M/G/k queue: Difference between revisions

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==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/K<sub>2</sub>/m Queue | journal = Advances in Applied Probability | volume = 12 | issue = 3 | pages = 799–823 | 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 = 544–552 | publisher = Applied Probability Trust | jstor = 3212698 | doi=10.1017/s0021900200096327}}</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 }}</ref><ref>{{cite journal | last1 = Boxma | first1 = O. J. | author-link1 = Onno Boxma | last2 =Cohen | first2 = J. W. | author-link2 = 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 = 1115–1127 | 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 }}</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| doi-access = free }}</ref>
 
:<math>E[W^{\text{M/G/}k}] = \frac{C^2+1}{2} \mathbb E [ W^{\text{M/M/}c}]</math>