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<h2 id="kendalls-notation">Kendall's notation</h2> <p>In <a href="/facts/Kendall%2527s_notation/zz5PJ7vs">Kendall's notation</a> for single queueing nodes, the random variable denoting bulk arrivals or service is denoted with a superscript, for example M<i>X</i>/M<i>Y</i>/1 denotes an <a href="/facts/M%2fM%2f1_queue/eTkKX6Ro">M/M/1 queue</a> where the arrivals are in batches determined by the random variable <i>X</i> and the services in bulk determined by the random variable <i>Y</i>. In a similar way, the <a href="/facts/GI%2fG%2f1_queue/pcfXFDIR">GI/G/1 queue</a> is extended to GI<i>X</i>/G<i>Y</i>/1.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> <h2 id="bulk-service">Bulk service</h2> <p>Customers arrive at random instants according to a <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a> and form a single queue, from the front of which batches of customers (typically with a fixed maximum size<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>) are served at a rate with independent distribution.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> The equilibrium distribution, mean and variance of queue length are known for this model.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>The optimal maximum size of batch, subject to operating cost constraints, can be modelled as a <a href="/facts/Markov_decision_process/cVlWIDrw">Markov decision process</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h2 id="bulk-arrival">Bulk arrival</h2> <p>Optimal service-provision procedures to minimize long run expected cost have been published.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <h2 id="waiting-time-distribution">Waiting Time Distribution</h2> <p>The waiting time distribution of bulk Poisson arrival is presented in.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Chiamsiri, Singha; Leonard, Michael S. (1981). "A Diffusion Approximation for Bulk Queues". Management Science. 27 (10): 1188–1199. doi:10.1287/mnsc.27.10.1188. JSTOR 2631086. <a href="/wiki/Management_Science_(journal)" target="_blank">/wiki/Management_Science_(journal)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Özden, Eda (2012). Discrete Time Analysis of Consolidated Transport Processes. KIT Scientific Publishing. p. 14. ISBN 978-3866448018. <a href="978-3866448018" target="_blank">978-3866448018</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Chaudhry, M. L.; Templeton, James G. C. (1983). A first course in bulk queues. Wiley. ISBN 978-0471862604. <a href="978-0471862604" target="_blank">978-0471862604</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Berg, Menachem; van der Duyn Schouten, Frank; Jansen, Jorg (1998). "Optimal Batch Provisioning to Customers Subject to a Delay-Limit". Management Science. 44 (5): 684–697. doi:10.1287/mnsc.44.5.684. JSTOR 2634473. <a href="/wiki/Management_Science_(journal)" target="_blank">/wiki/Management_Science_(journal)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Bailey, Norman T. J. (1954). "On Queueing Processes with Bulk Service". Journal of the Royal Statistical Society, Series B. 61 (1): 80–87. JSTOR 2984011. <a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" target="_blank">/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Deb, Rajat K. (1978). "Optimal Dispatching of a Finite Capacity Shuttle". Management Science. 24 (13): 1362–1372. doi:10.1287/mnsc.24.13.1362. JSTOR 2630642. <a href="/wiki/Management_Science_(journal)" target="_blank">/wiki/Management_Science_(journal)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Glazer, A.; Hassin, R. (1987). "Equilibrium Arrivals in Queues with Bulk Service at Scheduled Times". Transportation Science. 21 (4): 273–278. doi:10.1287/trsc.21.4.273. JSTOR 25768286. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Marcel F. Neuts (1967). "A General Class of Bulk Queues with Poisson Input" (PDF). The Annals of Mathematical Statistics. 38 (3): 759–770. doi:10.1214/aoms/1177698869. JSTOR 2238992. <a href="http://www.stat.purdue.edu/research/technical_reports/pdfs/1-527/tr-046.pdf" target="_blank">http://www.stat.purdue.edu/research/technical_reports/pdfs/1-527/tr-046.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Henderson, W.; Taylor, P. G. (1990). "Product form in networks of queues with batch arrivals and batch services". Queueing Systems. 6: 71–87. doi:10.1007/BF02411466. <a href="/wiki/Queueing_Systems" target="_blank">/wiki/Queueing_Systems</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Iglehart, Donald L.; Ward, Whitt (1970). "Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches" (PDF). Advances in Applied Probability. 2 (2): 355–369. doi:10.1017/s0001867800037435. JSTOR 1426324. Retrieved 30 Nov 2012. <a href="/wiki/Ward_Whitt" target="_blank">/wiki/Ward_Whitt</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Harrison, P. G.; Hayden, R. A.; Knottenbelt, W. (2013). "Product-forms in batch networks: Approximation and asymptotics" (PDF). Performance Evaluation. 70 (10): 822. CiteSeerX 10.1.1.352.5769. doi:10.1016/j.peva.2013.08.011. Archived from the original (PDF) on 2016-03-03. Retrieved 2015-09-04. <a href="/wiki/Peter_G._Harrison" target="_blank">/wiki/Peter_G._Harrison</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Chiamsiri, Singha; Leonard, Michael S. (1981). "A Diffusion Approximation for Bulk Queues". Management Science. 27 (10): 1188–1199. doi:10.1287/mnsc.27.10.1188. JSTOR 2631086. <a href="/wiki/Management_Science_(journal)" target="_blank">/wiki/Management_Science_(journal)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Downton, F. (1955). "Waiting Time in Bulk Service Queues". Journal of the Royal Statistical Society, Series B. 17 (2). Royal Statistical Society: 256–261. JSTOR 2983959. <a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" target="_blank">/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Bailey, Norman T. J. (1954). "On Queueing Processes with Bulk Service". Journal of the Royal Statistical Society, Series B. 61 (1): 80–87. JSTOR 2984011. <a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" target="_blank">/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Bailey, Norman T. J. (1954). "On Queueing Processes with Bulk Service". Journal of the Royal Statistical Society, Series B. 61 (1): 80–87. JSTOR 2984011. <a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" target="_blank">/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Deb, Rajat K.; Serfozo, Richard F. (1973). "Optimal Control of Batch Service Queues". Advances in Applied Probability. 5 (2): 340–361. doi:10.2307/1426040. JSTOR 1426040. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Berg, Menachem; van der Duyn Schouten, Frank; Jansen, Jorg (1998). "Optimal Batch Provisioning to Customers Subject to a Delay-Limit". Management Science. 44 (5): 684–697. doi:10.1287/mnsc.44.5.684. JSTOR 2634473. <a href="/wiki/Management_Science_(journal)" target="_blank">/wiki/Management_Science_(journal)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Medhi, Jyotiprasad (1975). "Waiting Time Distribution in a Poisson Queue with a General Bulk Service Rule". Management Science. 21 (7): 777–782. doi:10.1287/mnsc.21.7.777. JSTOR 2629773. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> </ol>