">,</span> <span class="identifier">single_lt_fit_rates</span><span class="special">);</span>
<span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">void</span> <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">ds_info</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">ds</span><span class="special">)</span>
<span class="special">{</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">secs_in_a_hour</span> <span class="special">=</span> <span class="number">3600</span><span class="special">;</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">secs_in_a_month</span> <span class="special">=</span> <span class="number">30</span> <span class="special">*</span> <span class="number">24</span> <span class="special">*</span> <span class="identifier">secs_in_a_hour</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"### "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"* Fitted Request Interarrival Time"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Mean (SD): "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" ("</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">") seconds."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - 99th Percentile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" seconds."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Probability that a VM will arrive within 30 minutes: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span><span class="special">,</span> <span class="identifier">secs_in_a_hour</span> <span class="special">/</span> <span class="number">2.0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Probability that a VM will arrive after 1 hour: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span><span class="special">,</span> <span class="identifier">secs_in_a_hour</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"* Fitted Multi-core VM Lifetime"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Mean (SD): "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" ("</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">") seconds."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - 99th Percentile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" seconds."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Probability that a VM will last for less than 1 month: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span><span class="special">,</span> <span class="identifier">secs_in_a_month</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Probability that a VM will last for more than 3 months: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span><span class="special">,</span> <span class="number">3.0</span><span class="special">*</span><span class="identifier">secs_in_a_month</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"* Fitted Single-core VM Lifetime"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Mean (SD): "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" ("</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">") seconds."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - 99th Percentile: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" seconds."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Probability that a VM will last for less than 1 month: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span><span class="special">,</span> <span class="identifier">secs_in_a_month</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">" - Probability that a VM will last for more than 3 months: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span><span class="special">,</span> <span class="number">3.0</span><span class="special">*</span><span class="identifier">secs_in_a_month</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds1</span><span class="special">());</span>
<span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds2</span><span class="special">());</span>
<span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds3</span><span class="special">());</span>
<span class="special">}</span>
</pre>
<p>
The resulting output (with floating-point precision set to 2) is:
</p>
<pre class="programlisting"><span class="special">###</span> <span class="identifier">DS1</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">2.2e+03</span> <span class="special">(</span><span class="number">8.1e+03</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">4.3e+04</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">within</span> <span class="number">30</span> <span class="identifier">minutes</span><span class="special">:</span> <span class="number">0.84</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">after</span> <span class="number">1</span> <span class="identifier">hour</span><span class="special">:</span> <span class="number">0.092</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Multi</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">2e+05</span> <span class="special">(</span><span class="number">3.9e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">1.8e+06</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">6.7e-08</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Single</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">3.2e+04</span> <span class="special">(</span><span class="number">1.4e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">7.4e+05</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">6.9e-12</span>
<span class="special">###</span> <span class="identifier">DS2</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">6.5e+04</span> <span class="special">(</span><span class="number">1.3e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">6.1e+05</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">within</span> <span class="number">30</span> <span class="identifier">minutes</span><span class="special">:</span> <span class="number">0.52</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">after</span> <span class="number">1</span> <span class="identifier">hour</span><span class="special">:</span> <span class="number">0.4</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Multi</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">1.8e+05</span> <span class="special">(</span><span class="number">6.4e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">3.3e+06</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">0.98</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">0.00028</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Single</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">6.2e+05</span> <span class="special">(</span><span class="number">1.6e+06</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">8e+06</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">0.91</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">0.011</span>
<span class="special">###</span> <span class="identifier">DS3</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">9.7e+03</span> <span class="special">(</span><span class="number">2.2e+04</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">1.1e+05</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">within</span> <span class="number">30</span> <span class="identifier">minutes</span><span class="special">:</span> <span class="number">0.53</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">after</span> <span class="number">1</span> <span class="identifier">hour</span><span class="special">:</span> <span class="number">0.36</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Multi</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">3.2e+04</span> <span class="special">(</span><span class="number">1e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">5.4e+05</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">1.9e-18</span>
<span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Single</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span>
<span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">4.3e+04</span> <span class="special">(</span><span class="number">1.6e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">8.4e+05</span> <span class="identifier">seconds</span><span class="special">.</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span>
<span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">9.3e-12</span>
</pre>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
The above results differ from the ones shown in Tables III, V, and VII
of (Wolski et al.,2013). We carefully double-checked them with Wolfram
Mathematica 10, which confirmed our results.
</p></td></tr>
</table></div>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h5"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.member_functions"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.member_functions">Member
Functions</a>
</h4>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h6"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.default_constructor"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.default_constructor">Default
Constructor</a>
</h5>
<pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">();</span>
</pre>
<p>
Constructs a <span class="emphasis"><em>1</em></span>-phase hyperexponential distribution
(i.e., an exponential distribution) with rate <code class="computeroutput"><span class="number">1</span></code>.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h7"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.constructor_from_iterators"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.constructor_from_iterators">Constructor
from Iterators</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">ProbIterT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateIterT</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbIterT</span> <span class="identifier">prob_first</span><span class="special">,</span> <span class="identifier">ProbIterT</span> <span class="identifier">prob_last</span><span class="special">,</span>
<span class="identifier">RateIterT</span> <span class="identifier">rate_first</span><span class="special">,</span> <span class="identifier">RateIterT</span> <span class="identifier">rate_last</span><span class="special">);</span>
</pre>
<p>
Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
vector</em></span> parameter given by the range defined by [<code class="computeroutput"><span class="identifier">prob_first</span></code>, <code class="computeroutput"><span class="identifier">prob_last</span></code>)
iterator pair, and <span class="emphasis"><em>rate vector</em></span> parameter given by
the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>,
<code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair.
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h8"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters">Parameters</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<code class="computeroutput"><span class="identifier">prob_first</span></code>, <code class="computeroutput"><span class="identifier">prob_last</span></code>: the range of non-negative
real elements representing the phase probabilities; elements are normalized
to sum to unity.
</li>
<li class="listitem">
<code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive
elements representing the rates.
</li>
</ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h9"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements">Type
Requirements</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">ProbIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT</span></code>: must meet the requirements
of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a>
concept.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h10"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example">Example</a>
</h6>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;</span> <span class="identifier">phase_prob</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;</span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span>
<span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">phase_prob</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">phase_prob</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">rates</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">rates</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h11"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_ranges_contain"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_ranges_contain">Construction
from Ranges/Containers</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">ProbRangeT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateRangeT</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbRangeT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">prob_range</span><span class="special">,</span>
<span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_range</span><span class="special">);</span>
</pre>
<p>
Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
vector</em></span> parameter given by the range defined by <code class="computeroutput"><span class="identifier">prob_range</span></code>, and <span class="emphasis"><em>rate vector</em></span>
parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>.
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
As an implementation detail, this constructor uses Boost's <a href="http://www.boost.org/doc/libs/release/libs/core/doc/html/core/enable_if.html" target="_top">enable_if/disable_if
mechanism</a> to disambiguate between this and other 2-argument constructors.
Refer to the source code for more details.
</p></td></tr>
</table></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h12"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters0">Parameters</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<code class="computeroutput"><span class="identifier">prob_range</span></code>: the range
of non-negative real elements representing the phase probabilities;
elements are normalized to sum to unity.
</li>
<li class="listitem">
<code class="computeroutput"><span class="identifier">rate_range</span></code>: the range
of positive real elements representing the rates.
</li>
</ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h13"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements0">Type
Requirements</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">ProbRangeT</span></code>, <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet the requirements
of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a>
concept: that includes native C++ arrays, standard library containers,
or a std::pair or iterators.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h14"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.examples0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.examples0">Examples</a>
</h6>
<pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;</span> <span class="identifier">phase_prob</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;</span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span>
<span class="identifier">hyperexponential</span> <span class="identifier">he1</span><span class="special">(</span><span class="identifier">phase_prob</span><span class="special">,</span> <span class="identifier">rates</span><span class="special">);</span> <span class="comment">// Construct from standard library container.</span>
<span class="keyword">double</span> <span class="identifier">phase_probs2</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">rates2</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span>
<span class="identifier">hyperexponential</span> <span class="identifier">he2</span><span class="special">(</span><span class="identifier">phase_probs2</span><span class="special">,</span> <span class="identifier">rates2</span><span class="special">);</span> <span class="comment">// Construct from native C++ array.</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h15"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_with_rates_iterator"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_with_rates_iterator">Construction
with rates-iterators (and all phase probabilities equal)</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RateIterT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateIterT2</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateIterT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_first</span><span class="special">,</span>
<span class="identifier">RateIterT2</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_last</span><span class="special">);</span>
</pre>
<p>
Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
parameter given by the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>,
<code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair,
and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase
probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
of the <span class="emphasis"><em>rate vector</em></span> and with each element set to <code class="computeroutput"><span class="number">1.0</span><span class="special">/</span><span class="identifier">n</span></code>).
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
As an implementation detail, this constructor uses Boost's <a href="http://www.boost.org/doc/libs/release/libs/core/doc/html/core/enable_if.html" target="_top">enable_if/disable_if
mechanism</a> to disambiguate between this and other 2-argument constructors.
Refer to the source code for more details.
</p></td></tr>
</table></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h16"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters1">Parameters</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive
elements representing the rates.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h17"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements1">Type
Requirements</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">RateIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT2</span></code>: must meet the requirements
of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a>
concept.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h18"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example0">Example</a>
</h6>
<pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;</span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span>
<span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">rates</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
<span class="identifier">BOOST_MATH_ASSERT</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// Phase probabilities will be equal and normalised to unity.</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h19"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_range"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_range">Construction
from a single range of rates (all phase probabilities will be equal)</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RateRangeT</span><span class="special">&gt;</span>
<span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">rate_range</span><span class="special">);</span>
</pre>
<p>
Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>,
and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase
probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
of the <span class="emphasis"><em>rate vector</em></span> and with each element set to <code class="computeroutput"><span class="number">1.0</span><span class="special">/</span><span class="identifier">n</span></code>).
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h20"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters2"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters2">Parameters</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">rate_range</span></code>: the range
of positive real elements representing the rates.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h21"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements2"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements2">Type
Requirements</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet
the requirements of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a>
concept: this includes native C++ array, standard library containers,
and a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> of iterators.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h22"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.examples1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.examples1">Examples</a>
</h6>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;</span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span>
<span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">);</span>
<span class="identifier">BOOST_MATH_ASSERT</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// Phase probabilities will be equal and normalised to unity.</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h23"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_initializer_li"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_initializer_li">Construction
from Initializer lists</a>
</h5>
<pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">l1</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">l2</span><span class="special">);</span>
</pre>
<p>
Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>rate
vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
defined by <code class="computeroutput"><span class="identifier">l2</span></code>.
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h24"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters3"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters3">Parameters</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list
of non-negative real elements representing the phase probabilities;
elements are normalized to sum to unity.
</li>
<li class="listitem">
<code class="computeroutput"><span class="identifier">l2</span></code>: the brace-init-list
of positive real elements representing the rates.
</li>
</ul></div>
<p>
The number of elements of the phase probabilities list and the rates list
must be the same.
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h25"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example1">Example</a>
</h6>
<pre class="programlisting"><span class="identifier">hyperexponential</span> <span class="identifier">he</span> <span class="special">=</span> <span class="special">{</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">},</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">}</span> <span class="special">};</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h26"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_initi"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_initi">Construction
from a single initializer list (all phase probabilities will be equal)</a>
</h5>
<pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">l1</span><span class="special">);</span>
</pre>
<p>
Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>phase
probability vector</em></span> set to the equal phase probabilities (i.e.,
to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
of the <span class="emphasis"><em>rate vector</em></span> and with each element set to <code class="computeroutput"><span class="number">1.0</span><span class="special">/</span><span class="identifier">n</span></code>).
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h27"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters4"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters4">Parameters</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list
of non-negative real elements representing the phase probabilities;
they are normalized to ensure that they sum to unity.
</li></ul></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h28"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example2"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example2">Example</a>
</h6>
<pre class="programlisting"><span class="identifier">hyperexponential</span> <span class="identifier">he</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span>
<span class="identifier">BOOST_MATH_ASSERT</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0.5</span><span class="special">);</span>
</pre>
<h5>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h29"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.accessors"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.accessors">Accessors</a>
</h5>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">num_phases</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Gets the number of phases of this distribution (the size of both the rate
and probability vectors).
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h30"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.return_value"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.return_value">Return
Value</a>
</h6>
<p>
An non-negative integer number representing the number of phases of this
distribution.
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">probabilities</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Gets the <span class="emphasis"><em>phase probability vector</em></span> parameter of this
distribution.
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
The returned probabilities are the <span class="bold"><strong>normalized</strong></span>
versions of the probability parameter values passed at construction time.
</p></td></tr>
</table></div>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h31"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.return_value0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.return_value0">Return
Value</a>
</h6>
<p>
A vector of non-negative real numbers representing the <span class="emphasis"><em>phase
probability vector</em></span> parameter of this distribution.
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">rates</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Gets the <span class="emphasis"><em>rate vector</em></span> parameter of this distribution.
</p>
<h6>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h32"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.return_value1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.return_value1">Return
Value</a>
</h6>
<p>
A vector of positive real numbers representing the <span class="emphasis"><em>rate vector</em></span>
parameter of this distribution.
</p>
<div class="warning"><table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td>
<th align="left">Warning</th>
</tr>
<tr><td align="left" valign="top">
<p>
The return type of these functions is a vector-by-value. This is deliberate
as we wish to hide the actual container used internally which may be
subject to future changes (for example to facilitate vectorization of
the cdf code etc). Users should note that some code that might otherwise
have been expected to work does not. For example, an attempt to output
the (normalized) probabilities:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">().</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">().</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">" "</span><span class="special">));</span>
</pre>
<p>
fails at compile or runtime because iterator types are incompatible,
but, for example,
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="char">' '</span> <span class="special">&lt;&lt;</span> <span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">1</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
outputs the expected values.
</p>
<p>
In general if you want to access a member of the returned container,
then assign to a variable first, and then access those members:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">();</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">t</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">t</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">" "</span><span class="special">));</span>
</pre>
</td></tr>
</table></div>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h33"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.non_member_accessor_functions"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.non_member_accessor_functions">Non-member
Accessor Functions</a>
</h4>
<p>
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
functions</a> that are generic to all distributions are supported:
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The formulae for calculating these are shown in the table below.
</p>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h34"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.accuracy"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.accuracy">Accuracy</a>
</h4>
<p>
The hyperexponential distribution is implemented in terms of the <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>
and as such should have very small errors, usually an <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>
or few.
</p>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h35"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.implementation"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.implementation">Implementation</a>
</h4>
<p>
In the following table:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(α<sub>1</sub>,...,α<sub>k</sub>)</em></span> is
the <span class="emphasis"><em>phase probability vector</em></span> parameter of the
<span class="emphasis"><em>k</em></span>-phase hyperexponential distribution,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(λ<sub>1</sub>,...,λ<sub>k</sub>)</em></span> is
the <span class="emphasis"><em>rate vector</em></span> parameter of the <span class="emphasis"><em>k</em></span>-phase
hyperexponential distribution,
</li>
<li class="listitem">
<span class="emphasis"><em>x</em></span> is the random variate.
</li>
</ul></div>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
Implementation Notes
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
support
</p>
</td>
<td>
<p>
<span class="emphasis"><em>x</em></span> ∈ [0,∞)
</p>
</td>
</tr>
<tr>
<td>
<p>
pdf
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_cdf.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
cdf complement
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_ccdf.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
No closed form available. Computed numerically.
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
No closed form available. Computed numerically.
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_mean.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_variance.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="number">0</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_skewness.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
kurtosis
</p>
</td>
<td>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_kurtosis.svg"></span>
</p></blockquote></div>
</td>
</tr>
<tr>
<td>
<p>
kurtosis excess
</p>
</td>
<td>
<p>
kurtosis <code class="computeroutput"><span class="special">-</span> <span class="number">3</span></code>
</p>
</td>
</tr>
</tbody>
</table></div>
<h4>
<a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h36"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.references"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.references">References</a>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
A.O. Allen, <span class="emphasis"><em>Probability, Statistics, and Queuing Theory with
Computer Science Applications, Second Edition</em></span>, Academic
Press, 1990.
</li>
<li class="listitem">
D.G. Feitelson, <span class="emphasis"><em>Workload Modeling for Computer Systems Performance
Evaluation</em></span>, Cambridge University Press, 2014
</li>
<li class="listitem">
A. Feldmann and W. Whitt, <span class="emphasis"><em>Fitting mixtures of exponentials
to long-tail distributions to analyze network performance models</em></span>,
Performance Evaluation 31(3-4):245, doi:10.1016/S0166-5316(97)00003-5,
1998.
</li>
<li class="listitem">
H.T. Papadopolous, C. Heavey and J. Browne, <span class="emphasis"><em>Queueing Theory
in Manufacturing Systems Analysis and Design</em></span>, Chapman &amp;
Hall/CRC, 1993, p. 35.
</li>
<li class="listitem">
R.F. Rosin, <span class="emphasis"><em>Determining a computing center environment</em></span>,
Communications of the ACM 8(7):463-468, 1965.
</li>
<li class="listitem">
K.S. Trivedi, <span class="emphasis"><em>Probability and Statistics with Reliability,
Queueing, and Computer Science Applications</em></span>, John Wiley
&amp; Sons, Inc., 2002.
</li>
<li class="listitem">
Wikipedia, <span class="emphasis"><em>Hyperexponential Distribution</em></span>, Online:
<a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">http://en.wikipedia.org/wiki/Hyperexponential_distribution</a>,
2014
</li>
<li class="listitem">
R. Wolski and J. Brevik, <span class="emphasis"><em>Using Parametric Models to Represent
Private Cloud Workloads</em></span>, IEEE TSC, PrePrint, DOI: <a href="http://doi.ieeecomputersociety.org/10.1109/TSC.2013.48" target="_top">10.1109/TSC.2013.48</a>,
2013.
</li>
<li class="listitem">
Wolfram Mathematica, <span class="emphasis"><em>Hyperexponential Distribution</em></span>,
Online: <a href="http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html" target="_top">http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html</a>,
2014.
</li>
</ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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</p>
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