M/G/k queue: Difference between revisions

Revision as of 20:06, 3 September 2015

Revision as of 05:29, 4 September 2015

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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 = 510-523510–523 | 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>{{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 journal | last1 = Hokstad | first1 = Per | year = 1980 | title = The Steady-State Solution of the M/K₂/m Queue | journal = Advances in Applied Probability | volume = 12 | issue = 3 | pages = 799-823799–823 | publisher = Applied Probability Trust | jstor = 1426432}}</ref><ref>{{cite journal | last1 = Köllerström | first1 = Julian | year = 1974 | title = Heavy Traffic Theory for Queues with Several Servers. I | journal = Journal of Applied Probability | volume = 11 | issue = 3 | pages = 544-552544–552 | publisher = Applied Probability Trust | jstor = 3212698}}</ref><ref>{{Cite journal | last1 = Nozaki | first1 = S. A. | last2 = Ross | first2 = S. M. | title = Approximations in Finite-Capacity Multi-Server Queues with Poisson Arrivals | journal = Journal of Applied Probability | volume = 15 | issue = 4 | pages = 826–834 | doi = 10.2307/3213437 | year = 1978 | pmid = | pmc = }}</ref><ref>{{cite journal | last1 = Boxma | first1 = O. J. | authorlink1 = Onno Boxma | last2 =Cohen | first2 = J. W. | authorlink2 = Wim Cohen | first3 = N. | last3 = Huffels | year = 1979 | title = Approximations of the Mean Waiting Time in an M/G/s Queueing System | journal = [[Operations Research (journal)|Operations Research]] | volume = 27 | issue = 6 | pages = 1115-11271115–1127 | publisher = INFORMS | jstor = 172087 | doi=10.1287/opre.27.6.1115}}</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>{{Cite journal | 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>{{Cite journal | 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>

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where ''C''² 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>{{Cite journal | 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">{{Cite journal | 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>{{Cite journal | 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>

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==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>{{Cite journal | 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>{{Cite journal | 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 journal | last1 = Hokstad | first1 = Per | year = 1979 | title = On the Steady-State Solution of the M/G/2 Queue | journal = Advances in Applied Probability | volume = 11 | issue = 1 | pages = 240-255240–255 | publisher = Applied Probability Trust | jstor = 1426776}}</ref> and waiting time distributions<ref>{{Cite journal | last1 = Boxma | first1 = O. J. | authorlink1 = Onno Boxma| last2 = Deng | first2 = Q. | last3 = Zwart | first3 = A. P. | journal = [[Queueing Systems]]| volume = 40 | pages = 5 | year = 2002 | doi = 10.1023/A:1017913826973 | title = Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers| pmid = | pmc = }}</ref> can be computed when the waiting time distribution has a rational Laplace transform.

==References==