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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 doi|10.1214/aoms/1177728975}}</ref> [[Agner Krarup Erlang]] first published on this model in 1909, starting the subject of [[queueing theory]].<ref>{{cite 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 | first = A. K. | last = Erlang | url = http://web.archive.org/web/20111001212934/http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf | year = 1909}}</ref> An extension of this model with more than one server is the [[M/D/c queue]].

==Model definition==

An M/D/1 queue is a stochastic process whose [[state space]] is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service.

* Arrivals occur at rate λ according to a [[Poisson process]] and move the process from state ''i'' to ''i'' + 1.
* Service times are deterministic time ''D'' (serving at rate ''μ'' = 1/''D'').
* A single server serves customers one at a time from the front of the queue, according to a [[first-come, first-served]] discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
* The buffer is of infinite size, so there is no limit on the number of customers it can contain.

==Delay==

Define ''ρ'' = ''λ''/''μ'' as the utilization; then the mean delay in the system in an M/D/1 queue is<ref>{{cite book|title=Wide Area Network Design:Concepts and Tools for Optimization|page=319|first=Robert S.|last=Cahn|year=1998|publisher=Morgan Kaufmann|isbn=1558604588}}</ref>
::<math>\frac{1}{2\mu}\cdot\frac{2-\rho}{1-\rho}.</math>
and in the queue:
::<math>\frac{1}{2\mu}\cdot\frac{\rho}{1-\rho}.</math>

==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 doi|10.1093/biomet/48.1-2.222}}</ref><ref>{{cite doi|10.1093/biomet/47.1-2.143}}</ref>

==Finite capacity==

===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 jstor|3215497}}</ref>

===Transient solution===

The transient solution of an M/D/1 queue of finite capacity N, often written M/D/1/N, was published by Garcia et al in 2002.<ref>{{cite jstor |3216008}}</ref>

==References==
{{Reflist}}

{{Queueing theory}}
{{Stochastic processes}}

{{DEFAULTSORT:M D 1 queue}}
[[Category:Stochastic processes]]
[[Category:Single queueing nodes]]

Monodomain model - Revision history

en>Fraulein451: added reference section