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mathematical model. For the wireless standard, see <a href="/wiki/802.11g" class="mw-redirect" title="802.11g">802.11g</a>.</div> <p>In <a href="/wiki/Queueing_theory" title="Queueing theory">queueing theory</a>, a discipline within the mathematical <a href="/wiki/Probability_theory" title="Probability theory">theory of probability</a>, a <b>G-network</b> (<b>generalized queueing network</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> often called a <b>Gelenbe network</b><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>) is an open network of G-queues first introduced by <a href="/wiki/Erol_Gelenbe" title="Erol Gelenbe">Erol Gelenbe</a> as a model for queueing systems with specific control functions, such as traffic re-routing or traffic destruction, as well as a model for <a href="/wiki/Neural_networks" class="mw-redirect" title="Neural networks">neural networks</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A G-queue is a network of queues with several types of novel and useful customers: </p> <ul><li><i>positive</i> customers, which arrive from other queues or arrive externally as Poisson arrivals, and obey standard service and routing disciplines as in conventional network models,</li> <li><i>negative</i> customers, which arrive from another queue, or which arrive externally as Poisson arrivals, and remove (or 'kill') customers in a non-empty queue, representing the need to remove traffic when the network is congested, including the removal of "batches" of customers</ref><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>"triggers", which arrive from other queues or from outside the network, and which displace customers and move them to other queues</li></ul> <p>A <a href="/wiki/Product-form_solution" title="Product-form solution">product-form solution</a> superficially similar in form to <a href="/wiki/Jackson%27s_theorem_(queueing_theory)" class="mw-redirect" title="Jackson's theorem (queueing theory)">Jackson's theorem</a>, but which requires the solution of a system of non-linear equations for the traffic flows, exists for the stationary distribution of G-networks while the traffic equations of a G-network are in fact surprisingly non-linear, and the model does not obey partial balance. This broke previous assumptions that partial balance was a necessary condition for a product-form solution. A powerful property of G-networks is that they are universal approximators for continuous and bounded functions, so that they can be used to approximate quite general input-output behaviours.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G-network&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">February 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A network of <i>m</i> interconnected queues is a <i>G-network</i> if </p> <ol><li>each queue has one server, who serves at rate <i>μ<sub>i</sub></i>,</li> <li>external arrivals of positive customers or of triggers or resets form <a href="/wiki/Poisson_processes" class="mw-redirect" title="Poisson processes">Poisson processes</a> of rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\Lambda _{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\Lambda _{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba25a90f3abebeef27485170bb919a68b9da38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.765ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\Lambda _{i}}}"></span> for positive customers, while triggers and resets, including negative customers, form a Poisson process of rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\lambda _{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\lambda _{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7756e0e3b03c4ef660bc3b5eb69a93738825892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.583ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\lambda _{i}}}"></span>,</li> <li>on completing service a customer moves from queue <i>i</i> to queue <i>j</i> as a positive customer with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {p_{ij}^{+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {p_{ij}^{+}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e66c0a5fce28f3bcffd69e90561a17e9b7a97a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.063ex; width:2.092ex; height:2.843ex;" alt="{\displaystyle \scriptstyle {p_{ij}^{+}}}"></span>, as a trigger or reset with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {p_{ij}^{-}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {p_{ij}^{-}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a68fffcf7315e3f3ab6103dba0166b031eb422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.063ex; width:2.092ex; height:2.843ex;" alt="{\displaystyle \scriptstyle {p_{ij}^{-}}}"></span> and departs the network with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {d_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {d_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8fe8db64c4958026428a8068e05bf9366c5dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.48ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {d_{i}}}"></span>,</li> <li>on arrival to a queue, a positive customer acts as usual and increases the queue length by 1,</li> <li>on arrival to a queue, the negative customer reduces the length of the queue by some random number (if there is at least one positive customer present at the queue), while a trigger moves a customer probabilistically to another queue and a reset sets the state of the queue to its steady-state if the queue is empty when the reset arrives. All triggers, negative customers and resets disappear after they have taken their action, so that they are in fact "control" signals in the network,</li></ol> <ul><li>note that normal customers leaving a queue can become triggers or resets and negative customers when they visit the next queue.</li></ul> <p>A queue in such a network is known as a <b>G-queue</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Stationary_distribution">Stationary distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G-network&action=edit&section=2" title="Edit section: Stationary distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Define the utilization at each node, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}={\frac {\lambda _{i}^{+}}{\mu _{i}+\lambda _{i}^{-}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}={\frac {\lambda _{i}^{+}}{\mu _{i}+\lambda _{i}^{-}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30758a631d1955f0bc17e3af24f34ff90a0053e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.844ex; height:6.843ex;" alt="{\displaystyle \rho _{i}={\frac {\lambda _{i}^{+}}{\mu _{i}+\lambda _{i}^{-}}}}"></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\lambda _{i}^{+},\lambda _{i}^{-}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\lambda _{i}^{+},\lambda _{i}^{-}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860b553dbd4a6835ccf9f9ad1d78403928edb126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.778ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {\lambda _{i}^{+},\lambda _{i}^{-}}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {i=1,\ldots ,m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {i=1,\ldots ,m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d7345ab9b6af127be71c823f4a2c7cef4fffc36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.951ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {i=1,\ldots ,m}}"></span> satisfy </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}^{+}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{+}+\Lambda _{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}^{+}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{+}+\Lambda _{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312c0ad6165b1c537102edc0f86a694cbf649823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.45ex; height:5.843ex;" alt="{\displaystyle \lambda _{i}^{+}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{+}+\Lambda _{i}\,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}^{-}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{-}+\lambda _{i}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}^{-}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{-}+\lambda _{i}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9818371ba1dfeb4eff20d6d3da3005615058622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.839ex; height:5.843ex;" alt="{\displaystyle \lambda _{i}^{-}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{-}+\lambda _{i}.\,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>Then writing (<i>n</i><sub>1</sub>, ... ,<i>n</i><sub>m</sub>) for the state of the network (with queue length <i>n</i><sub><i>i</i></sub> at node <i>i</i>), if a unique non-negative solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {(\lambda _{i}^{+},\lambda _{i}^{-})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {(\lambda _{i}^{+},\lambda _{i}^{-})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2acfac9be45483c383a82300f05589acaa7396f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.058ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {(\lambda _{i}^{+},\lambda _{i}^{-})}}"></span> exists to the above equations (<b><a href="#math_1">1</a></b>) and (<b><a href="#math_2">2</a></b>) such that <i>ρ</i><sub><i>i</i></sub> for all <i>i</i> then the stationary probability distribution π exists and is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n_{1},n_{2},\ldots ,n_{m})=\prod _{i=1}^{m}(1-\rho _{i})\rho _{i}^{n_{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n_{1},n_{2},\ldots ,n_{m})=\prod _{i=1}^{m}(1-\rho _{i})\rho _{i}^{n_{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb7edb256808674fd46f04cfc08ae5459255600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.895ex; height:6.843ex;" alt="{\displaystyle \pi (n_{1},n_{2},\ldots ,n_{m})=\prod _{i=1}^{m}(1-\rho _{i})\rho _{i}^{n_{i}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Proof">Proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G-network&action=edit&section=3" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">February 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>It is sufficient to show <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> satisfies the <a href="/wiki/Balance_equation" title="Balance equation">global balance equations</a> which, quite differently from Jackson networks are non-linear. We note that the model also allows for multiple classes. </p><p>G-networks have been used in a wide range of applications, including to represent Gene Regulatory Networks, the mix of control and payload in packet networks, neural networks, and the representation of colour images and medical images such as Magnetic Resonance Images. </p> <div class="mw-heading mw-heading2"><h2 id="Response_time_distribution">Response time distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G-network&action=edit&section=4" title="Edit section: Response time distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The response time is the length of time a customer spends in the system. The response time distribution for a single G-queue is known<sup id="cite_ref-joap-sojourn_9-0" class="reference"><a href="#cite_note-joap-sojourn-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> where customers are served using a <a href="/wiki/First_come_first_served" class="mw-redirect" title="First come first served">FCFS</a> discipline at rate <i>μ</i>, with positive arrivals at rate <i>λ</i><sup>+</sup> and negative arrivals at rate <i>λ</i><sup>−</sup> which kill customers from the end of the queue. The <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> of response time distribution in this situation is<sup id="cite_ref-joap-sojourn_9-1" class="reference"><a href="#cite_note-joap-sojourn-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-net-resp_10-0" class="reference"><a href="#cite_note-net-resp-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{\ast }(s)={\frac {\mu (1-\rho )}{\lambda ^{+}}}{\frac {s+\lambda +\mu (1-\rho )-{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}{\lambda ^{-}-\lambda ^{+}-\mu (1-\rho )-s+{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mi>s</mi> <mo>+</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mi>s</mi> <mo>+</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{\ast }(s)={\frac {\mu (1-\rho )}{\lambda ^{+}}}{\frac {s+\lambda +\mu (1-\rho )-{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}{\lambda ^{-}-\lambda ^{+}-\mu (1-\rho )-s+{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d527d32f75ce7219e9af3975fd37d00246bdec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:77.197ex; height:7.509ex;" alt="{\displaystyle W^{\ast }(s)={\frac {\mu (1-\rho )}{\lambda ^{+}}}{\frac {s+\lambda +\mu (1-\rho )-{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}{\lambda ^{-}-\lambda ^{+}-\mu (1-\rho )-s+{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}}}"></span></dd></dl> <p>where <i>λ</i> = <i>λ</i><sup>+</sup> + <i>λ</i><sup>−</sup> and <i>ρ</i> = <i>λ</i><sup>+</sup>/(<i>λ</i><sup>−</sup> + <i>μ</i>), requiring <i>ρ</i> < 1 for stability. </p><p>The response time for a tandem pair of G-queues (where customers who finish service at the first node immediately move to the second, then leave the network) is also known, and it is thought extensions to larger networks will be intractable.<sup id="cite_ref-net-resp_10-1" class="reference"><a href="#cite_note-net-resp-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G-network&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style 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"Sojourn Times in Single-Server Queues with Negative Customers". <i>Journal of Applied Probability</i>. <b>30</b> (4): 943–963. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3214524">10.2307/3214524</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3214524">3214524</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Probability&rft.atitle=Sojourn+Times+in+Single-Server+Queues+with+Negative+Customers&rft.volume=30&rft.issue=4&rft.pages=943-963&rft.date=1993&rft_id=info%3Adoi%2F10.2307%2F3214524&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3214524%23id-name%3DJSTOR&rft.aulast=Harrison&rft.aufirst=P.+G.&rft.au=Pitel%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG-network" class="Z3988"></span></span> </li> <li id="cite_note-net-resp-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-net-resp_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-net-resp_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarrison1998" class="citation conference cs1"><a href="/wiki/Peter_G._Harrison" title="Peter G. Harrison">Harrison, Peter G.</a> (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ClU2XG64UOIC&pg=PA9"><i>Response times in G-nets</i></a>. 13th International Symposium on Computer and Information Sciences (ISCIS 1998). pp. 9–16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9051994052" title="Special:BookSources/9051994052"><bdi>9051994052</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Response+times+in+G-nets&rft.pages=9-16&rft.date=1998&rft.isbn=9051994052&rft.aulast=Harrison&rft.aufirst=Peter+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DClU2XG64UOIC%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG-network" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Queueing_theory" title="Template:Queueing theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Queueing_theory" title="Template talk:Queueing theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Queueing_theory" title="Special:EditPage/Template:Queueing theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Queueing_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Single queueing nodes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/D/M/1_queue" title="D/M/1 queue">D/M/1 queue</a></li> <li><a href="/wiki/M/D/1_queue" title="M/D/1 queue">M/D/1 queue</a></li> <li><a href="/wiki/M/D/c_queue" title="M/D/c queue">M/D/c queue</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1 queue</a> <ul><li><a href="/wiki/Burke%27s_theorem" title="Burke's theorem">Burke's theorem</a></li></ul></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c queue</a></li> <li><a href="/wiki/M/M/%E2%88%9E_queue" title="M/M/∞ queue">M/M/∞ queue</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1 queue</a> <ul><li><a href="/wiki/Pollaczek%E2%80%93Khinchine_formula" title="Pollaczek–Khinchine formula">Pollaczek–Khinchine formula</a></li> <li><a href="/wiki/Matrix_analytic_method" title="Matrix analytic method">Matrix analytic method</a></li></ul></li> <li><a href="/wiki/M/G/k_queue" title="M/G/k queue">M/G/k queue</a></li> <li><a href="/wiki/G/M/1_queue" title="G/M/1 queue">G/M/1 queue</a></li> <li><a href="/wiki/G/G/1_queue" title="G/G/1 queue">G/G/1 queue</a> <ul><li><a href="/wiki/Kingman%27s_formula" title="Kingman's formula">Kingman's formula</a></li> <li><a href="/wiki/Lindley_equation" title="Lindley equation">Lindley equation</a></li></ul></li> <li><a href="/wiki/Fork%E2%80%93join_queue" title="Fork–join queue">Fork–join queue</a></li> <li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk queue</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Arrival processes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson point process</a></li> <li><a href="/wiki/Markovian_arrival_process" title="Markovian arrival process">Markovian arrival process</a></li> <li><a href="/wiki/Rational_arrival_process" title="Rational arrival process">Rational arrival process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Queueing networks</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jackson_network" title="Jackson network">Jackson network</a> <ul><li><a href="/wiki/Traffic_equations" title="Traffic equations">Traffic equations</a></li></ul></li> <li><a href="/wiki/Gordon%E2%80%93Newell_theorem" title="Gordon–Newell theorem">Gordon–Newell theorem</a> <ul><li><a href="/wiki/Mean_value_analysis" title="Mean value analysis">Mean value analysis</a></li> <li><a href="/wiki/Buzen%27s_algorithm" title="Buzen's algorithm">Buzen's algorithm</a></li></ul></li> <li><a href="/wiki/Kelly_network" title="Kelly network">Kelly network</a></li> <li><a class="mw-selflink selflink">G-network</a></li> <li><a href="/wiki/BCMP_network" title="BCMP network">BCMP network</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Service policies</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/FIFO_(computing_and_electronics)" title="FIFO (computing and electronics)">FIFO</a></li> <li><a href="/wiki/LIFO_(computing)" class="mw-redirect" title="LIFO (computing)">LIFO</a></li> <li><a href="/wiki/Processor_sharing" title="Processor sharing">Processor sharing</a></li> <li><a href="/wiki/Round-robin_scheduling" title="Round-robin scheduling">Round-robin</a></li> <li><a href="/wiki/Shortest_job_next" title="Shortest job next">Shortest job next</a></li> <li><a href="/wiki/Shortest_remaining_time" title="Shortest remaining time">Shortest remaining time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Continuous-time_Markov_chain" title="Continuous-time Markov chain">Continuous-time Markov chain</a></li> <li><a href="/wiki/Kendall%27s_notation" title="Kendall's notation">Kendall's notation</a></li> <li><a href="/wiki/Little%27s_law" title="Little's law">Little's law</a></li> <li><a href="/wiki/Product-form_solution" title="Product-form solution">Product-form solution</a> <ul><li><a href="/wiki/Balance_equation" title="Balance equation">Balance equation</a></li> <li><a href="/wiki/Quasireversibility" title="Quasireversibility">Quasireversibility</a></li> <li><a href="/wiki/Flow-equivalent_server_method" title="Flow-equivalent server method">Flow-equivalent server method</a></li></ul></li> <li><a href="/wiki/Arrival_theorem" title="Arrival theorem">Arrival theorem</a></li> <li><a href="/wiki/Decomposition_method_(queueing_theory)" title="Decomposition method (queueing theory)">Decomposition method</a></li> <li><a href="/wiki/Bene%C5%A1_method" title="Beneš method">Beneš method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fluid_limit" title="Fluid limit">Fluid limit</a></li> <li><a href="/wiki/Mean-field_theory" title="Mean-field theory">Mean-field theory</a></li> <li><a href="/wiki/Heavy_traffic_approximation" title="Heavy traffic approximation">Heavy traffic approximation</a> <ul><li><a href="/wiki/Reflected_Brownian_motion" title="Reflected Brownian motion">Reflected Brownian motion</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid queue</a></li> <li><a href="/wiki/Layered_queueing_network" title="Layered queueing network">Layered queueing network</a></li> <li><a href="/wiki/Polling_system" title="Polling system">Polling system</a></li> <li><a href="/wiki/Adversarial_queueing_network" title="Adversarial queueing network">Adversarial queueing network</a></li> <li><a href="/wiki/Loss_network" title="Loss network">Loss network</a></li> <li><a href="/wiki/Retrial_queue" title="Retrial queue">Retrial queue</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Data_buffer" title="Data buffer">Data buffer</a></li> <li><a href="/wiki/Erlang_(unit)" title="Erlang (unit)">Erlang (unit)</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a></li> <li><a href="/wiki/Flow_control_(data)" title="Flow control (data)">Flow control (data)</a></li> <li><a href="/wiki/Message_queue" title="Message queue">Message queue</a></li> <li><a href="/wiki/Network_congestion" title="Network congestion">Network congestion</a></li> <li><a href="/wiki/Network_scheduler" title="Network scheduler">Network scheduler</a></li> <li><a href="/wiki/Pipeline_(software)" title="Pipeline (software)">Pipeline (software)</a></li> <li><a href="/wiki/Quality_of_service" title="Quality of service">Quality of service</a></li> <li><a href="/wiki/Scheduling_(computing)" title="Scheduling (computing)">Scheduling (computing)</a></li> <li><a href="/wiki/Teletraffic_engineering" title="Teletraffic engineering">Teletraffic engineering</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Stochastic_processes" title="Template:Stochastic processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Stochastic_processes" title="Template talk:Stochastic processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes" style="font-size:114%;margin:0 4em"><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a class="mw-selflink selflink">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li> <li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Category</a></li></ul> </div></td></tr></tbody></table></div>    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