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| In [[probability theory]], a '''product-form solution''' is a particularly efficient form of solution for determining some metric of a system with distinct sub-components, where the metric for the collection of components can be written as a [[product (mathematics)|product]] of the metric across the different components. Using [[capital Pi notation]] a product-form solution has algebraic form
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| :<math>\text{P}(x_1,x_2,x_3,\ldots,x_n) = B \prod_{i=1}^n \text{P}(x_i)</math> | |
| where ''B'' is some constant. Solutions of this form are of interest as they are computationally inexpensive to evaluate for large values of ''n''. Such solutions in queueing networks are important for finding [[performance metric]]s in models of multiprogrammed and time-shared computer systems.
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| ==Equilibrium distributions==
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| The first product-form solutions were found for [[equilibrium distribution]]s of [[Markov chain]]s. Trivially, models composed of two or more [[Independence (probability theory)#Independent random variables|independent]] sub-components exhibit a product-form solution by the definition of independence. Initially the term was used in [[Queueing theory#Queueing networks|queueing networks]] where the sub-components would be individual queues. For example, [[Jackson's theorem (queueing theory)|Jackson's theorem]] gives the joint equilibrium distribution of an open queueing network as the product of the equilibrium distributions of the individual queues.<ref>{{Cite journal | first = James R. | last = Jackson | authorlink = James R. Jackson | title = Jobshop-like queueing systems | year = 1963 | pages = 131–142 | volume = 10 | issue = 1 | doi = 10.1287/mnsc.10.1.131 | journal = [[Management Science: A Journal of the Institute for Operations Research and the Management Sciences|Management Science]] }}</ref> After numerous extensions, chiefly the [[BCMP network]] it was thought [[local balance]] was a requirement for a product-form solution.<ref>{{Cite journal | first = Richard J. | last = Boucherie | first2 = N. M. | last2 = van Dijk | title = Local balance in queueing networks with positive and negative customers | doi = 10.1007/BF02033315 | journal = Annals of Operations Research | year = 1994 | pages = 463–492 | volume = 48 }}</ref><ref>{{Cite journal | first = K. Mani | last = Chandy | author-link = K.M. Chandy | first2 = J. H., Jr | last2 = Howard | first3 = D. F. | last3 = Towsley | title = Product form and local balance in queueing networks | doi = 10.1145/322003.322009 | journal = [[Journal of the ACM]] | year = 1977 | pages = 250–263 | volume = 24 }}</ref> [[Erol Gelenbe|Gelenbe]]'s [[G-network]] model showed this to not be the case.<ref>{{cite journal | doi = 10.2307/3214781 | title = G-Networks with triggered customer movement | first = Erol | last = Gelenbe | authorlink = Erol Gelenbe | journal = Journal of Applied Probability | volume = 30 | issue = 3 | year = 1993 | pages = 742–748 }}</ref> Product-form solutions are sometimes described as "stations are independent in equilibrium".<ref name="harrison-williams">{{cite journal | doi = 10.1214/aoap/1177005704 | title = Brownian models of feedforward queueing networks: quasireversibility and product-form solutions | first = J. M. | last = Harrison | authorlink = J. Michael Harrison | first2 = R. J. | last2 = Williams | journal = [[Annals of Applied Probability]] | volume = 2 | issue = 2 | year = 1992 | pages = 263–293 }}</ref> Product form solutions also exist in networks of [[bulk queue]]s.<ref>{{cite doi|10.1007/BF02411466}}</ref>
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| [[J. Michael Harrison|J.M. Harrison]] and R.J. Williams note that "virtually all of the models that have been successfully analyzed in classical queueing network theory are models having a so-called product-form stationary distribution"<ref name="harrison-williams" /> More recently, product-form solutions have been published for Markov process algebras (e.g. [[RCAT]] in [[PEPA]]<ref>{{cite doi|10.1016/S0166-5316(99)00005-X}}</ref><ref>{{cite doi|10.1016/S0304-3975(02)00375-4}}</ref>) and [[stochastic]] [[petri nets]].<ref>{{cite doi|10.1016/j.peva.2012.06.003}}</ref><ref>{{cite doi|10.1007/978-3-642-02424-5_8}}</ref> [[Martin Feinberg]]'s deficiency zero theorem gives a sufficient condition for [[chemical reaction network]]s to exhibit a product-form stationary distribution.<ref>{{cite doi|10.1007/s11538-010-9517-4}}</ref>
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| ==Sojourn time distributions==
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| The term ''product form'' has also been used to refer to the sojourn time distribution in a cyclic queueing system, where the time spent by jobs at ''M'' nodes is given as the product of time spent at each node.<ref>{{cite journal|title=The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues|journal=[[Journal of the ACM]]|first1=O. J.|last1=Boxma|authorlink1=Onno J. Boxma|first2=F. P.|last2=Kelly|authorlink2=Frank Kelly (mathematician)|first3=A. G.|last3=Konheim|volume=31|issue=1|date=January 1984|doi=10.1145/2422.322419}}</ref> In 1957 Reich showed the result for two [[M/M/1 queue]]s in tandem,<ref>{{cite jstor|2237237}}</ref> later extending this to ''n'' M/M/1 queues in tandem<ref>{{cite doi|10.1214/aoms/1177704275}}</ref> and it has been shown to apply to overtake–free paths in [[Jackson network]]s.<ref name="walrand" /> Walrand and Varaiya suggest that non-overtaking (where customers cannot overtake other customers by taking a different route through the network) may be a necessary condition for the result to hold.<ref name="walrand">{{cite jstor|1426753}}</ref> Mitrani offers exact solutions to some simple networks with overtaking, showing that none of these exhibit product-form sojourn time distributions.<ref>{{cite jstor|2345774}}</ref>
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| For closed networks, Chow showed a result to hold for two service nodes,<ref>{{cite journal|title=The Cycle Time Distribution of Exponential Cyclic Queues|journal=[[Journal of the ACM]]|first=We-Min|last=Chow|volume=27|issue=2|date=April 1980|doi=10.1145/322186.322193}}</ref> which was later generalised to a cycle of queues<ref>{{cite doi|10.1145/322358.322369}}</ref> and to overtake–free paths in [[Gordon–Newell network]]s.<ref>{{cite jstor|1426680}}</ref><ref>{{cite jstor|1426623}}</ref>
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| ==Extensions==
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| * Approximate product-form solutions are computed assuming independent marginal distributions, which can give a good approximation to the stationary distribution under some conditions.<ref>{{cite doi|10.1016/0166-5316(93)90017-O}}</ref><ref>{{cite doi|10.1016/0166-5316(92)90019-D}}</ref>
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| * Semi-product-form solutions are solutions where a distribution can be written as a product where terms have a limited functional dependency on the global state space, which can be approximated.<ref>{{cite doi|10.1007/978-3-642-15784-4_14}}</ref>
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| * Quasi-product-form solutions are either
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| **solutions which are not the product of marginal densities, but the marginal densities describe the distribution in a product-type manner<ref>{{cite doi|10.1287/moor.1070.0259}}</ref> or
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| ** approximate form for transient probability distributions which allows transient moments to be approximated.<ref>{{cite doi|10.1007/978-3-642-39408-9_3}}</ref>
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| ==References==
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| {{Reflist}}
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| {{Queueing theory}}
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| [[Category:Stochastic processes]]
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| [[Category:Queueing theory]]
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Hi there! :) My name is Joesph, I'm a student studying American Studies from Bodmin, Great Britain.
My website; microsoft office 2010 product key (Highly recommended Online site)