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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], an '''M/G/k queue''' is a queue model where arrivals are '''M'''arkovian (modulated by a [[Poisson process]]), service times have a '''G'''eneral [[probability distribution|distribution]] and there are ''k'' servers. The model name is written in [[Kendall's notation]], and is an extension of the [[M/M/c queue]], where service times must be [[exponential distribution|exponentially distributed]] and of the [[M/G/1 queue]] with a single server. Most performance metrics for this queueing system are not known and remain an [[open problem]].<ref>{{citeCite journal doi| last1 = Kingman | first1 = J. F. C. | authorlink1 = John Kingman | title = The first Erlang century—and the next | journal = [[Queueing Systems]] | volume = 63 | pages = 3–4 | year = 2009 | doi = 10.1007/s11134-009-9147-4}}</ref>
==Model definition==
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 jstor|1426474}}</ref>
Various approximations for the average queue size,<ref>{{citeCite journal doi| 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>{{citeCite journal doi| 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 jstor|169760}}</ref> and approximation by a [[reflected Brownian motion]]<ref>{{citeCite journal doi| 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">{{citeCite journal doi| 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>{{cite doi|10.1109/TPDS.2011.199}}</ref><ref>{{cite doi|10.1109/ICDCSW.2011.13}}</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==
There are numerous approximations for the average delay a job experiences.<ref name="cite jstor|169760"/><ref name="yao" /><ref>{{cite jstor|1426432}}</ref><ref>{{cite jstor|3212698}}</ref><ref>{{cite jstor|3213437}}</ref><ref>{{cite jstor|172087}}</ref> The first such was given in 1959 using a factor to adjust the mean waiting time in an [[M/M/c queue]]<ref name="gbdz" /><ref>{{citeCite journal doi| last1 = Lee | first1 = A. M. | last2 = Longton | first2 = P. A. | doi = 10.1057/jors.1959.5 | title = Queueing Processes Associated with Airline Passenger Check-in | journal = [[Journal of the Operational Research Society]]| volume = 10 | pages = 56 | year = 1959 | pmid = | pmc = }}</ref> This result is sometimes known as Kingman's law of congestion.<ref>{{citeCite journal doi| last1 = Gans | first1 = N. | last2 = Koole | first2 = G. | last3 = Mandelbaum | first3 = A. | doi = 10.1287/msom.5.2.79.16071 | title = Telephone Call Centers: Tutorial, Review, and Research Prospects | journal = [[Manufacturing & Service Operations Management]]| volume = 5 | issue = 2 | pages = 79 | year = 2003 | url = http://ie.technion.ac.il/Labs/Serveng/files/CCReview.pdf| pmid = | pmc = }}</ref>
:<math>E[W^{\text{M/G/}k}] = \frac{C^2+1}{2} \mathbb E [ W^{\text{M/M/}c}]</math>
where ''C''<sup>2</sup> is the [[coefficient of variation]] of the service time distribution. [[Ward Whitt]] described this approximation as “usually an excellent approximation, even given extra information about the service-time distribution."<ref>{{citeCite journal doi| last1 = Whitt | first1 = W. | authorlink1 = Ward Whitt| title = Approximations for the GI/G/m Queue| doi = 10.1111/j.1937-5956.1993.tb00094.x | journal = [[Production and Operations Management]]| volume = 2 | issue = 2 | pages = 114–161 | year = 2009 | url = http://www.columbia.edu/~ww2040/ApproxGIGm1993.pdf| pmid = | pmc = }}</ref>
However, it is known that no approximation using only the first two moments can be accurate in all cases.<ref name="gbdz">{{citeCite journal doi| last1 = Gupta | first1 = V. | last2 = Harchol-Balter | first2 = M. | last3 = Dai | first3 = J. G. | last4 = Zwart | first4 = B. | title = On the inapproximability of M/G/K: Why two moments of job size distribution are not enough | doi = 10.1007/s11134-009-9133-x | journal = [[Queueing Systems]]| volume = 64 | pages = 5 | year = 2009 | url = http://repository.cmu.edu/cgi/viewcontent.cgi?article=1867&context=compsci| pmid = | pmc = }}</ref>
A [[Markov–Krein]] characterisation has been shown to produce tight bounds on the mean waiting time.<ref>{{citeCite journal doi| last1 = Gupta | first1 = V. | last2 = Osogami | first2 = T. | doi = 10.1007/s11134-011-9248-8 | title = On Markov–Krein characterization of the mean waiting time in M/G/K and other queueing systems | journal = Queueing Systems | volume = 68 | issue = 3–4 | pages = 339 | year = 2011 | pmid = | pmc = }}</ref>
==Inter-departure times==
It is conjectured that the times between departures, given a departure leaves ''n'' customers in a queue, has a mean which as ''n'' tends to infinity is different from the intuitive 1/''μ'' result.<ref>{{citeCite journal doi| last1 = Veeger | first1 = C. | last2 = Kerner | first2 = Y. | last3 = Etman | first3 = P. | last4 = Adan | first4 = I. | title = Conditional inter-departure times from the M/G/s queue | doi = 10.1007/s11134-011-9240-3 | journal = [[Queueing Systems]]| volume = 68 | issue = 3–4 | pages = 353 | year = 2011 | pmid = | pmc = }}</ref>
==Two servers==
For an M/G/2 queue (the model with two servers) the problem of determining marginal probabilities can be reduced to solving a pair of [[integral equation]]s<ref>{{citeCite journal doi| last1 = Knessl | first1 = C. | last2 = Matkowsky | first2 = B. J. | last3 = Schuss | first3 = Z. | last4 = Tier | first4 = C. | title = An Integral Equation Approach to the M/G/2 Queue | doi = 10.1287/opre.38.3.506 | journal = [[Operations Research (journal)|Operations Research]]| volume = 38 | issue = 3 | pages = 506 | year = 1990 | jstor = 171363| pmid = | pmc = }}</ref> or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.<ref>{{citeCite journal doi| last1 = Cohen | first1 = J. W. | authorlink1 = Wim Cohen| title = On the M/G/2 queueing model | doi = 10.1016/0304-4149(82)90046-1 | journal = Stochastic Processes and their Applications | volume = 12 | issue = 3 | pages = 231–248 | year = 1982 | pmid = | pmc = }}</ref> The Laplace transform of queue length<ref>{{cite jstor|1426776}}</ref> and waiting time distributions<ref>{{cite doi|10.1023/A:1017913826973}}</ref> can be computed when the waiting time distribution has a rational Laplace transform.
==References==
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