en>David Eppstein: better stub classification — this isn't really combinatorics (what are we counting?) but it may be number theory.

2013-10-02T07:32:17Z

better stub classification — this isn't really combinatorics (what are we counting?) but it may be number theory.

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	~~Perhaps if you were taking your dog to~~ a ~~music festival~~, a ~~dog bikini might fit in, but~~ not ~~at the beach~~.<br>~~Although alligator meat sales~~ are ~~small compared with beef~~, ~~fish and other more familiar meats~~, ~~some specialty butcher shops in Southwest Florida are seeing~~ an ~~increased demand for alligator meat~~. ~~The standby~~ is ~~what most people here say when they ask about alligator meat at~~ a ~~restaurant or market~~.<br>~~http:~~//~~www~~.~~geneseecancerassistance~~.~~com/sale/uggs~~.~~asp?p~~=3 <br /> ~~http~~:/~~/www~~.~~geneseecancerassistance~~.~~com~~/~~sale/uggs~~.~~asp?p=15~~ <~~br /~~> ~~http:~~/~~/www~~.~~geneseecancerassistance~~.~~com~~/~~sale/uggs~~.~~asp?p~~=32 <br /> ~~http~~://~~www~~.~~geneseecancerassistance.com/sale~~/~~uggs~~.~~asp?p~~=~~121~~ <br /> ~~http://www~~.~~geneseecancerassistance~~.~~com~~/~~sale~~/~~uggs~~.~~asp?p~~=~~123~~ <br /> <br>~~http:~~/~~/gracechapel~~.kr/~~sermon1~~/~~436283~~<br>~~http:~~//~~vetopia.dothome.co.kr~~/xe/~~?document_srl~~=~~1354281~~<br><br>~~If you have any issues about wherever~~ in ~~addition~~ to ~~how you can use~~ [~~http:~~//~~www.bendtrapclub.com~~/~~cheap~~/~~ugg~~.~~asp Cheap Uggs Boots~~]~~, you are able to contact us from the website.~~		In [[queueing theory]], a discipline within the mathematical [[probability theory\|theory of probability]], the '''M/M/∞ queue''' is a multi-server [[queueing model]] where every arrival experiences immediate service and does not wait.<ref name="harrison">{{cite book\|first=Peter\|last=Harrison\|authorlink=Peter G. Harrison\|first2=Naresh M.\|last2=Patel\|title=Performance Modelling of Communication Networks and Computer Architectures\|publisher=Addison–Wesley\|year=1992\|page=173}}</ref> In [[Kendall's notation]] it describes a system where arrivals are governed by a [[Poisson process]], there are infinitely many servers, so jobs do not need to wait for a server. Each job has an exponentially distributed service time. It is a limit of the [[M/M/c queue]] model where the number of servers ''c'' becomes very large.

			The model can be used to model bound [[lazy deletion]] performance.<ref name="lock" />

			==Model definition==

			An M/M/∞ queue is a stochastic process whose [[state space]] is the set {0,1,2,3,...} where the value corresponds to the number of customers currently being served.

			* Arrivals occur at rate ''λ'' according to a [[Poisson process]] and move the process from state ''i'' to ''i'' + 1.
			* Service times have an [[exponential distribution]] with parameter ''μ'' and there are always sufficient servers such that every arriving job is served immediately. Transitions from state ''i'' to ''i'' − 1 are at rate ''iμ''

			The model has transition rate matrix
			:<math>Q=\begin{pmatrix}
			-\lambda & \lambda \\
			\mu & -(\mu+\lambda) & \lambda \\
			&2\mu & -(2\mu+\lambda) & \lambda \\
			&&3\mu & -(3\mu+\lambda) & \lambda \\
			&&&&\ddots
			\end{pmatrix}.</math>

			The [[state space]] diagram for this chain is as below.

			[[File:Mminfinity-statespace.svg]]

			==Transient solution==

			The transient distribution can be written using [[moment generating function]]s<ref>{{cite doi\|10.3844/jmssp.2010.1.3}}</ref> and formulas for transient means and variances computed by solving differential equations.<ref>{{cite doi\|10.3844/jmssp.2010.68.71}}</ref> Assuming the system starts in state 0 at time 0, then the probability the system is in state ''j'' at time ''t'' can be written as<ref>{{cite book \| edition = First edition \| title = Modeling and analysis of stochastic systems \| publisher = Chapman & Hall \| year = 1995 \| first = Vidyadhar G. \| last = Kulkarni \| isbn = 0412049910}}</ref>{{rp\|356}}
			::<math>p_{0j}(t) = \exp \left( -\frac{\lambda}{\mu}(1-e^{-\mu t}) \right) \frac{\left(\frac{\lambda}{\mu}(1-e^{-\mu t})\right)^j}{j!} \text{ for } j \geq 0</math>
			from which the mean queue length at time ''t'' can be computed (writing ''N''(''t'') for the number of customers in the system at time ''t'' given the system is empty at time zero)
			::<math>\mathbb E(N(t) \| N(0)=0) = \frac{\lambda}{\mu} (1-e^{-\mu t}) \text{ for } t \geq 0.</math>

			===Response time===
			The response time for each arriving job is a single exponential distribution with parameter ''μ''. The average response time is therefore 1/''μ''.<ref name="kleinrock">{{cite book \| title = Queueing Systems Volume 1: Theory \| first1=Leonard \| last1=Kleinrock \| authorlink = Leonard Kleinrock \| isbn = 0471491101 \| year=1975 \| pages=101–103, 404}}</ref>

			===Maximum queue length===
			Given the system is in equilibrium at time 0, we can compute the [[cumulative distribution function]] of the process maximum over a finite time horizon ''T'' in terms of [[Charlier polynomials]].<ref name="lock">{{cite doi\|10.1137/0216073}}</ref>

			===Congestion period===
			The congestion period is the length of time the process spends above a fixed level ''c'', starting timing from the instant the process transitions to state ''c'' + 1. This period has mean value<ref name="GuilleminFabrice">{{cite jstor\|3215073}}</ref>
			::<math>\frac{\mu}{\lambda}\sum_{i>c} \frac{c!}{i!}\left( \frac{\lambda}{\mu} \right)^{i-c}</math>
			and the Laplace transform can be expressed in terms of [[Kummer's function]].<ref>{{cite jstor\|1428137}}</ref>

			==Stationary analysis==

			The stationary probability mass function is a [[Poisson distribution]]<ref>{{cite book \| title =Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications \| page = 249 \| first1=Gunter \|last1= Bolch \|first2= Stefan \| last2= Greiner \| first3= Hermann \| last3= de Meer \| first4= Kishor Shridharbhai \|last4= Trivedi \| publisher = John Wiley & Sons \| year = 2006 \| isbn = 0471791563}}</ref>
			:: <math> \pi_k = \frac{(\lambda/\mu)^k e^{-\lambda/\mu}}{k!} \quad k \geq 0</math>
			so the mean number of jobs in the system is ''λ''/''μ''.

			==Heavy traffic==
			{{main\|heavy traffic approximation}}
			Writing ''N''<sub>''t''</sub> for the number of customers in the system at time t as ρ → ∞ the scaled process
			::<math>X_t = \frac{N_t - \lambda/\mu}{\sqrt{\lambda/\mu}}</math>
			converges to an [[Ornstein–Uhlenbeck process]] with [[normal distribution]] and correlation parameter 1, defined by the [[Itō calculus]] as<ref name="GuilleminFabrice" /><ref>{{cite doi\|10.1023/A:1012752719211}}</ref>
			::<math>dX_t = -Xdt+\sqrt{2}dW_t</math>
			where ''W'' is a [[standard Brownian motion]].

			==References==
			{{Reflist}}

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

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

en>David Eppstein: typo

2012-08-13T02:58:43Z

typo

New page

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en>David Eppstein: better stub classification — this isn't really combinatorics (what are we counting?) but it may be number theory.

en>David Eppstein: typo