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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>
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| 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>
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| ==Mean queue length==
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| 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>
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| :<math>L = \rho + \frac{\rho^2 + \lambda^2 \operatorname{Var}(S)}{2(1-\rho)}</math>
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| where | |
| *<math>\lambda</math> is the arrival rate of the [[Poisson process]]
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| *<math>1/\mu</math> is the mean of the service time distribution ''S''
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| *<math>\rho=\lambda/\mu</math> is the [[utilization]]
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| *Var(''S'') is the [[variance]] of the service time distribution ''S''.
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| 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>
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| ==Mean waiting time==
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| 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
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| :<math>L=\lambda W</math>
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| where
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| *''L'' is the mean queue length
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| *<math>\lambda</math> is the arrival rate of the [[Poisson process]]
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| *''W'' is the mean time spent at the queue both waiting and being serviced,
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| so
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| :<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>
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| :<math>W' = \frac{L}{\lambda} - \mu^{-1} = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)}.</math>
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| ==Queue length transform==
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| 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>
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| :<math>\pi(z) = \frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}</math>
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| 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>
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| ==Sojourn time transform==
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| 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>
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| :<math>W^\ast(s) = \frac{(1-\rho)s g(s)}{s-\lambda(1-g(s))}</math>
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| 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.
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| ==References==
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| {{Reflist}}
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| {{Queueing theory}}
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| {{DEFAULTSORT:Pollaczek-Khinchine formula}}
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| [[Category:Operations research]]
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| [[Category:Single queueing nodes]]
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