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  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist"></a><a class="link" href="hyperexponential_dist.html" title="Hyperexponential Distribution">Hyperexponential
  28         Distribution</a>
  29 </h4></div></div></div>
  30 <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>
  31 <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>
  32
  33 <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>
  34           <span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;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>
  35 <span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span><span class="special">;</span>
  36
  37 <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>
  38
  39 <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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  40 <span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  45
  46    <span class="comment">// Constructors:</span>
  47    <span class="identifier">hyperexponential_distribution</span><span class="special">();</span>  <span class="comment">// Default.</span>
  48
  49    <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>
  50    <span class="identifier">hyperexponential_distribution</span><span class="special">(</span>  <span class="comment">// Default equal probabilities.</span>
  51                                  <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>
  52                                  <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>
  53
  54    <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>
  55    <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>
  56                                  <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>
  57
  58    <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>
  59    <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>
  60                                  <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>
  61
  62    <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>
  63    <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>
  64
  65  <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>
  66    <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>
  67    <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>
  68  <span class="preprocessor">#endif</span>
  69
  70    <span class="comment">// Accessors:</span>
  71    <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>
  72    <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>
  73    <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>
  74 <span class="special">};</span>
  75
  76 <span class="special">}}</span> <span class="comment">// namespaces</span>
  77 </pre>
  78 <div class="note"><table border="0" summary="Note">
  79 <tr>
  80 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  81 <th align="left">Note</th>
  82 </tr>
  83 <tr><td align="left" valign="top"><p>
  84             An implementation-defined mechanism is provided to avoid ambiguity between
  85             constructors accepting ranges, iterators and constants as parameters.
  86             This should be transparent to the user. See below and the header file
  87             hyperexponential.hpp for details and explanatory comments.
  88           </p></td></tr>
  89 </table></div>
  90 <p>
  91           The class type <code class="computeroutput"><span class="identifier">hyperexponential_distribution</span></code>
  92           represents a <a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">hyperexponential
  93           distribution</a>.
  94         </p>
  95 <p>
  96           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
  97           probability distribution</a> obtained as a mixture of <span class="emphasis"><em>k</em></span>
  98           <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>s.
  99           It is also referred to as <span class="emphasis"><em>mixed exponential distribution</em></span>
 100           or parallel <span class="emphasis"><em>k-phase exponential distribution</em></span>.
 101         </p>
 102 <p>
 103           A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is characterized
 104           by two parameters, namely a <span class="emphasis"><em>phase probability vector</em></span>
 105           <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(&#945;<sub>1</sub>,...,&#945;<sub>k</sub>)</em></span> and a
 106           <span class="emphasis"><em>rate vector</em></span> <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(&#955;<sub>1</sub>,...,&#955;<sub>k</sub>)</em></span>.
 107         </p>
 108 <p>
 109           The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
 110           density function</a> for random variate <span class="emphasis"><em>x</em></span> in a
 111           hyperexponential distribution is given by:
 112         </p>
 113 <div class="blockquote"><blockquote class="blockquote"><p>
 114             <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span>
 115
 116           </p></blockquote></div>
 117 <p>
 118           The following graph illustrates the PDF of the hyperexponential distribution
 119           with five different parameters, namely:
 120         </p>
 121 <div class="orderedlist"><ol class="orderedlist" type="1">
 122 <li class="listitem">
 123               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(1.0)</em></span> (which degenerates to a simple
 124               exponential distribution),
 125             </li>
 126 <li class="listitem">
 127               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.1, 0.9)</em></span> and
 128               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 129             </li>
 130 <li class="listitem">
 131               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.9, 0.1)</em></span> and
 132               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 133             </li>
 134 <li class="listitem">
 135               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.2, 0.3, 0.5)</em></span>
 136               and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.0, 1.5)</em></span>,
 137             </li>
 138 <li class="listitem">
 139               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.5, 0.3, 0.2)</em></span>
 140               and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.0, 1.5)</em></span>.
 141             </li>
 142 </ol></div>
 143 <div class="blockquote"><blockquote class="blockquote"><p>
 144             <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf.svg" align="middle"></span>
 145
 146           </p></blockquote></div>
 147 <p>
 148           Also, the following graph illustrates the PDF of the hyperexponential distribution
 149           (solid lines) where only the <span class="emphasis"><em>phase probability vector</em></span>
 150           changes together with the PDF of the two limiting exponential distributions
 151           (dashed lines):
 152         </p>
 153 <div class="orderedlist"><ol class="orderedlist" type="1">
 154 <li class="listitem">
 155               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.1, 0.9)</em></span> and
 156               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 157             </li>
 158 <li class="listitem">
 159               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.6, 0.4)</em></span> and
 160               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 161             </li>
 162 <li class="listitem">
 163               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.9, 0.1)</em></span> and
 164               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 165             </li>
 166 <li class="listitem">
 167               Exponential distribution with parameter <span class="emphasis"><em>&#955;=0.5</em></span>,
 168             </li>
 169 <li class="listitem">
 170               Exponential distribution with parameter <span class="emphasis"><em>&#955;=1.5</em></span>.
 171             </li>
 172 </ol></div>
 173 <p>
 174           As expected, as the first element <span class="emphasis"><em>&#945;<sub>1</sub></em></span> of the <span class="emphasis"><em>phase
 175           probability vector</em></span> approaches to <span class="emphasis"><em>1</em></span> (or,
 176           equivalently, <span class="emphasis"><em>&#945;<sub>2</sub></em></span> approaches to <span class="emphasis"><em>0</em></span>),
 177           the resulting hyperexponential distribution nears the exponential distribution
 178           with parameter <span class="emphasis"><em>&#955;=0.5</em></span>. Conversely, as the first element
 179           <span class="emphasis"><em>&#945;<sub>2</sub></em></span> of the <span class="emphasis"><em>phase probability vector</em></span>
 180           approaches to <span class="emphasis"><em>1</em></span> (or, equivalently, <span class="emphasis"><em>&#945;<sub>1</sub></em></span>
 181           approaches to <span class="emphasis"><em>0</em></span>), the resulting hyperexponential distribution
 182           nears the exponential distribution with parameter <span class="emphasis"><em>&#955;=1.5</em></span>.
 183         </p>
 184 <div class="blockquote"><blockquote class="blockquote"><p>
 185             <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.svg" align="middle"></span>
 186
 187           </p></blockquote></div>
 188 <p>
 189           Finally, the following graph compares the PDF of the hyperexponential distribution
 190           with different number of phases but with the same mean value equal to
 191           <span class="emphasis"><em>2</em></span>:
 192         </p>
 193 <div class="orderedlist"><ol class="orderedlist" type="1">
 194 <li class="listitem">
 195               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(2.0)</em></span> (which degenerates to a simple
 196               exponential distribution),
 197             </li>
 198 <li class="listitem">
 199               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.5, 0.5)</em></span> and
 200               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.3, 1.5)</em></span>,
 201             </li>
 202 <li class="listitem">
 203               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1.0/3.0, 1.0/3.0, 1.0/3.0)</em></span>
 204               and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.2, 1.5, 3.0)</em></span>,
 205             </li>
 206 </ol></div>
 207 <div class="blockquote"><blockquote class="blockquote"><p>
 208             <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samemean.svg" align="middle"></span>
 209
 210           </p></blockquote></div>
 211 <p>
 212           As can be noted, even if the three distributions have the same mean value,
 213           the two hyperexponential distributions have a <span class="emphasis"><em>longer</em></span>
 214           tail with respect to the one of the exponential distribution. Indeed, the
 215           hyperexponential distribution has a larger variability than the exponential
 216           distribution, thus resulting in a <a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" target="_top">Coefficient
 217           of Variation</a> greater than <span class="emphasis"><em>1</em></span> (as opposed to
 218           the one of the exponential distribution which is exactly <span class="emphasis"><em>1</em></span>).
 219         </p>
 220 <h4>
 221 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h0"></a>
 222           <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>
 223         </h4>
 224 <p>
 225           A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is frequently
 226           used in <a href="http://en.wikipedia.org/wiki/Queueing_theory" target="_top">queueing
 227           theory</a> to model the distribution of the superposition of <span class="emphasis"><em>k</em></span>
 228           independent events, like, for instance, the service time distribution of
 229           a queueing station with <span class="emphasis"><em>k</em></span> servers in parallel where
 230           the <span class="emphasis"><em>i</em></span>-th server is chosen with probability <span class="emphasis"><em>&#945;<sub>i</sub></em></span>
 231           and its service time distribution is an exponential distribution with rate
 232           <span class="emphasis"><em>&#955;<sub>i</sub></em></span> (Allen,1990; Papadopolous et al.,1993; Trivedi,2002).
 233         </p>
 234 <p>
 235           For instance, CPUs service-time distribution in a computing system has
 236           often been observed to possess such a distribution (Rosin,1965). Also,
 237           the arrival of different types of customer to a single queueing station
 238           is often modeled as a hyperexponential distribution (Papadopolous et al.,1993).
 239           Similarly, if a product manufactured in several parallel assembly lines
 240           and the outputs are merged, the failure density of the overall product
 241           is likely to be hyperexponential (Trivedi,2002).
 242         </p>
 243 <p>
 244           Finally, since the hyperexponential distribution exhibits a high Coefficient
 245           of Variation (CoV), that is a CoV &gt; 1, it is especially suited to fit
 246           empirical data with large CoV (Feitelson,2014; Wolski et al.,2013) and
 247           to approximate <a href="http://en.wikipedia.org/wiki/Long_tail" target="_top">long-tail
 248           probability distributions</a> (Feldmann et al.,1998).
 249         </p>
 250 <h4>
 251 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h1"></a>
 252           <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
 253           distributions</a>
 254         </h4>
 255 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 256 <li class="listitem">
 257               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
 258               is simply an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
 259               Distribution</a>.
 260             </li>
 261 <li class="listitem">
 262               When the <span class="emphasis"><em>k</em></span> rates are all equal to <span class="emphasis"><em>&#955;</em></span>,
 263               the hyperexponential distribution is simple an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
 264               Distribution</a> with rate <span class="emphasis"><em>&#955;</em></span>.
 265             </li>
 266 </ul></div>
 267 <h4>
 268 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h2"></a>
 269           <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>
 270         </h4>
 271 <h5>
 272 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h3"></a>
 273           <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
 274           of Appliances</a>
 275         </h5>
 276 <p>
 277           Suppose a customer is buying an appliance and is choosing at random between
 278           an appliance with average lifetime of 10 years and an appliance with average
 279           lifetime of 12 years. Assuming the lifetime of this appliance follows an
 280           exponential distribution, the lifetime distribution of the purchased appliance
 281           can be modeled as a hyperexponential distribution with phase probability
 282           vector <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1/2,1/2)</em></span>
 283           and rate vector <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(1/10,1/12)</em></span>
 284           (Wolfram,2014).
 285         </p>
 286 <p>
 287           In the rest of this section, we provide an example C++ implementation for
 288           computing the average lifetime and the probability that the appliance will
 289           work for more than 15 years.
 290         </p>
 291 <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>
 292 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 293 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 294 <span class="special">{</span>
 295    <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>
 296
 297    <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>
 298
 299    <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>
 300       <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>
 301       <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>
 302    <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>
 303       <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>
 304       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 305 <span class="special">}</span>
 306 </pre>
 307 <p>
 308           The resulting output is:
 309         </p>
 310 <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>
 311 <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>
 312 </pre>
 313 <h5>
 314 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h4"></a>
 315           <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
 316           of Private Cloud Computing Systems</a>
 317         </h5>
 318 <p>
 319           <a href="http://en.wikipedia.org/wiki/Cloud_computing" target="_top">Cloud computing</a>
 320           has become a popular metaphor for dynamic and secure self-service access
 321           to computational and storage capabilities. In (Wolski et al.,2013), the
 322           authors analyze and model workloads gathered from enterprise-operated commercial
 323           <a href="http://en.wikipedia.org/wiki/Cloud_computing#Private_cloud" target="_top">private
 324           clouds</a> and show that 3-phase hyperexponential distributions (fitted
 325           using the <a href="http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm" target="_top">Expectation
 326           Maximization algorithm</a>) capture workload attributes accurately.
 327         </p>
 328 <p>
 329           In this type of computing system, user requests consist in demanding the
 330           provisioning of one or more <a href="http://en.wikipedia.org/wiki/Virtual_machine" target="_top">Virtual
 331           Machines</a> (VMs). In particular, in (Wolski et al.,2013) the workload
 332           experienced by each cloud system is a function of four distributions, one
 333           for each of the following workload attributes:
 334         </p>
 335 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 336 <li class="listitem">
 337               <span class="emphasis"><em>Request Interarrival Time</em></span>: the amount of time
 338               until the next request,
 339             </li>
 340 <li class="listitem">
 341               <span class="emphasis"><em>VM Lifetime</em></span>: the time duration over which a VM
 342               is provisioned to a physical machine,
 343             </li>
 344 <li class="listitem">
 345               <span class="emphasis"><em>Request Size</em></span>: the number of VMs in the request,
 346               and
 347             </li>
 348 <li class="listitem">
 349               <span class="emphasis"><em>Core Count</em></span>: the CPU core count requested for each
 350               VM.
 351             </li>
 352 </ul></div>
 353 <p>
 354           The authors assume that all VMs in a request have the same core count,
 355           but request sizes and core counts can vary from request to request. Moreover,
 356           all VMs within a request are assumed to have the same lifetime. Given these
 357           assumptions, the authors build a statistical model for the request interarrival
 358           time and VM lifetime attributes by fitting their respective data to a 3-phase
 359           hyperexponential distribution.
 360         </p>
 361 <p>
 362           In the following table, we show the sample mean and standard deviation
 363           (SD), in seconds, of the request interarrival time and of the VM lifetime
 364           distributions of the three datasets collected by authors:
 365         </p>
 366 <div class="informaltable"><table class="table">
 367 <colgroup>
 368 <col>
 369 <col>
 370 <col>
 371 <col>
 372 </colgroup>
 373 <thead><tr>
 374 <th>
 375                   <p>
 376                     Dataset
 377                   </p>
 378                 </th>
 379 <th>
 380                   <p>
 381                     Mean Request Interarrival Time (SD)
 382                   </p>
 383                 </th>
 384 <th>
 385                   <p>
 386                     Mean Multi-core VM Lifetime (SD)
 387                   </p>
 388                 </th>
 389 <th>
 390                   <p>
 391                     Mean Single-core VM Lifetime (SD)
 392                   </p>
 393                 </th>
 394 </tr></thead>
 395 <tbody>
 396 <tr>
 397 <td>
 398                   <p>
 399                     DS1
 400                   </p>
 401                 </td>
 402 <td>
 403                   <p>
 404                     2202.1 (2.2e+04)
 405                   </p>
 406                 </td>
 407 <td>
 408                   <p>
 409                     257173 (4.6e+05)
 410                   </p>
 411                 </td>
 412 <td>
 413                   <p>
 414                     28754.4 (1.6e+05)
 415                   </p>
 416                 </td>
 417 </tr>
 418 <tr>
 419 <td>
 420                   <p>
 421                     DS2
 422                   </p>
 423                 </td>
 424 <td>
 425                   <p>
 426                     41285.7 (1.1e+05)
 427                   </p>
 428                 </td>
 429 <td>
 430                   <p>
 431                     144669.0 (7.9e+05)
 432                   </p>
 433                 </td>
 434 <td>
 435                   <p>
 436                     599815.0 (1.7e+06)
 437                   </p>
 438                 </td>
 439 </tr>
 440 <tr>
 441 <td>
 442                   <p>
 443                     DS3
 444                   </p>
 445                 </td>
 446 <td>
 447                   <p>
 448                     11238.8 (3.0e+04)
 449                   </p>
 450                 </td>
 451 <td>
 452                   <p>
 453                     30739.2 (1.6e+05)
 454                   </p>
 455                 </td>
 456 <td>
 457                   <p>
 458                     44447.8 (2.2e+05)
 459                   </p>
 460                 </td>
 461 </tr>
 462 </tbody>
 463 </table></div>
 464 <p>
 465           Whereas in the following table we show the hyperexponential distribution
 466           parameters resulting from the fit:
 467         </p>
 468 <div class="informaltable"><table class="table">
 469 <colgroup>
 470 <col>
 471 <col>
 472 <col>
 473 <col>
 474 </colgroup>
 475 <thead><tr>
 476 <th>
 477                   <p>
 478                     Dataset
 479                   </p>
 480                 </th>
 481 <th>
 482                   <p>
 483                     Request Interarrival Time
 484                   </p>
 485                 </th>
 486 <th>
 487                   <p>
 488                     Multi-core VM Lifetime
 489                   </p>
 490                 </th>
 491 <th>
 492                   <p>
 493                     Single-core VM Lifetime
 494                   </p>
 495                 </th>
 496 </tr></thead>
 497 <tbody>
 498 <tr>
 499 <td>
 500                   <p>
 501                     DS1
 502                   </p>
 503                 </td>
 504 <td>
 505                   <p>
 506                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.34561,0.08648,0.56791),
 507                     <span class="bold"><strong>&#955;</strong></span>=(0.008,0.00005,0.02894)</em></span>
 508                   </p>
 509                 </td>
 510 <td>
 511                   <p>
 512                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.24667,0.37948,0.37385),
 513                     <span class="bold"><strong>&#955;</strong></span>=(0.00004,0.000002,0.00059)</em></span>
 514                   </p>
 515                 </td>
 516 <td>
 517                   <p>
 518                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.09325,0.22251,0.68424),
 519                     <span class="bold"><strong>&#955;</strong></span>=(0.000003,0.00109,0.00109)</em></span>
 520                   </p>
 521                 </td>
 522 </tr>
 523 <tr>
 524 <td>
 525                   <p>
 526                     DS2
 527                   </p>
 528                 </td>
 529 <td>
 530                   <p>
 531                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.38881,0.18227,0.42892),
 532                     <span class="bold"><strong>&#955;</strong></span>=(0.000006,0.05228,0.00081)</em></span>
 533                   </p>
 534                 </td>
 535 <td>
 536                   <p>
 537                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.42093,0.43960,0.13947),
 538                     <span class="bold"><strong>&#955;</strong></span>=(0.00186,0.00008,0.0000008)</em></span>
 539                   </p>
 540                 </td>
 541 <td>
 542                   <p>
 543                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.44885,0.30675,0.2444),
 544                     <span class="bold"><strong>&#955;</strong></span>=(0.00143,0.00005,0.0000004)</em></span>
 545                   </p>
 546                 </td>
 547 </tr>
 548 <tr>
 549 <td>
 550                   <p>
 551                     DS3
 552                   </p>
 553                 </td>
 554 <td>
 555                   <p>
 556                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.39442,0.24644,0.35914),
 557                     <span class="bold"><strong>&#955;</strong></span>=(0.00030,0.00003,0.00257)</em></span>
 558                   </p>
 559                 </td>
 560 <td>
 561                   <p>
 562                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.37621,0.14838,0.47541),
 563                     <span class="bold"><strong>&#955;</strong></span>=(0.00498,0.000005,0.00022)</em></span>
 564                   </p>
 565                 </td>
 566 <td>
 567                   <p>
 568                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.34131,0.12544,0.53325),
 569                     <span class="bold"><strong>&#955;</strong></span>=(0.000297,0.000003,0.00410)</em></span>
 570                   </p>
 571                 </td>
 572 </tr>
 573 </tbody>
 574 </table></div>
 575 <p>
 576           In the rest of this section, we provide an example C++ implementation for
 577           computing some statistical properties of the fitted distributions for each
 578           of the analyzed dataset.
 579         </p>
 580 <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>
 581 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 582 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">string</span><span class="special">&gt;</span>
 583
 584 <span class="keyword">struct</span> <span class="identifier">ds_info</span>
 585 <span class="special">{</span>
 586    <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">name</span><span class="special">;</span>
 587    <span class="keyword">double</span> <span class="identifier">iat_sample_mean</span><span class="special">;</span>
 588    <span class="keyword">double</span> <span class="identifier">iat_sample_sd</span><span class="special">;</span>
 589    <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>
 590    <span class="keyword">double</span> <span class="identifier">multi_lt_sample_mean</span><span class="special">;</span>
 591    <span class="keyword">double</span> <span class="identifier">multi_lt_sample_sd</span><span class="special">;</span>
 592    <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>
 593    <span class="keyword">double</span> <span class="identifier">single_lt_sample_mean</span><span class="special">;</span>
 594    <span class="keyword">double</span> <span class="identifier">single_lt_sample_sd</span><span class="special">;</span>
 595    <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>
 596 <span class="special">};</span>
 597
 598 <span class="comment">// DS1 dataset</span>
 599 <span class="identifier">ds_info</span> <span class="identifier">make_ds1</span><span class="special">()</span>
 600 <span class="special">{</span>
 601    <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
 602
 603    <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>
 604
 605    <span class="comment">// VM interarrival time distribution</span>
 606    <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>
 607    <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>
 608    <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>
 609    <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>
 610    <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>
 611
 612    <span class="comment">// Multi-core VM lifetime distribution</span>
 613    <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>
 614    <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>
 615    <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>
 616    <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>
 617    <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>
 618
 619    <span class="comment">// Single-core VM lifetime distribution</span>
 620    <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>
 621    <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>
 622    <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>
 623    <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>
 624    <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>
 625
 626    <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
 627 <span class="special">}</span>
 628
 629 <span class="comment">// DS2 dataset</span>
 630 <span class="identifier">ds_info</span> <span class="identifier">make_ds2</span><span class="special">()</span>
 631 <span class="special">{</span>
 632    <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
 633
 634    <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>
 635
 636    <span class="comment">// VM interarrival time distribution</span>
 637    <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>
 638    <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>
 639    <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>
 640    <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>
 641    <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>
 642
 643    <span class="comment">// Multi-core VM lifetime distribution</span>
 644    <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>
 645    <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>
 646    <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>
 647    <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>
 648    <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>
 649
 650    <span class="comment">// Single-core VM lifetime distribution</span>
 651    <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>
 652    <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>
 653    <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>
 654    <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>
 655    <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>
 656
 657    <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
 658 <span class="special">}</span>
 659
 660 <span class="comment">// DS3 dataset</span>
 661 <span class="identifier">ds_info</span> <span class="identifier">make_ds3</span><span class="special">()</span>
 662 <span class="special">{</span>
 663    <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
 664
 665    <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>
 666
 667    <span class="comment">// VM interarrival time distribution</span>
 668    <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>
 669    <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>
 670    <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>
 671    <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>
 672    <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>
 673
 674    <span class="comment">// Multi-core VM lifetime distribution</span>
 675    <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>
 676    <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>
 677    <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>
 678    <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>
 679    <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>
 680
 681    <span class="comment">// Single-core VM lifetime distribution</span>
 682    <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>
 683    <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>
 684    <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>
 685    <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>
 686    <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>
 687
 688    <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
 689 <span class="special">}</span>
 690
 691 <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>
 692 <span class="special">{</span>
 693    <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>
 694    <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>
 695
 696    <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>
 697    <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>
 698    <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>
 699    <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>
 700    <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>
 701    <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>
 702    <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>
 703    <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>
 704    <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>
 705    <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>
 706    <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>
 707    <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>
 708    <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>
 709    <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>
 710    <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>
 711    <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>
 712 <span class="special">}</span>
 713
 714 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 715 <span class="special">{</span>
 716    <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds1</span><span class="special">());</span>
 717
 718    <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds2</span><span class="special">());</span>
 719
 720    <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds3</span><span class="special">());</span>
 721 <span class="special">}</span>
 722 </pre>
 723 <p>
 724           The resulting output (with floating-point precision set to 2) is:
 725         </p>
 726 <pre class="programlisting"><span class="special">###</span> <span class="identifier">DS1</span>
 727 <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
 728  <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>
 729  <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>
 730  <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>
 731  <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>
 732 <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>
 733  <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>
 734  <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>
 735  <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>
 736  <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>
 737 <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>
 738  <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>
 739  <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>
 740  <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>
 741  <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>
 742 <span class="special">###</span> <span class="identifier">DS2</span>
 743 <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
 744  <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>
 745  <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>
 746  <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>
 747  <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>
 748 <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>
 749  <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>
 750  <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>
 751  <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>
 752  <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>
 753 <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>
 754  <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>
 755  <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>
 756  <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>
 757  <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>
 758 <span class="special">###</span> <span class="identifier">DS3</span>
 759 <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
 760  <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>
 761  <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>
 762  <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>
 763  <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>
 764 <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>
 765  <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>
 766  <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>
 767  <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>
 768  <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>
 769 <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>
 770  <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>
 771  <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>
 772  <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>
 773  <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>
 774 </pre>
 775 <div class="note"><table border="0" summary="Note">
 776 <tr>
 777 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 778 <th align="left">Note</th>
 779 </tr>
 780 <tr><td align="left" valign="top"><p>
 781             The above results differ from the ones shown in Tables III, V, and VII
 782             of (Wolski et al.,2013). We carefully double-checked them with Wolfram
 783             Mathematica 10, which confirmed our results.
 784           </p></td></tr>
 785 </table></div>
 786 <h4>
 787 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h5"></a>
 788           <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
 789           Functions</a>
 790         </h4>
 791 <h5>
 792 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h6"></a>
 793           <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
 794           Constructor</a>
 795         </h5>
 796 <pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">();</span>
 797 </pre>
 798 <p>
 799           Constructs a <span class="emphasis"><em>1</em></span>-phase hyperexponential distribution
 800           (i.e., an exponential distribution) with rate <code class="computeroutput"><span class="number">1</span></code>.
 801         </p>
 802 <h5>
 803 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h7"></a>
 804           <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
 805           from Iterators</a>
 806         </h5>
 807 <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>
 808 <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>
 809                               <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>
 810 </pre>
 811 <p>
 812           Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
 813           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>)
 814           iterator pair, and <span class="emphasis"><em>rate vector</em></span> parameter given by
 815           the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>,
 816           <code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair.
 817         </p>
 818 <h6>
 819 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h8"></a>
 820           <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>
 821         </h6>
 822 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 823 <li class="listitem">
 824               <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
 825               real elements representing the phase probabilities; elements are normalized
 826               to sum to unity.
 827             </li>
 828 <li class="listitem">
 829               <code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive
 830               elements representing the rates.
 831             </li>
 832 </ul></div>
 833 <h6>
 834 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h9"></a>
 835           <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
 836           Requirements</a>
 837         </h6>
 838 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 839               <code class="computeroutput"><span class="identifier">ProbIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT</span></code>: must meet the requirements
 840               of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a>
 841               concept.
 842             </li></ul></div>
 843 <h6>
 844 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h10"></a>
 845           <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>
 846         </h6>
 847 <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>
 848 <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>
 849
 850 <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>
 851 </pre>
 852 <h5>
 853 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h11"></a>
 854           <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
 855           from Ranges/Containers</a>
 856         </h5>
 857 <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>
 858 <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>
 859                               <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>
 860 </pre>
 861 <p>
 862           Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
 863           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>
 864           parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>.
 865         </p>
 866 <div class="note"><table border="0" summary="Note">
 867 <tr>
 868 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 869 <th align="left">Note</th>
 870 </tr>
 871 <tr><td align="left" valign="top"><p>
 872             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
 873             mechanism</a> to disambiguate between this and other 2-argument constructors.
 874             Refer to the source code for more details.
 875           </p></td></tr>
 876 </table></div>
 877 <h6>
 878 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h12"></a>
 879           <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>
 880         </h6>
 881 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 882 <li class="listitem">
 883               <code class="computeroutput"><span class="identifier">prob_range</span></code>: the range
 884               of non-negative real elements representing the phase probabilities;
 885               elements are normalized to sum to unity.
 886             </li>
 887 <li class="listitem">
 888               <code class="computeroutput"><span class="identifier">rate_range</span></code>: the range
 889               of positive real elements representing the rates.
 890             </li>
 891 </ul></div>
 892 <h6>
 893 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h13"></a>
 894           <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
 895           Requirements</a>
 896         </h6>
 897 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 898               <code class="computeroutput"><span class="identifier">ProbRangeT</span></code>, <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet the requirements
 899               of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a>
 900               concept: that includes native C++ arrays, standard library containers,
 901               or a std::pair or iterators.
 902             </li></ul></div>
 903 <h6>
 904 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h14"></a>
 905           <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>
 906         </h6>
 907 <pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span>
 908 <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>
 909 <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>
 910
 911 <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>
 912
 913 <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>
 914 <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>
 915
 916 <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>
 917 </pre>
 918 <h5>
 919 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h15"></a>
 920           <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
 921           with rates-iterators (and all phase probabilities equal)</a>
 922         </h5>
 923 <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>
 924 <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>
 925                               <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>
 926 </pre>
 927 <p>
 928           Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
 929           parameter given by the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>,
 930           <code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair,
 931           and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase
 932           probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
 933           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>).
 934         </p>
 935 <div class="note"><table border="0" summary="Note">
 936 <tr>
 937 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 938 <th align="left">Note</th>
 939 </tr>
 940 <tr><td align="left" valign="top"><p>
 941             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
 942             mechanism</a> to disambiguate between this and other 2-argument constructors.
 943             Refer to the source code for more details.
 944           </p></td></tr>
 945 </table></div>
 946 <h6>
 947 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h16"></a>
 948           <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>
 949         </h6>
 950 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 951               <code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive
 952               elements representing the rates.
 953             </li></ul></div>
 954 <h6>
 955 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h17"></a>
 956           <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
 957           Requirements</a>
 958         </h6>
 959 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 960               <code class="computeroutput"><span class="identifier">RateIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT2</span></code>: must meet the requirements
 961               of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a>
 962               concept.
 963             </li></ul></div>
 964 <h6>
 965 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h18"></a>
 966           <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>
 967         </h6>
 968 <pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span>
 969 <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>
 970
 971 <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>
 972
 973 <span class="identifier">BOOST_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>
 974 </pre>
 975 <h5>
 976 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h19"></a>
 977           <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
 978           from a single range of rates (all phase probabilities will be equal)</a>
 979         </h5>
 980 <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>
 981 <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>
 982 </pre>
 983 <p>
 984           Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
 985           parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>,
 986           and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase
 987           probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
 988           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>).
 989         </p>
 990 <h6>
 991 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h20"></a>
 992           <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>
 993         </h6>
 994 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 995               <code class="computeroutput"><span class="identifier">rate_range</span></code>: the range
 996               of positive real elements representing the rates.
 997             </li></ul></div>
 998 <h6>
 999 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h21"></a>
1000           <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
1001           Requirements</a>
1002         </h6>
1003 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
1004               <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet
1005               the requirements of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a>
1006               concept: this includes native C++ array, standard library containers,
1007               and a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> of iterators.
1008             </li></ul></div>
1009 <h6>
1010 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h22"></a>
1011           <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>
1012         </h6>
1013 <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>
1014
1015 <span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">);</span>
1016
1017 <span class="identifier">BOOST_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>
1018 </pre>
1019 <h5>
1020 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h23"></a>
1021           <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
1022           from Initializer lists</a>
1023         </h5>
1024 <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>
1025 </pre>
1026 <p>
1027           Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
1028           vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
1029           defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>rate
1030           vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
1031           defined by <code class="computeroutput"><span class="identifier">l2</span></code>.
1032         </p>
1033 <h6>
1034 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h24"></a>
1035           <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>
1036         </h6>
1037 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1038 <li class="listitem">
1039               <code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list
1040               of non-negative real elements representing the phase probabilities;
1041               elements are normalized to sum to unity.
1042             </li>
1043 <li class="listitem">
1044               <code class="computeroutput"><span class="identifier">l2</span></code>: the brace-init-list
1045               of positive real elements representing the rates.
1046             </li>
1047 </ul></div>
1048 <p>
1049           The number of elements of the phase probabilities list and the rates list
1050           must be the same.
1051         </p>
1052 <h6>
1053 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h25"></a>
1054           <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>
1055         </h6>
1056 <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>
1057 </pre>
1058 <h5>
1059 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h26"></a>
1060           <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
1061           from a single initializer list (all phase probabilities will be equal)</a>
1062         </h5>
1063 <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>
1064 </pre>
1065 <p>
1066           Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
1067           parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
1068           defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>phase
1069           probability vector</em></span> set to the equal phase probabilities (i.e.,
1070           to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
1071           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>).
1072         </p>
1073 <h6>
1074 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h27"></a>
1075           <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>
1076         </h6>
1077 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
1078               <code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list
1079               of non-negative real elements representing the phase probabilities;
1080               they are normalized to ensure that they sum to unity.
1081             </li></ul></div>
1082 <h6>
1083 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h28"></a>
1084           <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>
1085         </h6>
1086 <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>
1087
1088 <span class="identifier">BOOST_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>
1089 </pre>
1090 <h5>
1091 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h29"></a>
1092           <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>
1093         </h5>
1094 <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>
1095 </pre>
1096 <p>
1097           Gets the number of phases of this distribution (the size of both the rate
1098           and probability vectors).
1099         </p>
1100 <h6>
1101 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h30"></a>
1102           <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
1103           Value</a>
1104         </h6>
1105 <p>
1106           An non-negative integer number representing the number of phases of this
1107           distribution.
1108         </p>
1109 <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>
1110 </pre>
1111 <p>
1112           Gets the <span class="emphasis"><em>phase probability vector</em></span> parameter of this
1113           distribution.
1114         </p>
1115 <div class="note"><table border="0" summary="Note">
1116 <tr>
1117 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
1118 <th align="left">Note</th>
1119 </tr>
1120 <tr><td align="left" valign="top"><p>
1121             The returned probabilities are the <span class="bold"><strong>normalized</strong></span>
1122             versions of the probability parameter values passed at construction time.
1123           </p></td></tr>
1124 </table></div>
1125 <h6>
1126 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h31"></a>
1127           <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
1128           Value</a>
1129         </h6>
1130 <p>
1131           A vector of non-negative real numbers representing the <span class="emphasis"><em>phase
1132           probability vector</em></span> parameter of this distribution.
1133         </p>
1134 <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>
1135 </pre>
1136 <p>
1137           Gets the <span class="emphasis"><em>rate vector</em></span> parameter of this distribution.
1138         </p>
1139 <h6>
1140 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h32"></a>
1141           <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
1142           Value</a>
1143         </h6>
1144 <p>
1145           A vector of positive real numbers representing the <span class="emphasis"><em>rate vector</em></span>
1146           parameter of this distribution.
1147         </p>
1148 <div class="warning"><table border="0" summary="Warning">
1149 <tr>
1150 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td>
1151 <th align="left">Warning</th>
1152 </tr>
1153 <tr><td align="left" valign="top">
1154 <p>
1155             The return type of these functions is a vector-by-value. This is deliberate
1156             as we wish to hide the actual container used internally which may be
1157             subject to future changes (for example to facilitate vectorization of
1158             the cdf code etc). Users should note that some code that might otherwise
1159             have been expected to work does not. For example, an attempt to output
1160             the (normalized) probabilities:
1161           </p>
1162 <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>
1163 </pre>
1164 <p>
1165             fails at compile or runtime because iterator types are incompatible,
1166             but, for example,
1167           </p>
1168 <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>
1169 </pre>
1170 <p>
1171             outputs the expected values.
1172           </p>
1173 <p>
1174             In general if you want to access a member of the returned container,
1175             then assign to a variable first, and then access those members:
1176           </p>
1177 <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>
1178 <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>
1179 </pre>
1180 </td></tr>
1181 </table></div>
1182 <h4>
1183 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h33"></a>
1184           <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
1185           Accessor Functions</a>
1186         </h4>
1187 <p>
1188           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
1189           functions</a> that are generic to all distributions are supported:
1190           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
1191           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
1192           <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>,
1193           <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>,
1194           <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>,
1195           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
1196           <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>,
1197           <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>.
1198         </p>
1199 <p>
1200           The formulae for calculating these are shown in the table below.
1201         </p>
1202 <h4>
1203 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h34"></a>
1204           <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>
1205         </h4>
1206 <p>
1207           The hyperexponential distribution is implemented in terms of the <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>
1208           and as such should have very small errors, usually an <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>
1209           or few.
1210         </p>
1211 <h4>
1212 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h35"></a>
1213           <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>
1214         </h4>
1215 <p>
1216           In the following table:
1217         </p>
1218 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1219 <li class="listitem">
1220               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(&#945;<sub>1</sub>,...,&#945;<sub>k</sub>)</em></span> is
1221               the <span class="emphasis"><em>phase probability vector</em></span> parameter of the
1222               <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution,
1223             </li>
1224 <li class="listitem">
1225               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(&#955;<sub>1</sub>,...,&#955;<sub>k</sub>)</em></span> is
1226               the <span class="emphasis"><em>rate vector</em></span> parameter of the <span class="emphasis"><em>k</em></span>-phase
1227               hyperexponential distribution,
1228             </li>
1229 <li class="listitem">
1230               <span class="emphasis"><em>x</em></span> is the random variate.
1231             </li>
1232 </ul></div>
1233 <div class="informaltable"><table class="table">
1234 <colgroup>
1235 <col>
1236 <col>
1237 </colgroup>
1238 <thead><tr>
1239 <th>
1240                   <p>
1241                     Function
1242                   </p>
1243                 </th>
1244 <th>
1245                   <p>
1246                     Implementation Notes
1247                   </p>
1248                 </th>
1249 </tr></thead>
1250 <tbody>
1251 <tr>
1252 <td>
1253                   <p>
1254                     support
1255                   </p>
1256                 </td>
1257 <td>
1258                   <p>
1259                     <span class="emphasis"><em>x</em></span> &#8712; [0,&#8734;)
1260                   </p>
1261                 </td>
1262 </tr>
1263 <tr>
1264 <td>
1265                   <p>
1266                     pdf
1267                   </p>
1268                 </td>
1269 <td>
1270                   <div class="blockquote"><blockquote class="blockquote"><p>
1271                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span>
1272
1273                     </p></blockquote></div>
1274                 </td>
1275 </tr>
1276 <tr>
1277 <td>
1278                   <p>
1279                     cdf
1280                   </p>
1281                 </td>
1282 <td>
1283                   <div class="blockquote"><blockquote class="blockquote"><p>
1284                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_cdf.svg"></span>
1285
1286                     </p></blockquote></div>
1287                 </td>
1288 </tr>
1289 <tr>
1290 <td>
1291                   <p>
1292                     cdf complement
1293                   </p>
1294                 </td>
1295 <td>
1296                   <div class="blockquote"><blockquote class="blockquote"><p>
1297                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_ccdf.svg"></span>
1298
1299                     </p></blockquote></div>
1300                 </td>
1301 </tr>
1302 <tr>
1303 <td>
1304                   <p>
1305                     quantile
1306                   </p>
1307                 </td>
1308 <td>
1309                   <p>
1310                     No closed form available. Computed numerically.
1311                   </p>
1312                 </td>
1313 </tr>
1314 <tr>
1315 <td>
1316                   <p>
1317                     quantile from the complement
1318                   </p>
1319                 </td>
1320 <td>
1321                   <p>
1322                     No closed form available. Computed numerically.
1323                   </p>
1324                 </td>
1325 </tr>
1326 <tr>
1327 <td>
1328                   <p>
1329                     mean
1330                   </p>
1331                 </td>
1332 <td>
1333                   <div class="blockquote"><blockquote class="blockquote"><p>
1334                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_mean.svg"></span>
1335
1336                     </p></blockquote></div>
1337                 </td>
1338 </tr>
1339 <tr>
1340 <td>
1341                   <p>
1342                     variance
1343                   </p>
1344                 </td>
1345 <td>
1346                   <div class="blockquote"><blockquote class="blockquote"><p>
1347                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_variance.svg"></span>
1348
1349                     </p></blockquote></div>
1350                 </td>
1351 </tr>
1352 <tr>
1353 <td>
1354                   <p>
1355                     mode
1356                   </p>
1357                 </td>
1358 <td>
1359                   <p>
1360                     <code class="computeroutput"><span class="number">0</span></code>
1361                   </p>
1362                 </td>
1363 </tr>
1364 <tr>
1365 <td>
1366                   <p>
1367                     skewness
1368                   </p>
1369                 </td>
1370 <td>
1371                   <div class="blockquote"><blockquote class="blockquote"><p>
1372                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_skewness.svg"></span>
1373
1374                     </p></blockquote></div>
1375                 </td>
1376 </tr>
1377 <tr>
1378 <td>
1379                   <p>
1380                     kurtosis
1381                   </p>
1382                 </td>
1383 <td>
1384                   <div class="blockquote"><blockquote class="blockquote"><p>
1385                       <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_kurtosis.svg"></span>
1386
1387                     </p></blockquote></div>
1388                 </td>
1389 </tr>
1390 <tr>
1391 <td>
1392                   <p>
1393                     kurtosis excess
1394                   </p>
1395                 </td>
1396 <td>
1397                   <p>
1398                     kurtosis <code class="computeroutput"><span class="special">-</span> <span class="number">3</span></code>
1399                   </p>
1400                 </td>
1401 </tr>
1402 </tbody>
1403 </table></div>
1404 <h4>
1405 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h36"></a>
1406           <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>
1407         </h4>
1408 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1409 <li class="listitem">
1410               A.O. Allen, <span class="emphasis"><em>Probability, Statistics, and Queuing Theory with
1411               Computer Science Applications, Second Edition</em></span>, Academic
1412               Press, 1990.
1413             </li>
1414 <li class="listitem">
1415               D.G. Feitelson, <span class="emphasis"><em>Workload Modeling for Computer Systems Performance
1416               Evaluation</em></span>, Cambridge University Press, 2014
1417             </li>
1418 <li class="listitem">
1419               A. Feldmann and W. Whitt, <span class="emphasis"><em>Fitting mixtures of exponentials
1420               to long-tail distributions to analyze network performance models</em></span>,
1421               Performance Evaluation 31(3-4):245, doi:10.1016/S0166-5316(97)00003-5,
1422               1998.
1423             </li>
1424 <li class="listitem">
1425               H.T. Papadopolous, C. Heavey and J. Browne, <span class="emphasis"><em>Queueing Theory
1426               in Manufacturing Systems Analysis and Design</em></span>, Chapman &amp;
1427               Hall/CRC, 1993, p. 35.
1428             </li>
1429 <li class="listitem">
1430               R.F. Rosin, <span class="emphasis"><em>Determining a computing center environment</em></span>,
1431               Communications of the ACM 8(7):463-468, 1965.
1432             </li>
1433 <li class="listitem">
1434               K.S. Trivedi, <span class="emphasis"><em>Probability and Statistics with Reliability,
1435               Queueing, and Computer Science Applications</em></span>, John Wiley
1436               &amp; Sons, Inc., 2002.
1437             </li>
1438 <li class="listitem">
1439               Wikipedia, <span class="emphasis"><em>Hyperexponential Distribution</em></span>, Online:
1440               <a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">http://en.wikipedia.org/wiki/Hyperexponential_distribution</a>,
1441               2014
1442             </li>
1443 <li class="listitem">
1444               R. Wolski and J. Brevik, <span class="emphasis"><em>Using Parametric Models to Represent
1445               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>,
1446               2013.
1447             </li>
1448 <li class="listitem">
1449               Wolfram Mathematica, <span class="emphasis"><em>Hyperexponential Distribution</em></span>,
1450               Online: <a href="http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html" target="_top">http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html</a>,
1451               2014.
1452             </li>
1453 </ul></div>
1454 </div>
1455 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1456 <td align="left"></td>
1457 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
1458       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
1459       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
1460       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
1461       Daryle Walker and Xiaogang Zhang<p>
1462         Distributed under the Boost Software License, Version 1.0. (See accompanying
1463         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
1464       </p>
1465 </div></td>
1466 </tr></table>
1467 <hr>
1468 <div class="spirit-nav">
1469 <a accesskey="p" href="geometric_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="hypergeometric_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
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