G/G/1 queue

<h2 id="waiting-time">Waiting time</h2>
<a href="/facts/Kingman%2527s_formula/0cProUut">Kingman's formula</a> gives an approximation for the mean waiting time in a G/G/1 queue.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> <a href="/facts/Lindley%2527s_integral_equation/jyfVbf5M">Lindley's integral equation</a> is a relationship satisfied by the stationary waiting time distribution which can be solved using the <a href="/facts/Wiener%25E2%2580%2593Hopf_method/bvSfGfMz">Wiener–Hopf method</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="multiple-servers">Multiple servers</h2>
Few results are known for the general G/G/k model as it generalises the <a href="/facts/M%2fG%2fk_queue/N78At3Yo">M/G/k queue</a> for which few metrics are known. Bounds can be computed using <a href="/facts/Mean_value_analysis/AB5hVYI6">mean value analysis</a> techniques, adapting results from the <a href="/facts/M%2fM%2fc_queue/MXB6lmQ1">M/M/c queue</a> model, using <a href="/facts/Heavy_traffic_approximation/a3QNauD2">heavy traffic approximations</a>, empirical results<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: 189 <a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> or approximating distributions by <a href="/facts/Phase_type_distribution/LGDbCMKh">phase type distributions</a> and then using <a href="/facts/Matrix_analytic_method/I9SPfyKa">matrix analytic methods</a> to solve the approximate systems.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>: 201 
In a G/G/2 queue with <a href="/facts/Heavy-tailed_distribution/BHLSHPku">heavy-tailed</a> job sizes, the tail of the delay time distribution is known to behave like the tail of an exponential distribution squared under low loads and like the tail of an exponential distribution for high loads.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bhat, U. N. (2008). "The General Queue G/G/1 and Approximations". An Introduction to Queueing Theory. pp. 169–183. doi:10.1007/978-0-8176-4725-4_9. ISBN 978-0-8176-4724-7. <a href="978-0-8176-4724-7" target="_blank">978-0-8176-4724-7</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Foss, S. (2011). "The G/G/1 Queue". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0878. ISBN 9780470400531. <a href="9780470400531" target="_blank">9780470400531</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kendall, D. G. (1953). "Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain". The Annals of Mathematical Statistics. 24 (3): 338. doi:10.1214/aoms/1177728975. JSTOR 2236285. <a href="/wiki/David_George_Kendall" target="_blank">/wiki/David_George_Kendall</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Smith, W. L. (1953). "On the distribution of queueing times". Mathematical Proceedings of the Cambridge Philosophical Society. 49 (3): 449. Bibcode:1953PCPS...49..449S. doi:10.1017/S0305004100028620. <a href="/wiki/Wally_Smith_(mathematician)" target="_blank">/wiki/Wally_Smith_(mathematician)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Grassmann, Winfried; Tavakoli, Javad (June 2019). "The Distribution of the Line Length in a Discrete Time GI/G/1 Queue". Performance Evaluation. 131: 43–53. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kingman, J. F. C. (October 1961). "The single server queue in heavy traffic". Mathematical Proceedings of the Cambridge Philosophical Society. 57 (4): 902. Bibcode:1961PCPS...57..902K. doi:10.1017/S0305004100036094. JSTOR 2984229. <a href="/wiki/John_Kingman" target="_blank">/wiki/John_Kingman</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Prabhu, N. U. (1974). "Wiener-Hopf Techniques in Queueing Theory". Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems. Vol. 98. pp. 81–90. doi:10.1007/978-3-642-80838-8_5. ISBN 978-3-540-06763-4. <a href="978-3-540-06763-4" target="_blank">978-3-540-06763-4</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Gautam, Natarajan (2012). Analysis of Queues: Methods and Applications. CRC Press. ISBN 9781439806586. <a href="9781439806586" target="_blank">9781439806586</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Whitt, W. (2009). "Approximations for the GI/G/m Queue" (PDF). Production and Operations Management. 2 (2): 114–161. doi:10.1111/j.1937-5956.1993.tb00094.x. <a href="/wiki/Ward_Whitt" target="_blank">/wiki/Ward_Whitt</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Gautam, Natarajan (2012). Analysis of Queues: Methods and Applications. CRC Press. ISBN 9781439806586. <a href="9781439806586" target="_blank">9781439806586</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Harchol-Balter, M. (2012). "Task Assignment Policies for Server Farms". Performance Modeling and Design of Computer Systems. p. 408. doi:10.1017/CBO9781139226424.031. ISBN 9781139226424. <a href="9781139226424" target="_blank">9781139226424</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Whitt, W. (2000). "The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution" (PDF). Queueing Systems. 36: 71–87. doi:10.1023/A:1019143505968. <a href="/wiki/Ward_Whitt" target="_blank">/wiki/Ward_Whitt</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Foss, S.; Korshunov, D. (2006). "Heavy Tails in Multi-Server Queue". Queueing Systems. 52: 31. arXiv:1303.4705. doi:10.1007/s11134-006-3613-z. <a href="/wiki/Queueing_Systems" target="_blank">/wiki/Queueing_Systems</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
</ol>

G/G/1 queue open-in-new

G/G/1 queue