M/G/k queue: Difference between revisions

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Tijms ''et al.'' believe it is "not likely that computationally tractable methods can be developed to compute the exact numerical values of the steady-state probability in the M/G/''k'' queue."<ref name="tijms">{{Cite journal | last1 = Tijms | first1 = H. C. | last2 = Van Hoorn | first2 = M. H. | last3 = Federgruen | first3 = A. | title = Approximations for the Steady-State Probabilities in the M/G/c Queue | journal = Advances in Applied Probability | volume = 13 | issue = 1 | pages = 186–206 | doi = 10.2307/1426474 | jstor = 1426474| year = 1981 | pmid = | pmc = }}</ref>

Various approximations for the average queue size,<ref>{{Cite journal | last1 = Ma | first1 = B. N. W. | last2 = Mark | first2 = J. W. | doi = 10.1287/opre.43.1.158 | title = Approximation of the Mean Queue Length of an M/G/c Queueing System | journal = [[Operations Research (journal)|Operations Research]]| volume = 43 | pages = 158 | year = 1995 | jstor = 171768| pmid = | pmc = }}</ref> stationary distribution<ref>{{Cite journal | last1 = Breuer | first1 = L. | title = Continuity of the M/G/c queue | doi = 10.1007/s11134-008-9073-x | journal = [[Queueing Systems]]| volume = 58 | issue = 4 | pages = 321–331 | year = 2008 | pmid = | pmc = }}</ref><ref name="cite jstor|169760">{{cite journal | last1 = Hokstad | first1 = Per | year = 1978 | title = Approximations for the M/G/m Queue | journal = [[Operations Research: A Journal of the Institute for Operations Research and the Management Sciences|Operations Research]] | volume = 26 | issue = 3 | pages = 510–523 | publisher = INFORMS | jstor = 169760 | doi = 10.1287/opre.26.3.510}}</ref> and approximation by a [[reflected Brownian motion]]<ref>{{Cite journal | last1 = Kimura | first1 = T. | title = Diffusion Approximation for an M/G/m Queue | doi = 10.1287/opre.31.2.304 | journal = [[Operations Research (journal)|Operations Research]]| volume = 31 | issue = 2 | pages = 304–321 | year = 1983 | jstor = 170802| pmid = | pmc = }}</ref><ref name="yao">{{Cite journal | last1 = Yao | first1 = D. D. | title = Refining the Diffusion Approximation for the M/G/m Queue | doi = 10.1287/opre.33.6.1266 | journal = [[Operations Research (journal)|Operations Research]]| volume = 33 | issue = 6 | pages = 1266–1277 | year = 1985 | jstor = 170637| pmid = | pmc = }}</ref> have been offered by different authors. Recently a new approximate approach based on [[Laplace transform]] for steady state probabilities has been proposed by Hamzeh Khazaei ''et al.''.<ref>{{citeCite journal doi| last1 = Khazaei | first1 = H. | last2 = Misic | first2 = J. | last3 = Misic | first3 = V. B. | doi = 10.1109/TPDS.2011.199 | title = Performance Analysis of Cloud Computing Centers Using M/G/m/m+r Queuing Systems | journal = IEEE Transactions on Parallel and Distributed Systems | volume = 23 | issue = 5 | pages = 936 | year = 2012 | pmid = | pmc = }}</ref><ref>{{citeCite book doi| last1 = Khazaei | first1 = H. | last2 = Misic | first2 = J. | last3 = Misic | first3 = V. B. | doi = 10.1109/ICDCSW.2011.13 | chapter = Modelling of Cloud Computing Centers Using M/G/m Queues | title = 2011 31st International Conference on Distributed Computing Systems Workshops | pages = 87 | year = 2011 | isbn = 978-1-4577-0384-3 | pmid = | pmc = }}</ref> This new approach is yet accurate enough in cases of large number of servers and when the distribution of service time has a [[Coefficient of variation]] more than one.

==Average delay/waiting time==