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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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* A single server serves entities one at a time from the front of the queue, according to a [[first-come, first-served]] discipline. When the service is complete the entity leaves the queue and the number of entities in the system reduces by one.
* The buffer is of infinite size, so there is no limit on the number of entities it can contain.
===
The transition probability matrix for an M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below:<ref name=":1" />
<math>P=\begin{pmatrix} a_0 & a_1 & a_2 & a_3 & ... \\ a_0 & a_1 & a_2 & a_3 & ...\\ 0 & a_0 & a_1 & a_2 & ...\\ 0&0 & a_0 & a_1 & ...\\... & ... &...&... &...\\\end{pmatrix}</math> ， <math>a_n=\frac{\lambda^n}{n!}e^{-\lambda}</math>, n = 0,1,....
===
The following expressions present the classic performance metrics of a single server queuing system such as M/D/1, with:
* arrival rate <math>=\lambda</math>,
* service rate <math>=\mu</math>, and
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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]]
==References==
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