M/D/1 queue: Difference between revisions
removed Category:Kendall notation; added Category:Single queueing nodes using HotCat |
|||
Line 12: | Line 12: | ||
==Delay== |
==Delay== |
||
The mean delay in an M/D/1 queue |
The mean delay in an M/D/1 queue is<ref>{{cite book|title=Wide Area Network Design:Concepts and Tools for Optimization|page=329|first=Robert S.|last=Cahn|year=1998|publisher=Morgan Kaufmann|isbn=1558604588}}</ref> |
||
::<math>\frac{1}{\mu}\cdot\frac{2-\rho}{2-2\rho}.</math> |
::<math>\frac{1}{\mu}\cdot\frac{2-\rho}{2-2\rho}.</math> |
||
Revision as of 17:31, 8 May 2013
In queueing theory, a discipline within the mathematical 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.[1] Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory.[2][3]
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
The mean delay in an M/D/1 queue is[4]
Finite capacity
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.[5]
References
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1214/aoms/1177728975, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1214/aoms/1177728975
instead. - ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/s11134-009-9147-4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1007/s11134-009-9147-4
instead. - ^ "The theory of probabilities and telephone conversations" (PDF). Nyt Tidsskrift for Matematik B. 20: 33–39. 1909.
- ^ Cahn, Robert S. (1998). Wide Area Network Design:Concepts and Tools for Optimization. Morgan Kaufmann. p. 329. ISBN 1558604588.
- ^ Attention: This template ({{cite jstor}}) is deprecated. To cite the publication identified by jstor:3216008, please use {{cite journal}} with
|jstor=3216008
instead.