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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Counting process</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>A <b>counting process</b> is a <a href="Stochastic_process" title="Stochastic process">stochastic process</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{N(t),t\geq 0\}}">
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<mo fence="false" stretchy="false">{</mo>
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<annotation encoding="application/x-tex">{\displaystyle \{N(t),t\geq 0\}}</annotation>
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</math></span><img src="./857953211c726383aa504a42bbd3426d5ac4e361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.172ex; height:2.843ex;" alt="{\displaystyle \{N(t),t\geq 0\}}" loading="lazy"></span> with values that are non-negative, integer, and non-decreasing:
</p>
<ol><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(t)\geq 0.}">
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<mi>N</mi>
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<annotation encoding="application/x-tex">{\displaystyle N(t)\geq 0.}</annotation>
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</math></span><img src="./fbe15222fe002631917d93629c2012f6f99769ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.62ex; height:2.843ex;" alt="{\displaystyle N(t)\geq 0.}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(t)}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>N</mi>
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<annotation encoding="application/x-tex">{\displaystyle N(t)}</annotation>
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</math></span><img src="./4a8aa64e349180ee714adfb6620d3bb4610d97d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle N(t)}" loading="lazy"></span> is an integer.</li>
<li>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\leq t}">
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<annotation encoding="application/x-tex">{\displaystyle s\leq t}</annotation>
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</math></span><img src="./c3c12f663b9eb8759c0da898cc542d7cb3bcf5e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.029ex; height:2.176ex;" alt="{\displaystyle s\leq t}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(s)\leq N(t).}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>N</mi>
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<annotation encoding="application/x-tex">{\displaystyle N(s)\leq N(t).}</annotation>
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</math></span><img src="./6f09b9c887741e3f45063da29b6ae729a07ea44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.421ex; height:2.843ex;" alt="{\displaystyle N(s)\leq N(t).}" loading="lazy"></span></li></ol>
<p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s<t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
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<annotation encoding="application/x-tex">{\displaystyle s<t}</annotation>
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</math></span><img src="./7e2def280f2922bece4991a9bd1e06d83d093154.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.029ex; height:2.009ex;" alt="{\displaystyle s<t}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(t)-N(s)}">
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<mi>N</mi>
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<annotation encoding="application/x-tex">{\displaystyle N(t)-N(s)}</annotation>
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</math></span><img src="./286d6bc7a7408265cd13b26519685aabd3f92ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.516ex; height:2.843ex;" alt="{\displaystyle N(t)-N(s)}" loading="lazy"></span> is the number of events occurred during the interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s,t].}">
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<annotation encoding="application/x-tex">{\displaystyle (s,t].}</annotation>
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</math></span><img src="./1d1ec7ef02caed53f720f5996bd4d036dce7faa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.162ex; height:2.843ex;" alt="{\displaystyle (s,t].}" loading="lazy"></span> Examples of counting processes include <a href="Poisson_process" class="mw-redirect" title="Poisson process">Poisson processes</a> and <a href="Renewal_theory" title="Renewal theory">Renewal processes</a>.
</p><p>Counting processes deal with the number of occurrences of something over time. An example of a counting process is the number of job arrivals to a queue over time.
</p><p>If a process has the <a href="Markov_property" title="Markov property">Markov property</a>, it is said to be a Markov counting process.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Intensity_of_counting_processes" title="Intensity of counting processes">Intensity of counting processes</a></li>
<li><a href="Poisson_point_process" title="Poisson point process">Poisson point process</a> (example for a counting process)</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Ross, S.M. (1995) <i>Stochastic Processes</i>. Wiley. <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-12062-9</bdi></li>
<li>Higgins JJ, Keller-McNulty S (1995) <i>Concepts in Probability and Stochastic Modeling</i>. Wadsworth Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-23136-5</bdi></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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