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  24 </div>
  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;15.&#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;15.&#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 <p>
 114           <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span>
 115         </p>
 116 <p>
 117           The following graph illustrates the PDF of the hyperexponential distribution
 118           with five different parameters, namely:
 119         </p>
 120 <div class="orderedlist"><ol class="orderedlist" type="1">
 121 <li class="listitem">
 122               <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
 123               exponential distribution),
 124             </li>
 125 <li class="listitem">
 126               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.1, 0.9)</em></span> and
 127               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 128             </li>
 129 <li class="listitem">
 130               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.9, 0.1)</em></span> and
 131               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 132             </li>
 133 <li class="listitem">
 134               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.2, 0.3, 0.5)</em></span>
 135               and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.0, 1.5)</em></span>,
 136             </li>
 137 <li class="listitem">
 138               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.5, 0.3, 0.2)</em></span>
 139               and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.0, 1.5)</em></span>.
 140             </li>
 141 </ol></div>
 142 <p>
 143           <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf.svg" align="middle"></span>
 144         </p>
 145 <p>
 146           Also, the following graph illustrates the PDF of the hyperexponential distribution
 147           (solid lines) where only the <span class="emphasis"><em>phase probability vector</em></span>
 148           changes together with the PDF of the two limiting exponential distributions
 149           (dashed lines):
 150         </p>
 151 <div class="orderedlist"><ol class="orderedlist" type="1">
 152 <li class="listitem">
 153               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.1, 0.9)</em></span> and
 154               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 155             </li>
 156 <li class="listitem">
 157               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.6, 0.4)</em></span> and
 158               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 159             </li>
 160 <li class="listitem">
 161               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.9, 0.1)</em></span> and
 162               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
 163             </li>
 164 <li class="listitem">
 165               Exponential distribution with parameter <span class="emphasis"><em>&#955;=0.5</em></span>,
 166             </li>
 167 <li class="listitem">
 168               Exponential distribution with parameter <span class="emphasis"><em>&#955;=1.5</em></span>.
 169             </li>
 170 </ol></div>
 171 <p>
 172           As expected, as the first element <span class="emphasis"><em>&#945;<sub>1</sub></em></span> of the <span class="emphasis"><em>phase
 173           probability vector</em></span> approaches to <span class="emphasis"><em>1</em></span> (or,
 174           equivalently, <span class="emphasis"><em>&#945;<sub>2</sub></em></span> approaches to <span class="emphasis"><em>0</em></span>),
 175           the resulting hyperexponential distribution nears the exponential distribution
 176           with parameter <span class="emphasis"><em>&#955;=0.5</em></span>. Conversely, as the first element
 177           <span class="emphasis"><em>&#945;<sub>2</sub></em></span> of the <span class="emphasis"><em>phase probability vector</em></span>
 178           approaches to <span class="emphasis"><em>1</em></span> (or, equivalently, <span class="emphasis"><em>&#945;<sub>1</sub></em></span>
 179           approaches to <span class="emphasis"><em>0</em></span>), the resulting hyperexponential distribution
 180           nears the exponential distribution with parameter <span class="emphasis"><em>&#955;=1.5</em></span>.
 181         </p>
 182 <p>
 183           <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.svg" align="middle"></span>
 184         </p>
 185 <p>
 186           Finally, the following graph compares the PDF of the hyperexponential distribution
 187           with different number of phases but with the same mean value equal to
 188           <span class="emphasis"><em>2</em></span>:
 189         </p>
 190 <div class="orderedlist"><ol class="orderedlist" type="1">
 191 <li class="listitem">
 192               <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
 193               exponential distribution),
 194             </li>
 195 <li class="listitem">
 196               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.5, 0.5)</em></span> and
 197               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.3, 1.5)</em></span>,
 198             </li>
 199 <li class="listitem">
 200               <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>
 201               and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.2, 1.5, 3.0)</em></span>,
 202             </li>
 203 </ol></div>
 204 <p>
 205           <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samemean.svg" align="middle"></span>
 206         </p>
 207 <p>
 208           As can be noted, even if the three distributions have the same mean value,
 209           the two hyperexponential distributions have a <span class="emphasis"><em>longer</em></span>
 210           tail with respect to the one of the exponential distribution. Indeed, the
 211           hyperexponential distribution has a larger variability than the exponential
 212           distribution, thus resulting in a <a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" target="_top">Coefficient
 213           of Variation</a> greater than <span class="emphasis"><em>1</em></span> (as opposed to
 214           the one of the exponential distribution which is exactly <span class="emphasis"><em>1</em></span>).
 215         </p>
 216 <h4>
 217 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h0"></a>
 218           <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>
 219         </h4>
 220 <p>
 221           A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is frequently
 222           used in <a href="http://en.wikipedia.org/wiki/Queueing_theory" target="_top">queueing
 223           theory</a> to model the distribution of the superposition of <span class="emphasis"><em>k</em></span>
 224           independent events, like, for instance, the service time distribution of
 225           a queueing station with <span class="emphasis"><em>k</em></span> servers in parallel where
 226           the <span class="emphasis"><em>i</em></span>-th server is chosen with probability <span class="emphasis"><em>&#945;<sub>i</sub></em></span>
 227           and its service time distribution is an exponential distribution with rate
 228           <span class="emphasis"><em>&#955;<sub>i</sub></em></span> (Allen,1990; Papadopolous et al.,1993; Trivedi,2002).
 229         </p>
 230 <p>
 231           For instance, CPUs service-time distribution in a computing system has
 232           often been observed to possess such a distribution (Rosin,1965). Also,
 233           the arrival of different types of customer to a single queueing station
 234           is often modeled as a hyperexponential distribution (Papadopolous et al.,1993).
 235           Similarly, if a product manufactured in several parallel assembly lines
 236           and the outputs are merged, the failure density of the overall product
 237           is likely to be hyperexponential (Trivedi,2002).
 238         </p>
 239 <p>
 240           Finally, since the hyperexponential distribution exhibits a high Coefficient
 241           of Variation (CoV), that is a CoV &gt; 1, it is especially suited to fit
 242           empirical data with large CoV (Feitelson,2014; Wolski et al.,2013) and
 243           to approximate <a href="http://en.wikipedia.org/wiki/Long_tail" target="_top">long-tail
 244           probability distributions</a> (Feldmann et al.,1998).
 245         </p>
 246 <h4>
 247 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h1"></a>
 248           <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
 249           distributions</a>
 250         </h4>
 251 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 252 <li class="listitem">
 253               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
 254               is simply an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
 255               Distribution</a>.
 256             </li>
 257 <li class="listitem">
 258               When the <span class="emphasis"><em>k</em></span> rates are all equal to <span class="emphasis"><em>&#955;</em></span>,
 259               the hyperexponential distribution is simple an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
 260               Distribution</a> with rate <span class="emphasis"><em>&#955;</em></span>.
 261             </li>
 262 </ul></div>
 263 <h4>
 264 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h2"></a>
 265           <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>
 266         </h4>
 267 <h5>
 268 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h3"></a>
 269           <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
 270           of Appliances</a>
 271         </h5>
 272 <p>
 273           Suppose a customer is buying an appliance and is choosing at random between
 274           an appliance with average lifetime of 10 years and an appliance with average
 275           lifetime of 12 years. Assuming the lifetime of this appliance follows an
 276           exponential distribution, the lifetime distribution of the purchased appliance
 277           can be modeled as a hyperexponential distribution with phase probability
 278           vector <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1/2,1/2)</em></span>
 279           and rate vector <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(1/10,1/12)</em></span>
 280           (Wolfram,2014).
 281         </p>
 282 <p>
 283           In the rest of this section, we provide an example C++ implementation for
 284           computing the average lifetime and the probability that the appliance will
 285           work for more than 15 years.
 286         </p>
 287 <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>
 288 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 289 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 290 <span class="special">{</span>
 291    <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>
 292
 293    <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>
 294
 295    <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>
 296       <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>
 297       <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>
 298    <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>
 299       <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>
 300       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 301 <span class="special">}</span>
 302 </pre>
 303 <p>
 304           The resulting output is:
 305         </p>
 306 <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>
 307 <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>
 308 </pre>
 309 <h5>
 310 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h4"></a>
 311           <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
 312           of Private Cloud Computing Systems</a>
 313         </h5>
 314 <p>
 315           <a href="http://en.wikipedia.org/wiki/Cloud_computing" target="_top">Cloud computing</a>
 316           has become a popular metaphor for dynamic and secure self-service access
 317           to computational and storage capabilities. In (Wolski et al.,2013), the
 318           authors analyze and model workloads gathered from enterprise-operated commercial
 319           <a href="http://en.wikipedia.org/wiki/Cloud_computing#Private_cloud" target="_top">private
 320           clouds</a> and show that 3-phase hyperexponential distributions (fitted
 321           using the <a href="http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm" target="_top">Expectation
 322           Maximization algorithm</a>) capture workload attributes accurately.
 323         </p>
 324 <p>
 325           In this type of computing system, user requests consist in demanding the
 326           provisioning of one or more <a href="http://en.wikipedia.org/wiki/Virtual_machine" target="_top">Virtual
 327           Machines</a> (VMs). In particular, in (Wolski et al.,2013) the workload
 328           experienced by each cloud system is a function of four distributions, one
 329           for each of the following workload attributes:
 330         </p>
 331 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 332 <li class="listitem">
 333               <span class="emphasis"><em>Request Interarrival Time</em></span>: the amount of time
 334               until the next request,
 335             </li>
 336 <li class="listitem">
 337               <span class="emphasis"><em>VM Lifetime</em></span>: the time duration over which a VM
 338               is provisioned to a physical machine,
 339             </li>
 340 <li class="listitem">
 341               <span class="emphasis"><em>Request Size</em></span>: the number of VMs in the request,
 342               and
 343             </li>
 344 <li class="listitem">
 345               <span class="emphasis"><em>Core Count</em></span>: the CPU core count requested for each
 346               VM.
 347             </li>
 348 </ul></div>
 349 <p>
 350           The authors assume that all VMs in a request have the same core count,
 351           but request sizes and core counts can vary from request to request. Moreover,
 352           all VMs within a request are assumed to have the same lifetime. Given these
 353           assumptions, the authors build a statistical model for the request interarrival
 354           time and VM lifetime attributes by fitting their respective data to a 3-phase
 355           hyperexponential distribution.
 356         </p>
 357 <p>
 358           In the following table, we show the sample mean and standard deviation
 359           (SD), in seconds, of the request interarrival time and of the VM lifetime
 360           distributions of the three datasets collected by authors:
 361         </p>
 362 <div class="informaltable"><table class="table">
 363 <colgroup>
 364 <col>
 365 <col>
 366 <col>
 367 <col>
 368 </colgroup>
 369 <thead><tr>
 370 <th>
 371                   <p>
 372                     Dataset
 373                   </p>
 374                 </th>
 375 <th>
 376                   <p>
 377                     Mean Request Interarrival Time (SD)
 378                   </p>
 379                 </th>
 380 <th>
 381                   <p>
 382                     Mean Multi-core VM Lifetime (SD)
 383                   </p>
 384                 </th>
 385 <th>
 386                   <p>
 387                     Mean Single-core VM Lifetime (SD)
 388                   </p>
 389                 </th>
 390 </tr></thead>
 391 <tbody>
 392 <tr>
 393 <td>
 394                   <p>
 395                     DS1
 396                   </p>
 397                 </td>
 398 <td>
 399                   <p>
 400                     2202.1 (2.2e+04)
 401                   </p>
 402                 </td>
 403 <td>
 404                   <p>
 405                     257173 (4.6e+05)
 406                   </p>
 407                 </td>
 408 <td>
 409                   <p>
 410                     28754.4 (1.6e+05)
 411                   </p>
 412                 </td>
 413 </tr>
 414 <tr>
 415 <td>
 416                   <p>
 417                     DS2
 418                   </p>
 419                 </td>
 420 <td>
 421                   <p>
 422                     41285.7 (1.1e+05)
 423                   </p>
 424                 </td>
 425 <td>
 426                   <p>
 427                     144669.0 (7.9e+05)
 428                   </p>
 429                 </td>
 430 <td>
 431                   <p>
 432                     599815.0 (1.7e+06)
 433                   </p>
 434                 </td>
 435 </tr>
 436 <tr>
 437 <td>
 438                   <p>
 439                     DS3
 440                   </p>
 441                 </td>
 442 <td>
 443                   <p>
 444                     11238.8 (3.0e+04)
 445                   </p>
 446                 </td>
 447 <td>
 448                   <p>
 449                     30739.2 (1.6e+05)
 450                   </p>
 451                 </td>
 452 <td>
 453                   <p>
 454                     44447.8 (2.2e+05)
 455                   </p>
 456                 </td>
 457 </tr>
 458 </tbody>
 459 </table></div>
 460 <p>
 461           Whereas in the following table we show the hyperexponential distribution
 462           parameters resulting from the fit:
 463         </p>
 464 <div class="informaltable"><table class="table">
 465 <colgroup>
 466 <col>
 467 <col>
 468 <col>
 469 <col>
 470 </colgroup>
 471 <thead><tr>
 472 <th>
 473                   <p>
 474                     Dataset
 475                   </p>
 476                 </th>
 477 <th>
 478                   <p>
 479                     Request Interarrival Time
 480                   </p>
 481                 </th>
 482 <th>
 483                   <p>
 484                     Multi-core VM Lifetime
 485                   </p>
 486                 </th>
 487 <th>
 488                   <p>
 489                     Single-core VM Lifetime
 490                   </p>
 491                 </th>
 492 </tr></thead>
 493 <tbody>
 494 <tr>
 495 <td>
 496                   <p>
 497                     DS1
 498                   </p>
 499                 </td>
 500 <td>
 501                   <p>
 502                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.34561,0.08648,0.56791),
 503                     <span class="bold"><strong>&#955;</strong></span>=(0.008,0.00005,0.02894)</em></span>
 504                   </p>
 505                 </td>
 506 <td>
 507                   <p>
 508                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.24667,0.37948,0.37385),
 509                     <span class="bold"><strong>&#955;</strong></span>=(0.00004,0.000002,0.00059)</em></span>
 510                   </p>
 511                 </td>
 512 <td>
 513                   <p>
 514                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.09325,0.22251,0.68424),
 515                     <span class="bold"><strong>&#955;</strong></span>=(0.000003,0.00109,0.00109)</em></span>
 516                   </p>
 517                 </td>
 518 </tr>
 519 <tr>
 520 <td>
 521                   <p>
 522                     DS2
 523                   </p>
 524                 </td>
 525 <td>
 526                   <p>
 527                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.38881,0.18227,0.42892),
 528                     <span class="bold"><strong>&#955;</strong></span>=(0.000006,0.05228,0.00081)</em></span>
 529                   </p>
 530                 </td>
 531 <td>
 532                   <p>
 533                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.42093,0.43960,0.13947),
 534                     <span class="bold"><strong>&#955;</strong></span>=(0.00186,0.00008,0.0000008)</em></span>
 535                   </p>
 536                 </td>
 537 <td>
 538                   <p>
 539                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.44885,0.30675,0.2444),
 540                     <span class="bold"><strong>&#955;</strong></span>=(0.00143,0.00005,0.0000004)</em></span>
 541                   </p>
 542                 </td>
 543 </tr>
 544 <tr>
 545 <td>
 546                   <p>
 547                     DS3
 548                   </p>
 549                 </td>
 550 <td>
 551                   <p>
 552                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.39442,0.24644,0.35914),
 553                     <span class="bold"><strong>&#955;</strong></span>=(0.00030,0.00003,0.00257)</em></span>
 554                   </p>
 555                 </td>
 556 <td>
 557                   <p>
 558                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.37621,0.14838,0.47541),
 559                     <span class="bold"><strong>&#955;</strong></span>=(0.00498,0.000005,0.00022)</em></span>
 560                   </p>
 561                 </td>
 562 <td>
 563                   <p>
 564                     <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.34131,0.12544,0.53325),
 565                     <span class="bold"><strong>&#955;</strong></span>=(0.000297,0.000003,0.00410)</em></span>
 566                   </p>
 567                 </td>
 568 </tr>
 569 </tbody>
 570 </table></div>
 571 <p>
 572           In the rest of this section, we provide an example C++ implementation for
 573           computing some statistical properties of the fitted distributions for each
 574           of the analyzed dataset.
 575         </p>
 576 <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>
 577 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 578 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">string</span><span class="special">&gt;</span>
 579
 580 <span class="keyword">struct</span> <span class="identifier">ds_info</span>
 581 <span class="special">{</span>
 582    <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">name</span><span class="special">;</span>
 583    <span class="keyword">double</span> <span class="identifier">iat_sample_mean</span><span class="special">;</span>
 584    <span class="keyword">double</span> <span class="identifier">iat_sample_sd</span><span class="special">;</span>
 585    <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>
 586    <span class="keyword">double</span> <span class="identifier">multi_lt_sample_mean</span><span class="special">;</span>
 587    <span class="keyword">double</span> <span class="identifier">multi_lt_sample_sd</span><span class="special">;</span>
 588    <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>
 589    <span class="keyword">double</span> <span class="identifier">single_lt_sample_mean</span><span class="special">;</span>
 590    <span class="keyword">double</span> <span class="identifier">single_lt_sample_sd</span><span class="special">;</span>
 591    <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>
 592 <span class="special">};</span>
 593
 594 <span class="comment">// DS1 dataset</span>
 595 <span class="identifier">ds_info</span> <span class="identifier">make_ds1</span><span class="special">()</span>
 596 <span class="special">{</span>
 597    <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
 598
 599    <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>
 600
 601    <span class="comment">// VM interarrival time distribution</span>
 602    <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>
 603    <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>
 604    <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>
 605    <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>
 606    <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>
 607
 608    <span class="comment">// Multi-core VM lifetime distribution</span>
 609    <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>
 610    <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>
 611    <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>
 612    <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>
 613    <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>
 614
 615    <span class="comment">// Single-core VM lifetime distribution</span>
 616    <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>
 617    <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>
 618    <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>
 619    <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>
 620    <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>
 621
 622    <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
 623 <span class="special">}</span>
 624
 625 <span class="comment">// DS2 dataset</span>
 626 <span class="identifier">ds_info</span> <span class="identifier">make_ds2</span><span class="special">()</span>
 627 <span class="special">{</span>
 628    <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
 629
 630    <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>
 631
 632    <span class="comment">// VM interarrival time distribution</span>
 633    <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>
 634    <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>
 635    <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>
 636    <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>
 637    <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>
 638
 639    <span class="comment">// Multi-core VM lifetime distribution</span>
 640    <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>
 641    <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>
 642    <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>
 643    <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>
 644    <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>
 645
 646    <span class="comment">// Single-core VM lifetime distribution</span>
 647    <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>
 648    <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>
 649    <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>
 650    <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>
 651    <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>
 652
 653    <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
 654 <span class="special">}</span>
 655
 656 <span class="comment">// DS3 dataset</span>
 657 <span class="identifier">ds_info</span> <span class="identifier">make_ds3</span><span class="special">()</span>
 658 <span class="special">{</span>
 659    <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span>
 660
 661    <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>
 662
 663    <span class="comment">// VM interarrival time distribution</span>
 664    <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>
 665    <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>
 666    <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>
 667    <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>
 668    <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>
 669
 670    <span class="comment">// Multi-core VM lifetime distribution</span>
 671    <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>
 672    <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>
 673    <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>
 674    <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>
 675    <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>
 676
 677    <span class="comment">// Single-core VM lifetime distribution</span>
 678    <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>
 679    <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>
 680    <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>
 681    <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>
 682    <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>
 683
 684    <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span>
 685 <span class="special">}</span>
 686
 687 <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>
 688 <span class="special">{</span>
 689    <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>
 690    <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>
 691
 692    <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>
 693    <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>
 694    <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>
 695    <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>
 696    <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>
 697    <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>
 698    <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>
 699    <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>
 700    <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>
 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 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>
 702    <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>
 703    <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>
 704    <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>
 705    <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>
 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 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>
 707    <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>
 708 <span class="special">}</span>
 709
 710 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 711 <span class="special">{</span>
 712    <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds1</span><span class="special">());</span>
 713
 714    <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds2</span><span class="special">());</span>
 715
 716    <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds3</span><span class="special">());</span>
 717 <span class="special">}</span>
 718 </pre>
 719 <p>
 720           The resulting output (with floating-point precision set to 2) is:
 721         </p>
 722 <pre class="programlisting"><span class="special">###</span> <span class="identifier">DS1</span>
 723 <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
 724  <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>
 725  <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>
 726  <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>
 727  <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>
 728 <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>
 729  <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>
 730  <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>
 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">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>
 732  <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>
 733 <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>
 734  <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>
 735  <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>
 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">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>
 737  <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>
 738 <span class="special">###</span> <span class="identifier">DS2</span>
 739 <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
 740  <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>
 741  <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>
 742  <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>
 743  <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>
 744 <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>
 745  <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>
 746  <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>
 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">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>
 748  <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>
 749 <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>
 750  <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>
 751  <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>
 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">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>
 753  <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>
 754 <span class="special">###</span> <span class="identifier">DS3</span>
 755 <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span>
 756  <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>
 757  <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>
 758  <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>
 759  <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>
 760 <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>
 761  <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>
 762  <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>
 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">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>
 764  <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>
 765 <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>
 766  <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>
 767  <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>
 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">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>
 769  <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>
 770 </pre>
 771 <div class="note"><table border="0" summary="Note">
 772 <tr>
 773 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 774 <th align="left">Note</th>
 775 </tr>
 776 <tr><td align="left" valign="top"><p>
 777             The above results differ from the ones shown in Tables III, V, and VII
 778             of (Wolski et al.,2013). We carefully double-checked them with Wolfram
 779             Mathematica 10, which confirmed our results.
 780           </p></td></tr>
 781 </table></div>
 782 <h4>
 783 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h5"></a>
 784           <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
 785           Functions</a>
 786         </h4>
 787 <h5>
 788 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h6"></a>
 789           <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
 790           Constructor</a>
 791         </h5>
 792 <pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">();</span>
 793 </pre>
 794 <p>
 795           Constructs a <span class="emphasis"><em>1</em></span>-phase hyperexponential distribution
 796           (i.e., an exponential distribution) with rate <code class="computeroutput"><span class="number">1</span></code>.
 797         </p>
 798 <h5>
 799 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h7"></a>
 800           <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
 801           from Iterators</a>
 802         </h5>
 803 <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>
 804 <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>
 805                               <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>
 806 </pre>
 807 <p>
 808           Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
 809           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>)
 810           iterator pair, and <span class="emphasis"><em>rate vector</em></span> parameter given by
 811           the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>,
 812           <code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair.
 813         </p>
 814 <h6>
 815 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h8"></a>
 816           <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>
 817         </h6>
 818 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 819 <li class="listitem">
 820               <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
 821               real elements representing the phase probabilities; elements are normalized
 822               to sum to unity.
 823             </li>
 824 <li class="listitem">
 825               <code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive
 826               elements representing the rates.
 827             </li>
 828 </ul></div>
 829 <h6>
 830 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h9"></a>
 831           <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
 832           Requirements</a>
 833         </h6>
 834 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 835               <code class="computeroutput"><span class="identifier">ProbIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT</span></code>: must meet the requirements
 836               of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a>
 837               concept.
 838             </li></ul></div>
 839 <h6>
 840 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h10"></a>
 841           <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>
 842         </h6>
 843 <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>
 844 <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>
 845
 846 <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>
 847 </pre>
 848 <h5>
 849 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h11"></a>
 850           <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
 851           from Ranges/Containers</a>
 852         </h5>
 853 <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>
 854 <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>
 855                               <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>
 856 </pre>
 857 <p>
 858           Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
 859           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>
 860           parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>.
 861         </p>
 862 <div class="note"><table border="0" summary="Note">
 863 <tr>
 864 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 865 <th align="left">Note</th>
 866 </tr>
 867 <tr><td align="left" valign="top"><p>
 868             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
 869             mechanism</a> to disambiguate between this and other 2-argument constructors.
 870             Refer to the source code for more details.
 871           </p></td></tr>
 872 </table></div>
 873 <h6>
 874 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h12"></a>
 875           <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>
 876         </h6>
 877 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 878 <li class="listitem">
 879               <code class="computeroutput"><span class="identifier">prob_range</span></code>: the range
 880               of non-negative real elements representing the phase probabilities;
 881               elements are normalized to sum to unity.
 882             </li>
 883 <li class="listitem">
 884               <code class="computeroutput"><span class="identifier">rate_range</span></code>: the range
 885               of positive real elements representing the rates.
 886             </li>
 887 </ul></div>
 888 <h6>
 889 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h13"></a>
 890           <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
 891           Requirements</a>
 892         </h6>
 893 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 894               <code class="computeroutput"><span class="identifier">ProbRangeT</span></code>, <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet the requirements
 895               of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a>
 896               concept: that includes native C++ arrays, standard library containers,
 897               or a std::pair or iterators.
 898             </li></ul></div>
 899 <h6>
 900 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h14"></a>
 901           <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>
 902         </h6>
 903 <pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span>
 904 <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>
 905 <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>
 906
 907 <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>
 908
 909 <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>
 910 <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>
 911
 912 <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>
 913 </pre>
 914 <h5>
 915 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h15"></a>
 916           <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
 917           with rates-iterators (and all phase probabilities equal)</a>
 918         </h5>
 919 <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>
 920 <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>
 921                               <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>
 922 </pre>
 923 <p>
 924           Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
 925           parameter given by the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>,
 926           <code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair,
 927           and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase
 928           probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
 929           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>).
 930         </p>
 931 <div class="note"><table border="0" summary="Note">
 932 <tr>
 933 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 934 <th align="left">Note</th>
 935 </tr>
 936 <tr><td align="left" valign="top"><p>
 937             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
 938             mechanism</a> to disambiguate between this and other 2-argument constructors.
 939             Refer to the source code for more details.
 940           </p></td></tr>
 941 </table></div>
 942 <h6>
 943 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h16"></a>
 944           <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>
 945         </h6>
 946 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 947               <code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive
 948               elements representing the rates.
 949             </li></ul></div>
 950 <h6>
 951 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h17"></a>
 952           <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
 953           Requirements</a>
 954         </h6>
 955 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 956               <code class="computeroutput"><span class="identifier">RateIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT2</span></code>: must meet the requirements
 957               of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a>
 958               concept.
 959             </li></ul></div>
 960 <h6>
 961 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h18"></a>
 962           <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>
 963         </h6>
 964 <pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span>
 965 <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>
 966
 967 <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>
 968
 969 <span class="identifier">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>
 970 </pre>
 971 <h5>
 972 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h19"></a>
 973           <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
 974           from a single range of rates (all phase probabilities will be equal)</a>
 975         </h5>
 976 <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>
 977 <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>
 978 </pre>
 979 <p>
 980           Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
 981           parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>,
 982           and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase
 983           probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
 984           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>).
 985         </p>
 986 <h6>
 987 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h20"></a>
 988           <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>
 989         </h6>
 990 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 991               <code class="computeroutput"><span class="identifier">rate_range</span></code>: the range
 992               of positive real elements representing the rates.
 993             </li></ul></div>
 994 <h6>
 995 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h21"></a>
 996           <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
 997           Requirements</a>
 998         </h6>
 999 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
1000               <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet
1001               the requirements of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a>
1002               concept: this includes native C++ array, standard library containers,
1003               and a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> of iterators.
1004             </li></ul></div>
1005 <h6>
1006 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h22"></a>
1007           <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>
1008         </h6>
1009 <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>
1010
1011 <span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">);</span>
1012
1013 <span class="identifier">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>
1014 </pre>
1015 <h5>
1016 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h23"></a>
1017           <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
1018           from Initializer lists</a>
1019         </h5>
1020 <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>
1021 </pre>
1022 <p>
1023           Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability
1024           vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
1025           defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>rate
1026           vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
1027           defined by <code class="computeroutput"><span class="identifier">l2</span></code>.
1028         </p>
1029 <h6>
1030 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h24"></a>
1031           <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>
1032         </h6>
1033 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1034 <li class="listitem">
1035               <code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list
1036               of non-negative real elements representing the phase probabilities;
1037               elements are normalized to sum to unity.
1038             </li>
1039 <li class="listitem">
1040               <code class="computeroutput"><span class="identifier">l2</span></code>: the brace-init-list
1041               of positive real elements representing the rates.
1042             </li>
1043 </ul></div>
1044 <p>
1045           The number of elements of the phase probabilities list and the rates list
1046           must be the same.
1047         </p>
1048 <h6>
1049 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h25"></a>
1050           <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>
1051         </h6>
1052 <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>
1053 </pre>
1054 <h5>
1055 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h26"></a>
1056           <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
1057           from a single initializer list (all phase probabilities will be equal)</a>
1058         </h5>
1059 <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>
1060 </pre>
1061 <p>
1062           Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span>
1063           parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a>
1064           defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>phase
1065           probability vector</em></span> set to the equal phase probabilities (i.e.,
1066           to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code>
1067           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>).
1068         </p>
1069 <h6>
1070 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h27"></a>
1071           <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>
1072         </h6>
1073 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
1074               <code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list
1075               of non-negative real elements representing the phase probabilities;
1076               they are normalized to ensure that they sum to unity.
1077             </li></ul></div>
1078 <h6>
1079 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h28"></a>
1080           <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>
1081         </h6>
1082 <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>
1083
1084 <span class="identifier">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>
1085 </pre>
1086 <h5>
1087 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h29"></a>
1088           <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>
1089         </h5>
1090 <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>
1091 </pre>
1092 <p>
1093           Gets the number of phases of this distribution (the size of both the rate
1094           and probability vectors).
1095         </p>
1096 <h6>
1097 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h30"></a>
1098           <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
1099           Value</a>
1100         </h6>
1101 <p>
1102           An non-negative integer number representing the number of phases of this
1103           distribution.
1104         </p>
1105 <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>
1106 </pre>
1107 <p>
1108           Gets the <span class="emphasis"><em>phase probability vector</em></span> parameter of this
1109           distribution.
1110         </p>
1111 <div class="note"><table border="0" summary="Note">
1112 <tr>
1113 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
1114 <th align="left">Note</th>
1115 </tr>
1116 <tr><td align="left" valign="top"><p>
1117             The returned probabilities are the <span class="bold"><strong>normalized</strong></span>
1118             versions of the probability parameter values passed at construction time.
1119           </p></td></tr>
1120 </table></div>
1121 <h6>
1122 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h31"></a>
1123           <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
1124           Value</a>
1125         </h6>
1126 <p>
1127           A vector of non-negative real numbers representing the <span class="emphasis"><em>phase
1128           probability vector</em></span> parameter of this distribution.
1129         </p>
1130 <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>
1131 </pre>
1132 <p>
1133           Gets the <span class="emphasis"><em>rate vector</em></span> parameter of this distribution.
1134         </p>
1135 <h6>
1136 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h32"></a>
1137           <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
1138           Value</a>
1139         </h6>
1140 <p>
1141           A vector of positive real numbers representing the <span class="emphasis"><em>rate vector</em></span>
1142           parameter of this distribution.
1143         </p>
1144 <div class="warning"><table border="0" summary="Warning">
1145 <tr>
1146 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td>
1147 <th align="left">Warning</th>
1148 </tr>
1149 <tr><td align="left" valign="top">
1150 <p>
1151             The return type of these functions is a vector-by-value. This is deliberate
1152             as we wish to hide the actual container used internally which may be
1153             subject to future changes (for example to facilitate vectorization of
1154             the cdf code etc). Users should note that some code that might otherwise
1155             have been expected to work does not. For example, an attempt to output
1156             the (normalized) probabilities:
1157           </p>
1158 <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>
1159 </pre>
1160 <p>
1161             fails at compile or runtime because iterator types are incompatible,
1162             but, for example,
1163           </p>
1164 <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>
1165 </pre>
1166 <p>
1167             outputs the expected values.
1168           </p>
1169 <p>
1170             In general if you want to access a member of the returned container,
1171             then assign to a variable first, and then access those members:
1172           </p>
1173 <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>
1174 <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>
1175 </pre>
1176 </td></tr>
1177 </table></div>
1178 <h4>
1179 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h33"></a>
1180           <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
1181           Accessor Functions</a>
1182         </h4>
1183 <p>
1184           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
1185           functions</a> that are generic to all distributions are supported:
1186           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
1187           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
1188           <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>,
1189           <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>,
1190           <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>,
1191           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
1192           <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>,
1193           <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>.
1194         </p>
1195 <p>
1196           The formulae for calculating these are shown in the table below.
1197         </p>
1198 <h4>
1199 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h34"></a>
1200           <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>
1201         </h4>
1202 <p>
1203           The hyperexponential distribution is implemented in terms of the <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>
1204           and as such should have very small errors, usually an <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>
1205           or few.
1206         </p>
1207 <h4>
1208 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h35"></a>
1209           <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>
1210         </h4>
1211 <p>
1212           In the following table:
1213         </p>
1214 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1215 <li class="listitem">
1216               <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(&#945;<sub>1</sub>,...,&#945;<sub>k</sub>)</em></span> is
1217               the <span class="emphasis"><em>phase probability vector</em></span> parameter of the
1218               <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution,
1219             </li>
1220 <li class="listitem">
1221               <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(&#955;<sub>1</sub>,...,&#955;<sub>k</sub>)</em></span> is
1222               the <span class="emphasis"><em>rate vector</em></span> parameter of the <span class="emphasis"><em>k</em></span>-phase
1223               hyperexponential distribution,
1224             </li>
1225 <li class="listitem">
1226               <span class="emphasis"><em>x</em></span> is the random variate.
1227             </li>
1228 </ul></div>
1229 <div class="informaltable"><table class="table">
1230 <colgroup>
1231 <col>
1232 <col>
1233 </colgroup>
1234 <thead><tr>
1235 <th>
1236                   <p>
1237                     Function
1238                   </p>
1239                 </th>
1240 <th>
1241                   <p>
1242                     Implementation Notes
1243                   </p>
1244                 </th>
1245 </tr></thead>
1246 <tbody>
1247 <tr>
1248 <td>
1249                   <p>
1250                     support
1251                   </p>
1252                 </td>
1253 <td>
1254                   <p>
1255                     <span class="emphasis"><em>x</em></span> &#8712; [0,&#8734;)
1256                   </p>
1257                 </td>
1258 </tr>
1259 <tr>
1260 <td>
1261                   <p>
1262                     pdf
1263                   </p>
1264                 </td>
1265 <td>
1266                   <p>
1267                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span>
1268                   </p>
1269                 </td>
1270 </tr>
1271 <tr>
1272 <td>
1273                   <p>
1274                     cdf
1275                   </p>
1276                 </td>
1277 <td>
1278                   <p>
1279                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_cdf.svg"></span>
1280                   </p>
1281                 </td>
1282 </tr>
1283 <tr>
1284 <td>
1285                   <p>
1286                     cdf complement
1287                   </p>
1288                 </td>
1289 <td>
1290                   <p>
1291                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_ccdf.svg"></span>
1292                   </p>
1293                 </td>
1294 </tr>
1295 <tr>
1296 <td>
1297                   <p>
1298                     quantile
1299                   </p>
1300                 </td>
1301 <td>
1302                   <p>
1303                     No closed form available. Computed numerically.
1304                   </p>
1305                 </td>
1306 </tr>
1307 <tr>
1308 <td>
1309                   <p>
1310                     quantile from the complement
1311                   </p>
1312                 </td>
1313 <td>
1314                   <p>
1315                     No closed form available. Computed numerically.
1316                   </p>
1317                 </td>
1318 </tr>
1319 <tr>
1320 <td>
1321                   <p>
1322                     mean
1323                   </p>
1324                 </td>
1325 <td>
1326                   <p>
1327                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_mean.svg"></span>
1328                   </p>
1329                 </td>
1330 </tr>
1331 <tr>
1332 <td>
1333                   <p>
1334                     variance
1335                   </p>
1336                 </td>
1337 <td>
1338                   <p>
1339                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_variance.svg"></span>
1340                   </p>
1341                 </td>
1342 </tr>
1343 <tr>
1344 <td>
1345                   <p>
1346                     mode
1347                   </p>
1348                 </td>
1349 <td>
1350                   <p>
1351                     <code class="computeroutput"><span class="number">0</span></code>
1352                   </p>
1353                 </td>
1354 </tr>
1355 <tr>
1356 <td>
1357                   <p>
1358                     skewness
1359                   </p>
1360                 </td>
1361 <td>
1362                   <p>
1363                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_skewness.svg"></span>
1364                   </p>
1365                 </td>
1366 </tr>
1367 <tr>
1368 <td>
1369                   <p>
1370                     kurtosis
1371                   </p>
1372                 </td>
1373 <td>
1374                   <p>
1375                     <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_kurtosis.svg"></span>
1376                   </p>
1377                 </td>
1378 </tr>
1379 <tr>
1380 <td>
1381                   <p>
1382                     kurtosis excess
1383                   </p>
1384                 </td>
1385 <td>
1386                   <p>
1387                     kurtosis <code class="computeroutput"><span class="special">-</span> <span class="number">3</span></code>
1388                   </p>
1389                 </td>
1390 </tr>
1391 </tbody>
1392 </table></div>
1393 <h4>
1394 <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h36"></a>
1395           <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>
1396         </h4>
1397 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1398 <li class="listitem">
1399               A.O. Allen, <span class="emphasis"><em>Probability, Statistics, and Queuing Theory with
1400               Computer Science Applications, Second Edition</em></span>, Academic
1401               Press, 1990.
1402             </li>
1403 <li class="listitem">
1404               D.G. Feitelson, <span class="emphasis"><em>Workload Modeling for Computer Systems Performance
1405               Evaluation</em></span>, Cambridge University Press, 2014
1406             </li>
1407 <li class="listitem">
1408               A. Feldmann and W. Whitt, <span class="emphasis"><em>Fitting mixtures of exponentials
1409               to long-tail distributions to analyze network performance models</em></span>,
1410               Performance Evaluation 31(3-4):245, doi:10.1016/S0166-5316(97)00003-5,
1411               1998.
1412             </li>
1413 <li class="listitem">
1414               H.T. Papadopolous, C. Heavey and J. Browne, <span class="emphasis"><em>Queueing Theory
1415               in Manufacturing Systems Analysis and Design</em></span>, Chapman &amp;
1416               Hall/CRC, 1993, p. 35.
1417             </li>
1418 <li class="listitem">
1419               R.F. Rosin, <span class="emphasis"><em>Determining a computing center environment</em></span>,
1420               Communications of the ACM 8(7):463-468, 1965.
1421             </li>
1422 <li class="listitem">
1423               K.S. Trivedi, <span class="emphasis"><em>Probability and Statistics with Reliability,
1424               Queueing, and Computer Science Applications</em></span>, John Wiley
1425               &amp; Sons, Inc., 2002.
1426             </li>
1427 <li class="listitem">
1428               Wikipedia, <span class="emphasis"><em>Hyperexponential Distribution</em></span>, Online:
1429               <a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">http://en.wikipedia.org/wiki/Hyperexponential_distribution</a>,
1430               2014
1431             </li>
1432 <li class="listitem">
1433               R. Wolski and J. Brevik, <span class="emphasis"><em>Using Parametric Models to Represent
1434               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>,
1435               2013.
1436             </li>
1437 <li class="listitem">
1438               Wolfram Mathematica, <span class="emphasis"><em>Hyperexponential Distribution</em></span>,
1439               Online: <a href="http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html" target="_top">http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html</a>,
1440               2014.
1441             </li>
1442 </ul></div>
1443 </div>
1444 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1445 <td align="left"></td>
1446 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
1447       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
1448       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
1449       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
1450         Distributed under the Boost Software License, Version 1.0. (See accompanying
1451         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>)
1452       </p>
1453 </div></td>
1454 </tr></table>
1455 <hr>
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