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| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist"></a><a class="link" href="hyperexponential_dist.html" title="Hyperexponential Distribution">Hyperexponential |
| Distribution</a> |
| </h4></div></div></div> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span></pre> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter 14. 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<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">hyperexponential_distribution</span><span class="special"><></span> <span class="identifier">hyperexponential</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> |
| |
| <span class="comment">// Constructors:</span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">();</span> <span class="comment">// Default.</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span> <span class="comment">// Default equal probabilities.</span> |
| <span class="identifier">RateIterT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_first</span><span class="special">,</span> |
| <span class="identifier">RateIterT2</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_last</span><span class="special">);</span> <span class="comment">// Rates using Iterators.</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbIterT</span> <span class="identifier">prob_first</span><span class="special">,</span> <span class="identifier">ProbIterT</span> <span class="identifier">prob_last</span><span class="special">,</span> |
| <span class="identifier">RateIterT</span> <span class="identifier">rate_first</span><span class="special">,</span> <span class="identifier">RateIterT</span> <span class="identifier">rate_last</span><span class="special">);</span> <span class="comment">// Iterators.</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbRangeT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">prob_range</span><span class="special">,</span> |
| <span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_range</span><span class="special">);</span> <span class="comment">// Ranges.</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RateRangeT</span><span class="special">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_range</span><span class="special">);</span> |
| |
| <span class="preprocessor">#if</span> <span class="special">!</span><span class="identifier">defined</span><span class="special">(</span><span class="identifier">BOOST_NO_CXX11_HDR_INITIALIZER_LIST</span><span class="special">)</span> <span class="comment">// C++11 initializer lists supported.</span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></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"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">l2</span><span class="special">);</span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">l1</span><span class="special">);</span> |
| <span class="preprocessor">#endif</span> |
| |
| <span class="comment">// Accessors:</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">num_phases</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">probabilities</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">rates</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| An implementation-defined mechanism is provided to avoid ambiguity between |
| constructors accepting ranges, iterators and constants as parameters. |
| This should be transparent to the user. See below and the header file |
| hyperexponential.hpp for details and explanatory comments. |
| </p></td></tr> |
| </table></div> |
| <p> |
| The class type <code class="computeroutput"><span class="identifier">hyperexponential_distribution</span></code> |
| represents a <a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">hyperexponential |
| distribution</a>. |
| </p> |
| <p> |
| A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is a <a href="http://en.wikipedia.org/wiki/Continuous_probability_distribution" target="_top">continuous |
| probability distribution</a> obtained as a mixture of <span class="emphasis"><em>k</em></span> |
| <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>s. |
| It is also referred to as <span class="emphasis"><em>mixed exponential distribution</em></span> |
| or parallel <span class="emphasis"><em>k-phase exponential distribution</em></span>. |
| </p> |
| <p> |
| A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is characterized |
| by two parameters, namely a <span class="emphasis"><em>phase probability vector</em></span> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(α<sub>1</sub>,...,α<sub>k</sub>)</em></span> and a |
| <span class="emphasis"><em>rate vector</em></span> <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(λ<sub>1</sub>,...,λ<sub>k</sub>)</em></span>. |
| </p> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability |
| density function</a> for random variate <span class="emphasis"><em>x</em></span> in a |
| hyperexponential distribution is given by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span> |
| </p> |
| <p> |
| The following graph illustrates the PDF of the hyperexponential distribution |
| with five different parameters, namely: |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(1.0)</em></span> (which degenerates to a simple |
| exponential distribution), |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.1, 0.9)</em></span> and |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.9, 0.1)</em></span> and |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.2, 0.3, 0.5)</em></span> |
| and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.0, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.5, 0.3, 0.2)</em></span> |
| and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.0, 1.5)</em></span>. |
| </li> |
| </ol></div> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf.svg" align="middle"></span> |
| </p> |
| <p> |
| Also, the following graph illustrates the PDF of the hyperexponential distribution |
| (solid lines) where only the <span class="emphasis"><em>phase probability vector</em></span> |
| changes together with the PDF of the two limiting exponential distributions |
| (dashed lines): |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.1, 0.9)</em></span> and |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.6, 0.4)</em></span> and |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.9, 0.1)</em></span> and |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| Exponential distribution with parameter <span class="emphasis"><em>λ=0.5</em></span>, |
| </li> |
| <li class="listitem"> |
| Exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>. |
| </li> |
| </ol></div> |
| <p> |
| As expected, as the first element <span class="emphasis"><em>α<sub>1</sub></em></span> of the <span class="emphasis"><em>phase |
| probability vector</em></span> approaches to <span class="emphasis"><em>1</em></span> (or, |
| equivalently, <span class="emphasis"><em>α<sub>2</sub></em></span> approaches to <span class="emphasis"><em>0</em></span>), |
| the resulting hyperexponential distribution nears the exponential distribution |
| with parameter <span class="emphasis"><em>λ=0.5</em></span>. Conversely, as the first element |
| <span class="emphasis"><em>α<sub>2</sub></em></span> of the <span class="emphasis"><em>phase probability vector</em></span> |
| approaches to <span class="emphasis"><em>1</em></span> (or, equivalently, <span class="emphasis"><em>α<sub>1</sub></em></span> |
| approaches to <span class="emphasis"><em>0</em></span>), the resulting hyperexponential distribution |
| nears the exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.svg" align="middle"></span> |
| </p> |
| <p> |
| Finally, the following graph compares the PDF of the hyperexponential distribution |
| with different number of phases but with the same mean value equal to |
| <span class="emphasis"><em>2</em></span>: |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(2.0)</em></span> (which degenerates to a simple |
| exponential distribution), |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.5, 0.5)</em></span> and |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.3, 1.5)</em></span>, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0/3.0, 1.0/3.0, 1.0/3.0)</em></span> |
| and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.2, 1.5, 3.0)</em></span>, |
| </li> |
| </ol></div> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samemean.svg" align="middle"></span> |
| </p> |
| <p> |
| As can be noted, even if the three distributions have the same mean value, |
| the two hyperexponential distributions have a <span class="emphasis"><em>longer</em></span> |
| tail with respect to the one of the exponential distribution. Indeed, the |
| hyperexponential distribution has a larger variability than the exponential |
| distribution, thus resulting in a <a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" target="_top">Coefficient |
| of Variation</a> greater than <span class="emphasis"><em>1</em></span> (as opposed to |
| the one of the exponential distribution which is exactly <span class="emphasis"><em>1</em></span>). |
| </p> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h0"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.applications"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.applications">Applications</a> |
| </h4> |
| <p> |
| A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is frequently |
| used in <a href="http://en.wikipedia.org/wiki/Queueing_theory" target="_top">queueing |
| theory</a> to model the distribution of the superposition of <span class="emphasis"><em>k</em></span> |
| independent events, like, for instance, the service time distribution of |
| a queueing station with <span class="emphasis"><em>k</em></span> servers in parallel where |
| the <span class="emphasis"><em>i</em></span>-th server is chosen with probability <span class="emphasis"><em>α<sub>i</sub></em></span> |
| and its service time distribution is an exponential distribution with rate |
| <span class="emphasis"><em>λ<sub>i</sub></em></span> (Allen,1990; Papadopolous et al.,1993; Trivedi,2002). |
| </p> |
| <p> |
| For instance, CPUs service-time distribution in a computing system has |
| often been observed to possess such a distribution (Rosin,1965). Also, |
| the arrival of different types of customer to a single queueing station |
| is often modeled as a hyperexponential distribution (Papadopolous et al.,1993). |
| Similarly, if a product manufactured in several parallel assembly lines |
| and the outputs are merged, the failure density of the overall product |
| is likely to be hyperexponential (Trivedi,2002). |
| </p> |
| <p> |
| Finally, since the hyperexponential distribution exhibits a high Coefficient |
| of Variation (CoV), that is a CoV > 1, it is especially suited to fit |
| empirical data with large CoV (Feitelson,2014; Wolski et al.,2013) and |
| to approximate <a href="http://en.wikipedia.org/wiki/Long_tail" target="_top">long-tail |
| probability distributions</a> (Feldmann et al.,1998). |
| </p> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h1"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.related_distributions"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.related_distributions">Related |
| distributions</a> |
| </h4> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| When the number of phases <span class="emphasis"><em>k</em></span> is equal to <code class="computeroutput"><span class="number">1</span></code>, the hyperexponential distribution |
| is simply an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential |
| Distribution</a>. |
| </li> |
| <li class="listitem"> |
| When the <span class="emphasis"><em>k</em></span> rates are all equal to <span class="emphasis"><em>λ</em></span>, |
| the hyperexponential distribution is simple an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential |
| Distribution</a> with rate <span class="emphasis"><em>λ</em></span>. |
| </li> |
| </ul></div> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h2"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.examples"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.examples">Examples</a> |
| </h4> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h3"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.lifetime_of_appliances"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.lifetime_of_appliances">Lifetime |
| of Appliances</a> |
| </h5> |
| <p> |
| Suppose a customer is buying an appliance and is choosing at random between |
| an appliance with average lifetime of 10 years and an appliance with average |
| lifetime of 12 years. Assuming the lifetime of this appliance follows an |
| exponential distribution, the lifetime distribution of the purchased appliance |
| can be modeled as a hyperexponential distribution with phase probability |
| vector <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1/2,1/2)</em></span> |
| and rate vector <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(1/10,1/12)</em></span> |
| (Wolfram,2014). |
| </p> |
| <p> |
| In the rest of this section, we provide an example C++ implementation for |
| computing the average lifetime and the probability that the appliance will |
| work for more than 15 years. |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10.0</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12.0</span> <span class="special">};</span> |
| |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">);</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Average lifetime: "</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">mean</span><span class="special">(</span><span class="identifier">he</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="string">" years"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that the appliance will work for more than 15 years: "</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">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">he</span><span class="special">,</span> <span class="number">15.0</span><span class="special">))</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| The resulting output is: |
| </p> |
| <pre class="programlisting"><span class="identifier">Average</span> <span class="identifier">lifetime</span><span class="special">:</span> <span class="number">11</span> <span class="identifier">years</span> |
| <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">the</span> <span class="identifier">appliance</span> <span class="identifier">will</span> <span class="identifier">work</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">15</span> <span class="identifier">years</span><span class="special">:</span> <span class="number">0.254817</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h4"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.workloads_of_private_cloud_compu"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.workloads_of_private_cloud_compu">Workloads |
| of Private Cloud Computing Systems</a> |
| </h5> |
| <p> |
| <a href="http://en.wikipedia.org/wiki/Cloud_computing" target="_top">Cloud computing</a> |
| has become a popular metaphor for dynamic and secure self-service access |
| to computational and storage capabilities. In (Wolski et al.,2013), the |
| authors analyze and model workloads gathered from enterprise-operated commercial |
| <a href="http://en.wikipedia.org/wiki/Cloud_computing#Private_cloud" target="_top">private |
| clouds</a> and show that 3-phase hyperexponential distributions (fitted |
| using the <a href="http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm" target="_top">Expectation |
| Maximization algorithm</a>) capture workload attributes accurately. |
| </p> |
| <p> |
| In this type of computing system, user requests consist in demanding the |
| provisioning of one or more <a href="http://en.wikipedia.org/wiki/Virtual_machine" target="_top">Virtual |
| Machines</a> (VMs). In particular, in (Wolski et al.,2013) the workload |
| experienced by each cloud system is a function of four distributions, one |
| for each of the following workload attributes: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <span class="emphasis"><em>Request Interarrival Time</em></span>: the amount of time |
| until the next request, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em>VM Lifetime</em></span>: the time duration over which a VM |
| is provisioned to a physical machine, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em>Request Size</em></span>: the number of VMs in the request, |
| and |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em>Core Count</em></span>: the CPU core count requested for each |
| VM. |
| </li> |
| </ul></div> |
| <p> |
| The authors assume that all VMs in a request have the same core count, |
| but request sizes and core counts can vary from request to request. Moreover, |
| all VMs within a request are assumed to have the same lifetime. Given these |
| assumptions, the authors build a statistical model for the request interarrival |
| time and VM lifetime attributes by fitting their respective data to a 3-phase |
| hyperexponential distribution. |
| </p> |
| <p> |
| In the following table, we show the sample mean and standard deviation |
| (SD), in seconds, of the request interarrival time and of the VM lifetime |
| distributions of the three datasets collected by authors: |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Dataset |
| </p> |
| </th> |
| <th> |
| <p> |
| Mean Request Interarrival Time (SD) |
| </p> |
| </th> |
| <th> |
| <p> |
| Mean Multi-core VM Lifetime (SD) |
| </p> |
| </th> |
| <th> |
| <p> |
| Mean Single-core VM Lifetime (SD) |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| DS1 |
| </p> |
| </td> |
| <td> |
| <p> |
| 2202.1 (2.2e+04) |
| </p> |
| </td> |
| <td> |
| <p> |
| 257173 (4.6e+05) |
| </p> |
| </td> |
| <td> |
| <p> |
| 28754.4 (1.6e+05) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| DS2 |
| </p> |
| </td> |
| <td> |
| <p> |
| 41285.7 (1.1e+05) |
| </p> |
| </td> |
| <td> |
| <p> |
| 144669.0 (7.9e+05) |
| </p> |
| </td> |
| <td> |
| <p> |
| 599815.0 (1.7e+06) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| DS3 |
| </p> |
| </td> |
| <td> |
| <p> |
| 11238.8 (3.0e+04) |
| </p> |
| </td> |
| <td> |
| <p> |
| 30739.2 (1.6e+05) |
| </p> |
| </td> |
| <td> |
| <p> |
| 44447.8 (2.2e+05) |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| Whereas in the following table we show the hyperexponential distribution |
| parameters resulting from the fit: |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Dataset |
| </p> |
| </th> |
| <th> |
| <p> |
| Request Interarrival Time |
| </p> |
| </th> |
| <th> |
| <p> |
| Multi-core VM Lifetime |
| </p> |
| </th> |
| <th> |
| <p> |
| Single-core VM Lifetime |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| DS1 |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.34561,0.08648,0.56791), |
| <span class="bold"><strong>λ</strong></span>=(0.008,0.00005,0.02894)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.24667,0.37948,0.37385), |
| <span class="bold"><strong>λ</strong></span>=(0.00004,0.000002,0.00059)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.09325,0.22251,0.68424), |
| <span class="bold"><strong>λ</strong></span>=(0.000003,0.00109,0.00109)</em></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| DS2 |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.38881,0.18227,0.42892), |
| <span class="bold"><strong>λ</strong></span>=(0.000006,0.05228,0.00081)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.42093,0.43960,0.13947), |
| <span class="bold"><strong>λ</strong></span>=(0.00186,0.00008,0.0000008)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.44885,0.30675,0.2444), |
| <span class="bold"><strong>λ</strong></span>=(0.00143,0.00005,0.0000004)</em></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| DS3 |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.39442,0.24644,0.35914), |
| <span class="bold"><strong>λ</strong></span>=(0.00030,0.00003,0.00257)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.37621,0.14838,0.47541), |
| <span class="bold"><strong>λ</strong></span>=(0.00498,0.000005,0.00022)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.34131,0.12544,0.53325), |
| <span class="bold"><strong>λ</strong></span>=(0.000297,0.000003,0.00410)</em></span> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| In the rest of this section, we provide an example C++ implementation for |
| computing some statistical properties of the fitted distributions for each |
| of the analyzed dataset. |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">string</span><span class="special">></span> |
| |
| <span class="keyword">struct</span> <span class="identifier">ds_info</span> |
| <span class="special">{</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">name</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">iat_sample_mean</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">iat_sample_sd</span><span class="special">;</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">iat_he</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">multi_lt_sample_mean</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">multi_lt_sample_sd</span><span class="special">;</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">multi_lt_he</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">single_lt_sample_mean</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">single_lt_sample_sd</span><span class="special">;</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span> <span class="identifier">single_lt_he</span><span class="special">;</span> |
| <span class="special">};</span> |
| |
| <span class="comment">// DS1 dataset</span> |
| <span class="identifier">ds_info</span> <span class="identifier">make_ds1</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span> |
| |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">=</span> <span class="string">"DS1"</span><span class="special">;</span> |
| |
| <span class="comment">// VM interarrival time distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.34561</span><span class="special">,</span><span class="number">0.08648</span><span class="special">,</span><span class="number">0.56791</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.0008</span><span class="special">,</span><span class="number">0.00005</span><span class="special">,</span><span class="number">0.02894</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_mean</span> <span class="special">=</span> <span class="number">2202.1</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_sd</span> <span class="special">=</span> <span class="number">2.2e+4</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">iat_fit_probs</span><span class="special">,</span> <span class="identifier">iat_fit_rates</span><span class="special">);</span> |
| |
| <span class="comment">// Multi-core VM lifetime distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.24667</span><span class="special">,</span><span class="number">0.37948</span><span class="special">,</span><span class="number">0.37385</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.00004</span><span class="special">,</span><span class="number">0.000002</span><span class="special">,</span><span class="number">0.00059</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_mean</span> <span class="special">=</span> <span class="number">257173</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_sd</span> <span class="special">=</span> <span class="number">4.6e+5</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">multi_lt_fit_probs</span><span class="special">,</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">);</span> |
| |
| <span class="comment">// Single-core VM lifetime distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.09325</span><span class="special">,</span><span class="number">0.22251</span><span class="special">,</span><span class="number">0.68424</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.000003</span><span class="special">,</span><span class="number">0.00109</span><span class="special">,</span><span class="number">0.00109</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_mean</span> <span class="special">=</span> <span class="number">28754.4</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_sd</span> <span class="special">=</span> <span class="number">1.6e+5</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">single_lt_fit_probs</span><span class="special">,</span> <span class="identifier">single_lt_fit_rates</span><span class="special">);</span> |
| |
| <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span> |
| <span class="special">}</span> |
| |
| <span class="comment">// DS2 dataset</span> |
| <span class="identifier">ds_info</span> <span class="identifier">make_ds2</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span> |
| |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">=</span> <span class="string">"DS2"</span><span class="special">;</span> |
| |
| <span class="comment">// VM interarrival time distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.38881</span><span class="special">,</span><span class="number">0.18227</span><span class="special">,</span><span class="number">0.42892</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.000006</span><span class="special">,</span><span class="number">0.05228</span><span class="special">,</span><span class="number">0.00081</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_mean</span> <span class="special">=</span> <span class="number">41285.7</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_sd</span> <span class="special">=</span> <span class="number">1.1e+05</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">iat_fit_probs</span><span class="special">,</span> <span class="identifier">iat_fit_rates</span><span class="special">);</span> |
| |
| <span class="comment">// Multi-core VM lifetime distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.42093</span><span class="special">,</span><span class="number">0.43960</span><span class="special">,</span><span class="number">0.13947</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.00186</span><span class="special">,</span><span class="number">0.00008</span><span class="special">,</span><span class="number">0.0000008</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_mean</span> <span class="special">=</span> <span class="number">144669.0</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_sd</span> <span class="special">=</span> <span class="number">7.9e+05</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">multi_lt_fit_probs</span><span class="special">,</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">);</span> |
| |
| <span class="comment">// Single-core VM lifetime distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.44885</span><span class="special">,</span><span class="number">0.30675</span><span class="special">,</span><span class="number">0.2444</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.00143</span><span class="special">,</span><span class="number">0.00005</span><span class="special">,</span><span class="number">0.0000004</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_mean</span> <span class="special">=</span> <span class="number">599815.0</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_sd</span> <span class="special">=</span> <span class="number">1.7e+06</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">single_lt_fit_probs</span><span class="special">,</span> <span class="identifier">single_lt_fit_rates</span><span class="special">);</span> |
| |
| <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span> |
| <span class="special">}</span> |
| |
| <span class="comment">// DS3 dataset</span> |
| <span class="identifier">ds_info</span> <span class="identifier">make_ds3</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">ds_info</span> <span class="identifier">ds</span><span class="special">;</span> |
| |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special">=</span> <span class="string">"DS3"</span><span class="special">;</span> |
| |
| <span class="comment">// VM interarrival time distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.39442</span><span class="special">,</span><span class="number">0.24644</span><span class="special">,</span><span class="number">0.35914</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">iat_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.00030</span><span class="special">,</span><span class="number">0.00003</span><span class="special">,</span><span class="number">0.00257</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_mean</span> <span class="special">=</span> <span class="number">11238.8</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_sample_sd</span> <span class="special">=</span> <span class="number">3.0e+04</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">iat_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">iat_fit_probs</span><span class="special">,</span> <span class="identifier">iat_fit_rates</span><span class="special">);</span> |
| |
| <span class="comment">// Multi-core VM lifetime distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.37621</span><span class="special">,</span><span class="number">0.14838</span><span class="special">,</span><span class="number">0.47541</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.00498</span><span class="special">,</span><span class="number">0.000005</span><span class="special">,</span><span class="number">0.00022</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_mean</span> <span class="special">=</span> <span class="number">30739.2</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_sample_sd</span> <span class="special">=</span> <span class="number">1.6e+05</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">multi_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">multi_lt_fit_probs</span><span class="special">,</span> <span class="identifier">multi_lt_fit_rates</span><span class="special">);</span> |
| |
| <span class="comment">// Single-core VM lifetime distribution</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_probs</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.34131</span><span class="special">,</span><span class="number">0.12544</span><span class="special">,</span><span class="number">0.53325</span><span class="special">};</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">single_lt_fit_rates</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.000297</span><span class="special">,</span><span class="number">0.000003</span><span class="special">,</span><span class="number">0.00410</span><span class="special">};</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_mean</span> <span class="special">=</span> <span class="number">44447.8</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_sample_sd</span> <span class="special">=</span> <span class="number">2.2e+05</span><span class="special">;</span> |
| <span class="identifier">ds</span><span class="special">.</span><span class="identifier">single_lt_he</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">hyperexponential</span><span class="special">(</span><span class="identifier">single_lt_fit_probs</span><span class="special">,</span> <span class="identifier">single_lt_fit_rates</span><span class="special">);</span> |
| |
| <span class="keyword">return</span> <span class="identifier">ds</span><span class="special">;</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">void</span> <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">ds_info</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">ds</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">secs_in_a_hour</span> <span class="special">=</span> <span class="number">3600</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">secs_in_a_month</span> <span class="special">=</span> <span class="number">30</span><span class="special">*</span><span class="number">24</span><span class="special">*</span><span class="identifier">secs_in_a_hour</span><span class="special">;</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"### "</span> <span class="special"><<</span> <span class="identifier">ds</span><span class="special">.</span><span class="identifier">name</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"* Fitted Request Interarrival Time"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Mean (SD): "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">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"><<</span> <span class="string">" ("</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">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"><<</span> <span class="string">") seconds."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - 99th Percentile: "</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">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"><<</span> <span class="string">" seconds."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Probability that a VM will arrive within 30 minutes: "</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">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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Probability that a VM will arrive after 1 hour: "</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">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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"* Fitted Multi-core VM Lifetime"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Mean (SD): "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">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"><<</span> <span class="string">" ("</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">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"><<</span> <span class="string">") seconds."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - 99th Percentile: "</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">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"><<</span> <span class="string">" seconds."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Probability that a VM will last for less than 1 month: "</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">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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Probability that a VM will last for more than 3 months: "</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">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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"* Fitted Single-core VM Lifetime"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Mean (SD): "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">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"><<</span> <span class="string">" ("</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">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"><<</span> <span class="string">") seconds."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - 99th Percentile: "</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">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"><<</span> <span class="string">" seconds."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Probability that a VM will last for less than 1 month: "</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">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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" - Probability that a VM will last for more than 3 months: "</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">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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds1</span><span class="special">());</span> |
| |
| <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds2</span><span class="special">());</span> |
| |
| <span class="identifier">print_fitted</span><span class="special">(</span><span class="identifier">make_ds3</span><span class="special">());</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| The resulting output (with floating-point precision set to 2) is: |
| </p> |
| <pre class="programlisting"><span class="special">###</span> <span class="identifier">DS1</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">2.2e+03</span> <span class="special">(</span><span class="number">8.1e+03</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">4.3e+04</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">within</span> <span class="number">30</span> <span class="identifier">minutes</span><span class="special">:</span> <span class="number">0.84</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">after</span> <span class="number">1</span> <span class="identifier">hour</span><span class="special">:</span> <span class="number">0.092</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Multi</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">2e+05</span> <span class="special">(</span><span class="number">3.9e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">1.8e+06</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">6.7e-08</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Single</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">3.2e+04</span> <span class="special">(</span><span class="number">1.4e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">7.4e+05</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">6.9e-12</span> |
| <span class="special">###</span> <span class="identifier">DS2</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">6.5e+04</span> <span class="special">(</span><span class="number">1.3e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">6.1e+05</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">within</span> <span class="number">30</span> <span class="identifier">minutes</span><span class="special">:</span> <span class="number">0.52</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">after</span> <span class="number">1</span> <span class="identifier">hour</span><span class="special">:</span> <span class="number">0.4</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Multi</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">1.8e+05</span> <span class="special">(</span><span class="number">6.4e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">3.3e+06</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">0.98</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">0.00028</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Single</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">6.2e+05</span> <span class="special">(</span><span class="number">1.6e+06</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">8e+06</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">0.91</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">0.011</span> |
| <span class="special">###</span> <span class="identifier">DS3</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Request</span> <span class="identifier">Interarrival</span> <span class="identifier">Time</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">9.7e+03</span> <span class="special">(</span><span class="number">2.2e+04</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">1.1e+05</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">within</span> <span class="number">30</span> <span class="identifier">minutes</span><span class="special">:</span> <span class="number">0.53</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">arrive</span> <span class="identifier">after</span> <span class="number">1</span> <span class="identifier">hour</span><span class="special">:</span> <span class="number">0.36</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Multi</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">3.2e+04</span> <span class="special">(</span><span class="number">1e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">5.4e+05</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">1.9e-18</span> |
| <span class="special">*</span> <span class="identifier">Fitted</span> <span class="identifier">Single</span><span class="special">-</span><span class="identifier">core</span> <span class="identifier">VM</span> <span class="identifier">Lifetime</span> |
| <span class="special">-</span> <span class="identifier">Mean</span> <span class="special">(</span><span class="identifier">SD</span><span class="special">):</span> <span class="number">4.3e+04</span> <span class="special">(</span><span class="number">1.6e+05</span><span class="special">)</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="number">99</span><span class="identifier">th</span> <span class="identifier">Percentile</span><span class="special">:</span> <span class="number">8.4e+05</span> <span class="identifier">seconds</span><span class="special">.</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">less</span> <span class="identifier">than</span> <span class="number">1</span> <span class="identifier">month</span><span class="special">:</span> <span class="number">1</span> |
| <span class="special">-</span> <span class="identifier">Probability</span> <span class="identifier">that</span> <span class="identifier">a</span> <span class="identifier">VM</span> <span class="identifier">will</span> <span class="identifier">last</span> <span class="keyword">for</span> <span class="identifier">more</span> <span class="identifier">than</span> <span class="number">3</span> <span class="identifier">months</span><span class="special">:</span> <span class="number">9.3e-12</span> |
| </pre> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| The above results differ from the ones shown in Tables III, V, and VII |
| of (Wolski et al.,2013). We carefully double-checked them with Wolfram |
| Mathematica 10, which confirmed our results. |
| </p></td></tr> |
| </table></div> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h5"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.member_functions"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.member_functions">Member |
| Functions</a> |
| </h4> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h6"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.default_constructor"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.default_constructor">Default |
| Constructor</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">();</span> |
| </pre> |
| <p> |
| Constructs a <span class="emphasis"><em>1</em></span>-phase hyperexponential distribution |
| (i.e., an exponential distribution) with rate <code class="computeroutput"><span class="number">1</span></code>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h7"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.constructor_from_iterators"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.constructor_from_iterators">Constructor |
| from Iterators</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbIterT</span> <span class="identifier">prob_first</span><span class="special">,</span> <span class="identifier">ProbIterT</span> <span class="identifier">prob_last</span><span class="special">,</span> |
| <span class="identifier">RateIterT</span> <span class="identifier">rate_first</span><span class="special">,</span> <span class="identifier">RateIterT</span> <span class="identifier">rate_last</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability |
| vector</em></span> parameter given by the range defined by [<code class="computeroutput"><span class="identifier">prob_first</span></code>, <code class="computeroutput"><span class="identifier">prob_last</span></code>) |
| iterator pair, and <span class="emphasis"><em>rate vector</em></span> parameter given by |
| the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>, |
| <code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair. |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h8"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters">Parameters</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">prob_first</span></code>, <code class="computeroutput"><span class="identifier">prob_last</span></code>: the range of non-negative |
| real elements representing the phase probabilities; elements are normalized |
| to sum to unity. |
| </li> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive |
| elements representing the rates. |
| </li> |
| </ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h9"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements">Type |
| Requirements</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">ProbIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT</span></code>: must meet the requirements |
| of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a> |
| concept. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h10"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example">Example</a> |
| </h6> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">phase_prob</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">};</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span> |
| |
| <span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">phase_prob</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">phase_prob</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">rates</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">rates</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h11"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_ranges_contain"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_ranges_contain">Construction |
| from Ranges/Containers</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">ProbRangeT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">prob_range</span><span class="special">,</span> |
| <span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_range</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability |
| vector</em></span> parameter given by the range defined by <code class="computeroutput"><span class="identifier">prob_range</span></code>, and <span class="emphasis"><em>rate vector</em></span> |
| parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>. |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| As an implementation detail, this constructor uses Boost's <a href="http://www.boost.org/doc/libs/release/libs/core/doc/html/core/enable_if.html" target="_top">enable_if/disable_if |
| mechanism</a> to disambiguate between this and other 2-argument constructors. |
| Refer to the source code for more details. |
| </p></td></tr> |
| </table></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h12"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters0">Parameters</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">prob_range</span></code>: the range |
| of non-negative real elements representing the phase probabilities; |
| elements are normalized to sum to unity. |
| </li> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">rate_range</span></code>: the range |
| of positive real elements representing the rates. |
| </li> |
| </ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h13"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements0">Type |
| Requirements</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">ProbRangeT</span></code>, <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet the requirements |
| of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a> |
| concept: that includes native C++ arrays, standard library containers, |
| or a std::pair or iterators. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h14"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.examples0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.examples0">Examples</a> |
| </h6> |
| <pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">phase_prob</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">};</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span> |
| |
| <span class="identifier">hyperexponential</span> <span class="identifier">he1</span><span class="special">(</span><span class="identifier">phase_prob</span><span class="special">,</span> <span class="identifier">rates</span><span class="special">);</span> <span class="comment">// Construct from standard library container.</span> |
| |
| <span class="keyword">double</span> <span class="identifier">phase_probs2</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">};</span> |
| <span class="keyword">double</span> <span class="identifier">rates2</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span> |
| |
| <span class="identifier">hyperexponential</span> <span class="identifier">he2</span><span class="special">(</span><span class="identifier">phase_probs2</span><span class="special">,</span> <span class="identifier">rates2</span><span class="special">);</span> <span class="comment">// Construct from native C++ array.</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h15"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_with_rates_iterator"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_with_rates_iterator">Construction |
| with rates-iterators (and all phase probabilities equal)</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateIterT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_first</span><span class="special">,</span> |
| <span class="identifier">RateIterT2</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_last</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span> |
| parameter given by the range defined by the [<code class="computeroutput"><span class="identifier">rate_first</span></code>, |
| <code class="computeroutput"><span class="identifier">rate_last</span></code>) iterator pair, |
| and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase |
| probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code> |
| of the <span class="emphasis"><em>rate vector</em></span> and with each element set to <code class="computeroutput"><span class="number">1.0</span><span class="special">/</span><span class="identifier">n</span></code>). |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| As an implementation detail, this constructor uses Boost's <a href="http://www.boost.org/doc/libs/release/libs/core/doc/html/core/enable_if.html" target="_top">enable_if/disable_if |
| mechanism</a> to disambiguate between this and other 2-argument constructors. |
| Refer to the source code for more details. |
| </p></td></tr> |
| </table></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h16"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters1">Parameters</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">rate_first</span></code>, <code class="computeroutput"><span class="identifier">rate_last</span></code>: the range of positive |
| elements representing the rates. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h17"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements1">Type |
| Requirements</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">RateIterT</span></code>, <code class="computeroutput"><span class="identifier">RateIterT2</span></code>: must meet the requirements |
| of the <a href="http://en.cppreference.com/w/cpp/concept/InputIterator" target="_top">InputIterator</a> |
| concept. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h18"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example0">Example</a> |
| </h6> |
| <pre class="programlisting"><span class="comment">// We could be using any standard library container here... vector, deque, array, list etc:</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span> |
| |
| <span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">rates</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> |
| |
| <span class="identifier">assert</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// Phase probabilities will be equal and normalised to unity.</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h19"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_range"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_range">Construction |
| from a single range of rates (all phase probabilities will be equal)</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RateRangeT</span><span class="special">></span> |
| <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateRangeT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_range</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span> |
| parameter given by the range defined by <code class="computeroutput"><span class="identifier">rate_range</span></code>, |
| and <span class="emphasis"><em>phase probability vector</em></span> set to the equal phase |
| probabilities (i.e., to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code> |
| of the <span class="emphasis"><em>rate vector</em></span> and with each element set to <code class="computeroutput"><span class="number">1.0</span><span class="special">/</span><span class="identifier">n</span></code>). |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h20"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters2"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters2">Parameters</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">rate_range</span></code>: the range |
| of positive real elements representing the rates. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h21"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements2"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.type_requirements2">Type |
| Requirements</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">RateRangeT</span></code>: must meet |
| the requirements of the <a href="http://www.boost.org/doc/libs/release/libs/range/doc/html/range/concepts.html" target="_top">Range</a> |
| concept: this includes native C++ array, standard library containers, |
| and a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> of iterators. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h22"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.examples1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.examples1">Examples</a> |
| </h6> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">rates</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span> |
| |
| <span class="identifier">hyperexponential</span> <span class="identifier">he</span><span class="special">(</span><span class="identifier">rates</span><span class="special">);</span> |
| |
| <span class="identifier">assert</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0.5</span><span class="special">);</span> <span class="comment">// Phase probabilities will be equal and normalised to unity.</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h23"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_initializer_li"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_initializer_li">Construction |
| from Initializer lists</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></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"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">l2</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a hyperexponential distribution with <span class="emphasis"><em>phase probability |
| vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a> |
| defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>rate |
| vector</em></span> parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a> |
| defined by <code class="computeroutput"><span class="identifier">l2</span></code>. |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h24"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters3"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters3">Parameters</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list |
| of non-negative real elements representing the phase probabilities; |
| elements are normalized to sum to unity. |
| </li> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">l2</span></code>: the brace-init-list |
| of positive real elements representing the rates. |
| </li> |
| </ul></div> |
| <p> |
| The number of elements of the phase probabilities list and the rates list |
| must be the same. |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h25"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example1">Example</a> |
| </h6> |
| <pre class="programlisting"><span class="identifier">hyperexponential</span> <span class="identifier">he</span> <span class="special">=</span> <span class="special">{</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.5</span> <span class="special">},</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">}</span> <span class="special">};</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h26"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_initi"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_from_a_single_initi">Construction |
| from a single initializer list (all phase probabilities will be equal)</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">l1</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a hyperexponential distribution with <span class="emphasis"><em>rate vector</em></span> |
| parameter given by the <a href="http://en.cppreference.com/w/cpp/language/list_initialization" target="_top">brace-init-list</a> |
| defined by <code class="computeroutput"><span class="identifier">l1</span></code>, and <span class="emphasis"><em>phase |
| probability vector</em></span> set to the equal phase probabilities (i.e., |
| to a vector of the same length <code class="computeroutput"><span class="identifier">n</span></code> |
| of the <span class="emphasis"><em>rate vector</em></span> and with each element set to <code class="computeroutput"><span class="number">1.0</span><span class="special">/</span><span class="identifier">n</span></code>). |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h27"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.parameters4"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.parameters4">Parameters</a> |
| </h6> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| <code class="computeroutput"><span class="identifier">l1</span></code>: the brace-init-list |
| of non-negative real elements representing the phase probabilities; |
| they are normalized to ensure that they sum to unity. |
| </li></ul></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h28"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.example2"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.example2">Example</a> |
| </h6> |
| <pre class="programlisting"><span class="identifier">hyperexponential</span> <span class="identifier">he</span> <span class="special">=</span> <span class="special">{</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">10</span><span class="special">,</span> <span class="number">1.0</span> <span class="special">/</span> <span class="number">12</span> <span class="special">};</span> |
| |
| <span class="identifier">assert</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">()[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0.5</span><span class="special">);</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h29"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.accessors"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.accessors">Accessors</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">num_phases</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Gets the number of phases of this distribution (the size of both the rate |
| and probability vectors). |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h30"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.return_value"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.return_value">Return |
| Value</a> |
| </h6> |
| <p> |
| An non-negative integer number representing the number of phases of this |
| distribution. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">probabilities</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Gets the <span class="emphasis"><em>phase probability vector</em></span> parameter of this |
| distribution. |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| The returned probabilities are the <span class="bold"><strong>normalized</strong></span> |
| versions of the probability parameter values passed at construction time. |
| </p></td></tr> |
| </table></div> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h31"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.return_value0"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.return_value0">Return |
| Value</a> |
| </h6> |
| <p> |
| A vector of non-negative real numbers representing the <span class="emphasis"><em>phase |
| probability vector</em></span> parameter of this distribution. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="identifier">rates</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Gets the <span class="emphasis"><em>rate vector</em></span> parameter of this distribution. |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h32"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.return_value1"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.return_value1">Return |
| Value</a> |
| </h6> |
| <p> |
| A vector of positive real numbers representing the <span class="emphasis"><em>rate vector</em></span> |
| parameter of this distribution. |
| </p> |
| <div class="warning"><table border="0" summary="Warning"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td> |
| <th align="left">Warning</th> |
| </tr> |
| <tr><td align="left" valign="top"> |
| <p> |
| The return type of these functions is a vector-by-value. This is deliberate |
| as we wish to hide the actual container used internally which may be |
| subject to future changes (for example to facilitate vectorization of |
| the cdf code etc). Users should note that some code that might otherwise |
| have been expected to work does not. For example, an attempt to output |
| the (normalized) probabilities: |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">().</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">().</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">" "</span><span class="special">));</span> |
| </pre> |
| <p> |
| fails at compile or runtime because iterator types are incompatible, |
| but, for example, |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</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="char">' '</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">1</span><span class="special">]</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| outputs the expected values. |
| </p> |
| <p> |
| In general if you want to access a member of the returned container, |
| then assign to a variable first, and then access those members: |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">he</span><span class="special">.</span><span class="identifier">probabilities</span><span class="special">();</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">t</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">t</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">" "</span><span class="special">));</span> |
| </pre> |
| </td></tr> |
| </table></div> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h33"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.non_member_accessor_functions"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.non_member_accessor_functions">Non-member |
| Accessor Functions</a> |
| </h4> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor |
| functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. |
| </p> |
| <p> |
| The formulae for calculating these are shown in the table below. |
| </p> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h34"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.accuracy"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.accuracy">Accuracy</a> |
| </h4> |
| <p> |
| The hyperexponential distribution is implemented in terms of the <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a> |
| and as such should have very small errors, usually an <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a> |
| or few. |
| </p> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h35"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.implementation"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.implementation">Implementation</a> |
| </h4> |
| <p> |
| In the following table: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(α<sub>1</sub>,...,α<sub>k</sub>)</em></span> is |
| the <span class="emphasis"><em>phase probability vector</em></span> parameter of the |
| <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(λ<sub>1</sub>,...,λ<sub>k</sub>)</em></span> is |
| the <span class="emphasis"><em>rate vector</em></span> parameter of the <span class="emphasis"><em>k</em></span>-phase |
| hyperexponential distribution, |
| </li> |
| <li class="listitem"> |
| <span class="emphasis"><em>x</em></span> is the random variate. |
| </li> |
| </ul></div> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| support |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>x</em></span> ∈ [0,∞) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_cdf.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_ccdf.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| No closed form available. Computed numerically. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| No closed form available. Computed numerically. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_mean.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_variance.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="number">0</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_skewness.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_kurtosis.svg"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| kurtosis <code class="computeroutput"><span class="special">-</span> <span class="number">3</span></code> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <h4> |
| <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h36"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.references"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.references">References</a> |
| </h4> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| A.O. Allen, <span class="emphasis"><em>Probability, Statistics, and Queuing Theory with |
| Computer Science Applications, Second Edition</em></span>, Academic |
| Press, 1990. |
| </li> |
| <li class="listitem"> |
| D.G. Feitelson, <span class="emphasis"><em>Workload Modeling for Computer Systems Performance |
| Evaluation</em></span>, Cambridge University Press, 2014 |
| </li> |
| <li class="listitem"> |
| A. Feldmann and W. Whitt, <span class="emphasis"><em>Fitting mixtures of exponentials |
| to long-tail distributions to analyze network performance models</em></span>, |
| Performance Evaluation 31(3-4):245, doi:10.1016/S0166-5316(97)00003-5, |
| 1998. |
| </li> |
| <li class="listitem"> |
| H.T. Papadopolous, C. Heavey and J. Browne, <span class="emphasis"><em>Queueing Theory |
| in Manufacturing Systems Analysis and Design</em></span>, Chapman & |
| Hall/CRC, 1993, p. 35. |
| </li> |
| <li class="listitem"> |
| R.F. Rosin, <span class="emphasis"><em>Determining a computing center environment</em></span>, |
| Communications of the ACM 8(7):463-468, 1965. |
| </li> |
| <li class="listitem"> |
| K.S. Trivedi, <span class="emphasis"><em>Probability and Statistics with Reliability, |
| Queueing, and Computer Science Applications</em></span>, John Wiley |
| & Sons, Inc., 2002. |
| </li> |
| <li class="listitem"> |
| Wikipedia, <span class="emphasis"><em>Hyperexponential Distribution</em></span>, Online: |
| <a href="http://en.wikipedia.org/wiki/Hyperexponential_distribution" target="_top">http://en.wikipedia.org/wiki/Hyperexponential_distribution</a>, |
| 2014 |
| </li> |
| <li class="listitem"> |
| R. Wolski and J. Brevik, <span class="emphasis"><em>Using Parametric Models to Represent |
| Private Cloud Workloads</em></span>, IEEE TSC, PrePrint, DOI: <a href="http://doi.ieeecomputersociety.org/10.1109/TSC.2013.48" target="_top">10.1109/TSC.2013.48</a>, |
| 2013. |
| </li> |
| <li class="listitem"> |
| Wolfram Mathematica, <span class="emphasis"><em>Hyperexponential Distribution</em></span>, |
| Online: <a href="http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html" target="_top">http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html</a>, |
| 2014. |
| </li> |
| </ul></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
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