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The Figure "Stage Space Diagram of M/D/1 Queue" is appropriate for MM1, but misleading for MD1. According to the trasition matrix there are many more links availbale. The transition rate is given for the intervals mu (=1) and corresponds to the probabilities of new arrivals within "\mu" time periods(Poisson). If the stage stay the same, then exactly one arrival was observed while serving one unit. |
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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], an '''M/D/1 queue''' represents the queue length in a system having a single server, 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. |
==Model definition==
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=== Relations for Mean Waiting Time in M/M/1 and M/D/1 queues ===
For an equilibrium M/G/1 queue, the expected value of the time W spent by a customer in the queue are given by Pollaczek-Khintchine formula as below:<ref name=":0">{{Cite book|title=Introduction to Queuing Theory|last=Cooper|first=Robert B.|publisher=Elsevier Science Publishing Co.|year=1981|isbn=0-444-00379-7
<math>E(W)=\frac{\rho\tau}{2(1-\rho)}\left(1+\frac{\sigma^2}{\tau^2}\right)</math>
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==Stationary distribution==
The number of jobs in the queue can be written as [[M/G/1 queue|M/G/1 type Markov chain]] and the stationary distribution found for state ''i'' (written π<sub>''i''</sub>) in the case ''D'' = 1 to be<ref name=":1">{{cite journal| url = http://kashiwa.nagaokaut.ac.jp/members/nakagawa/ronbun/029.pdf | journal = Journal of the Operations Research Society of Japan | volume = 48 | year = 2005 | issue = 2 | pages = 111–122 | title = On the Series Expansion for the Stationary Probabilities of an M/D/1 queue | first = Kenji | last = Nakagawa | doi=10.15807/jorsj.48.111| doi-access = free }}</ref>
:<math>\begin{align}\pi_0 &= 1-\lambda \\
\pi_1 &= (1-\lambda)(e^\lambda - 1)\\
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==Busy period==
The busy period is the time period measured from the instant a first customer arrives at an empty queue to the time when the queue is again empty. This time period is equal to ''D'' times the number of customers served. If ''ρ'' < 1, then the number of customers served during a busy period of the queue has a [[Borel distribution]] with parameter ''ρ''.<ref>{{Cite journal | last1 = Tanner | first1 = J. C. | doi = 10.1093/biomet/48.1-2.222 | title = A derivation of the Borel distribution | journal = [[Biometrika]]| volume = 48 | pages = 222–224 | year = 1961
==Finite capacity==
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===Stationary distribution===
A stationary distribution for the number of customers in the queue and mean queue length can be computed using [[probability generating function]]s.<ref>{{cite journal | last1 = Brun | first1 = Olivier | last2 =Garcia | first2 = Jean-Marie | year = 2000 | title = Analytical Solution of Finite Capacity M/D/1 Queues | journal = Journal of Applied Probability | volume = 37 | issue = 4 | pages = 1092–1098 | publisher = [[Applied Probability Trust]] | jstor = 3215497 | doi = 10.1239/jap/1014843086
===Transient solution===
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==Application==
Includes applications in wide area [[network planning and design|network design]],<ref>{{cite journal|last1=Kotobi|first1=Khashayar|last2=Bilén|first2=Sven G.|title=Spectrum sharing via hybrid cognitive players evaluated by an M/D/1 queuing model|journal=EURASIP Journal on Wireless Communications and Networking|volume=2017|date=2017|pages=85 |doi=10.1186/s13638-017-0871-x|url=https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-017-0871-x |
==References==
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