Packing dimension - Revision history

77.246.192.42: /* Definitions */

2014-12-23T19:17:36Z

Definitions

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	~~In [[queueing theory]]~~, ~~a discipline within the mathematical [[probability theory\|theory of probability]], the~~ '''Pollaczek–Khinchine formula''' states a relationship between the queue length and service time distribution Laplace transforms for an [[M/G/1 queue]] (where jobs arrive according to a [[Poisson process]] and have general service time distribution). ~~The term is also used to refer to the relationships between the mean queue length~~ and ~~mean waiting/service time~~ in ~~such a model.<ref>{{cite doi\|10.1007/0-387-21525-5_8}}</ref>~~		I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my full name. My wife and I live in Kentucky. Distributing production is where her primary earnings comes from. I am really fond of to go to karaoke but I've been taking on new things recently.<br><br>My web site - [http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 phone psychic]

	The formula was first published by [[Felix Pollaczek]] in 1930<ref>{{cite journal\|last=Pollaczek\|first=F.\|authorlink=Felix Pollaczek\|year=1930\|title=Über eine Aufgabe der Wahrscheinlichkeitstheorie\|journal=[[Mathematische Zeitschrift]]\|volume=32\|pages=64–100\|doi=10.1007/BF01194620}}</ref> and recast in probabilistic terms by [[Aleksandr Khinchin]]<ref>{{cite journal\|last=Khintchine\|first=A. Y\|authorlink=Aleksandr Khinchin\|year=1932\|title=Mathematical theory of a stationary queue\|journal=[[Matematicheskii Sbornik]]\|volume=39\|number=4\|pages=73–84\|url=http://mi.mathnet.ru/rus/msb/v39/i4/p73\|accessdate=2011-07-14}}</ref> two years later.<ref>{{cite journal\|title=Review: J. W. Cohen, The Single Server Queue\|first=Lajos\|last=Takács\|authorlink=Lajos Takács\|journal=[[Annals of Mathematical Statistics]]\|volume=42\|issue=6\|year=1971\|pages=2162–2164\|doi=10.1214/aoms/1177693087}}</ref><ref>{{cite doi\|10.1007/s11134-009-9147-4}}</ref> In [[ruin theory]] the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt).~~<ref>{{cite doi\|10.1002/9780470317044.ch5}}</ref>~~

	~~==Mean queue length==~~
	~~The formula states that the mean queue length ''L''~~ is ~~given by<ref>{{cite book\|title=Probability Models\|page=192\|last=Haigh\|first=John\|publisher=Springer\|year=2002\|isbn=1-85233-431-2}}</ref>~~

	~~:<math>L = \rho + \frac{\rho^2 + \lambda^2 \operatorname{Var}(S)}{2(1-\rho)}</math>~~

	where
	*<math>\lambda</math> is the arrival rate of the [[Poisson process]]
	*<math>1/\mu</math> is the mean of the service time distribution ''S''
	*<math>\rho=\lambda/\mu</math> is the [[utilization]]
	*Var(''S'') is the [[variance]] of the service time distribution ''S''.

	For the mean queue length to be finite it is necessary that <math>\rho < 1</math> as otherwise jobs arrive faster than they leave the queue. "Traffic intensity," ranges between 0 and 1, and is the mean fraction of ~~time that the server is busy. If the arrival rate <math>\lambda_a</math> is greater than or equal~~ to ~~the service rate <math>\lambda_s</math>, the queuing delay becomes infinite. The variance term enters the expression due~~ to ~~[[Palm calculus\|Feller~~'~~s paradox]].<ref>{{cite journal\|title=Some Reflections~~ on the Renewal-Theory Paradox in Queueing Theory\|first1=Robert B.\|last1=Cooper\|first2=Shun-Chen\|last2=Niu\|first3=Mandyam M.\|last3=Srinivasan\|journal=Journal of Applied Mathematics and Stochastic Analysis\|volume=11\|number=3\|year=1998\|pages=355–368\|url=http://www.cse.fau.edu/~bob/publications/CNS.4h.~~pdf\|accessdate=2011-07-14}}~~<~~/ref~~>

	~~==Mean waiting time==~~
	~~If we write ''W'' for the mean time a customer spends in the queue, then~~ <~~math~~>~~W=W'+\mu^{~~-~~1}</math> where <math>W'</math> is the mean waiting time (time spent in the queue waiting for service) and <math>\mu</math> is the service rate. Using~~ [~~[Little's law]], which states that~~
	~~:<math>L=\lambda W</math>~~
	~~where~~
	*''L'' is the mean queue length
	*<math>\lambda</math> is the arrival rate of the [[Poisson process]]
	*''W'' is the mean time spent at the queue both waiting and being serviced,
	so
	:~~<math>W = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)} + \mu^{-1}.</math>~~
	We can write an expression for the mean waiting time as<ref>{{cite book\|first=Peter G.\|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=228\|isbn=0-201-54419-9}}</ref>
	~~:<math>W' = \frac{L}{\lambda} - \mu^{-1} = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)}.</math>~~

	~~==Queue length transform==~~

	~~Writing π(''z'') for the [[probability-generating function]] of the number of customers in the queue<ref name="diagle">{{cite doi\|10.1007~~/~~0-387-22859-4_5}}<~~/~~ref>~~

	~~:<math>\pi(z) = \frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}</math>~~

	~~where g(''s'') is the Laplace transform of the service time probability density function~~.~~<ref>{{cite doi\|10.1088/0967-1846/3/1/003}}</ref>~~

	~~==Sojourn time transform==~~

	~~Writing W<sup>*</sup>(''s'') for the [[Laplace–Stieltjes transform]] transform of the waiting time distribution,<ref name="diagle">{{cite doi\|10~~.~~1007/0-387-22859-4_5}}</ref>~~

	~~:<math>W^\ast(s) = \frac{(1-\rho)s g(s)}{s-\lambda(1-g(s))}<~~/~~math>~~

	~~where again g(''s'') is the Laplace transform of serivice time probability density function~~. ~~''n''th moments can be obtained by differentiating the transform ''n'' times, multiplying by (-1)<sup>''n''</sup> and evaluating at ''s'' ~~=&~~nbsp;0.~~

	~~==References=~~=

	~~{{Reflist}}~~

	~~{{Queueing theory}}~~

	~~{{DEFAULTSORT:Pollaczek-Khinchine formula}}~~
	~~[[Category:Operations research]]~~
	~~[[Category:Single queueing nodes]~~]

139.82.115.10: /* An example */

2012-06-15T13:47:09Z

An example

New page

In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], the '''Pollaczek–Khinchine formula''' states a relationship between the queue length and service time distribution Laplace transforms for an [[M/G/1 queue]] (where jobs arrive according to a [[Poisson process]] and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model.<ref>{{cite doi|10.1007/0-387-21525-5_8}}</ref>

The formula was first published by [[Felix Pollaczek]] in 1930<ref>{{cite journal|last=Pollaczek|first=F.|authorlink=Felix Pollaczek|year=1930|title=Über eine Aufgabe der Wahrscheinlichkeitstheorie|journal=[[Mathematische Zeitschrift]]|volume=32|pages=64–100|doi=10.1007/BF01194620}}</ref> and recast in probabilistic terms by [[Aleksandr Khinchin]]<ref>{{cite journal|last=Khintchine|first=A. Y|authorlink=Aleksandr Khinchin|year=1932|title=Mathematical theory of a stationary queue|journal=[[Matematicheskii Sbornik]]|volume=39|number=4|pages=73–84|url=http://mi.mathnet.ru/rus/msb/v39/i4/p73|accessdate=2011-07-14}}</ref> two years later.<ref>{{cite journal|title=Review: J. W. Cohen, The Single Server Queue|first=Lajos|last=Takács|authorlink=Lajos Takács|journal=[[Annals of Mathematical Statistics]]|volume=42|issue=6|year=1971|pages=2162–2164|doi=10.1214/aoms/1177693087}}</ref><ref>{{cite doi|10.1007/s11134-009-9147-4}}</ref> In [[ruin theory]] the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt).<ref>{{cite doi|10.1002/9780470317044.ch5}}</ref>

==Mean queue length==
The formula states that the mean queue length ''L'' is given by<ref>{{cite book|title=Probability Models|page=192|last=Haigh|first=John|publisher=Springer|year=2002|isbn=1-85233-431-2}}</ref>

:<math>L = \rho + \frac{\rho^2 + \lambda^2 \operatorname{Var}(S)}{2(1-\rho)}</math>

where
*<math>\lambda</math> is the arrival rate of the [[Poisson process]]
*<math>1/\mu</math> is the mean of the service time distribution ''S''
*<math>\rho=\lambda/\mu</math> is the [[utilization]]
*Var(''S'') is the [[variance]] of the service time distribution ''S''.

For the mean queue length to be finite it is necessary that <math>\rho < 1</math> as otherwise jobs arrive faster than they leave the queue. "Traffic intensity," ranges between 0 and 1, and is the mean fraction of time that the server is busy. If the arrival rate <math>\lambda_a</math> is greater than or equal to the service rate <math>\lambda_s</math>, the queuing delay becomes infinite. The variance term enters the expression due to [[Palm calculus|Feller's paradox]].<ref>{{cite journal|title=Some Reflections on the Renewal-Theory Paradox in Queueing Theory|first1=Robert B.|last1=Cooper|first2=Shun-Chen|last2=Niu|first3=Mandyam M.|last3=Srinivasan|journal=Journal of Applied Mathematics and Stochastic Analysis|volume=11|number=3|year=1998|pages=355–368|url=http://www.cse.fau.edu/~bob/publications/CNS.4h.pdf|accessdate=2011-07-14}}</ref>

==Mean waiting time==
If we write ''W'' for the mean time a customer spends in the queue, then <math>W=W'+\mu^{-1}</math> where <math>W'</math> is the mean waiting time (time spent in the queue waiting for service) and <math>\mu</math> is the service rate. Using [[Little's law]], which states that
:<math>L=\lambda W</math>
where
*''L'' is the mean queue length
*<math>\lambda</math> is the arrival rate of the [[Poisson process]]
*''W'' is the mean time spent at the queue both waiting and being serviced,
so
:<math>W = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)} + \mu^{-1}.</math>
We can write an expression for the mean waiting time as<ref>{{cite book|first=Peter G.|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=228|isbn=0-201-54419-9}}</ref>
:<math>W' = \frac{L}{\lambda} - \mu^{-1} = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)}.</math>

==Queue length transform==

Writing π(''z'') for the [[probability-generating function]] of the number of customers in the queue<ref name="diagle">{{cite doi|10.1007/0-387-22859-4_5}}</ref>

:<math>\pi(z) = \frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}</math>

where g(''s'') is the Laplace transform of the service time probability density function.<ref>{{cite doi|10.1088/0967-1846/3/1/003}}</ref>

==Sojourn time transform==

Writing W<sup>*</sup>(''s'') for the [[Laplace–Stieltjes transform]] transform of the waiting time distribution,<ref name="diagle">{{cite doi|10.1007/0-387-22859-4_5}}</ref>

:<math>W^\ast(s) = \frac{(1-\rho)s g(s)}{s-\lambda(1-g(s))}</math>

where again g(''s'') is the Laplace transform of serivice time probability density function. ''n''th moments can be obtained by differentiating the transform ''n'' times, multiplying by (-1)<sup>''n''</sup> and evaluating at ''s'' = 0.

==References==

{{Reflist}}

{{Queueing theory}}

{{DEFAULTSORT:Pollaczek-Khinchine formula}}
[[Category:Operations research]]
[[Category:Single queueing nodes]]