M/D/c queue: Difference between revisions

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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], an '''M/D/c queue''' represents the queue length in a system having ''c'' servers, where arrivals are determined by a [[Poisson process]] and job service times are fixed (deterministic). The model name is written in [[Kendall's notation]].<ref>{{Cite journal | last1 = Kendall | first1 = D. G. | authorlink1author-link1 = David George Kendall| title = Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain | doi = 10.1214/aoms/1177728975 | jstor = 2236285| journal = The Annals of Mathematical Statistics | volume = 24 | issue = 3 | pages = 338338–354 | year = 1953| urldoi-access = http://projecteuclid.org/euclid.aoms/1177728975 | pmid = | pmc =free }}</ref> [[Agner Krarup Erlang]] first published on this model in 1909, starting the subject of [[queueing theory]].<ref>{{Cite journal | last1 = Kingman | first1 = J. F. C. | authorlink1author-link1 = John Kingman | title = The first Erlang century—and the next | journal = [[Queueing Systems]] | volume = 63 | issue = 1–4 | pages = 3–4 | year = 2009 | doi = 10.1007/s11134-009-9147-4}}</ref><ref>{{cite journal | title = The theory of probabilities and telephone conversations | journal = Nyt Tidsskrift for Matematik B | volume = 20 | pages = 33–39 | archiveurlarchive-url = https://web.archive.org/web/20120207184053/http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf |archivedatearchive-date=2012-02-07| year = 1909| url = http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf}}</ref> The model is an extension of the [[M/D/1 queue]] which has only a single server.

==Model definition==

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==Waiting time distribution==

Erlang showed that when ''ρ''&nbsp;=&nbsp;(''λ''&nbsp;''D'')/''c''&nbsp;<&nbsp;1, the waiting time distribution has distribution F(''y'') given by<ref name="franx">{{Cite journal | last1 = Franx | first1 = G. J. | title = A simple solution for the M/D/c waiting time distribution | doi = 10.1016/S0167-6377(01)00108-0 | journal = Operations Research Letters | volume = 29 | issue = 5 | pages = 221–229 | year = 2001 | pmid = | pmc = }}</ref>

::<math>F(y) = \int_0^\infty F(x+y-D)\frac{\lambda^c x^{c-1}}{(c-1)!} e^{-\lambda x} \text{d} x, \quad y \geq 0 \quad c \in \mathbb N.</math>