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en>Hyacinth: /* References */ ==Further reading==, uncited

2014-08-15T10:46:08Z

References: ==Further reading==, uncited

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	~~In [[queueing theory]]~~, ~~a discipline within the mathematical [[probability theory\|theory of probability]], the~~ '~~''arrival theorem'''<ref>{{cite doi\|10.1007/0-387-21525-5_4}}</ref> (also referred~~ to as the '''random observer property''', '''ROP''' or '''job observer property'''<ref>{{cite book\|title=Sample-path Analysis of Queueing Systems\|first=Muhammad\|last=El-Taha\|publisher=Springer\|year=1999\|isbn=0-7923-8210-2\|page=94}}</ref>) states that "upon arrival at a station, a job observes the system as if in ~~steady state at an arbitrary instant for~~ the ~~system without that job~~.~~"<ref name="dijk">{{cite doi\|10~~.~~1016~~/~~0169-7552(93)90073-D}}<~~/~~ref>~~		They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. If you have any sort of concerns regarding where and ways to utilize [http://babahdwallpaper.com/profile/quuhk wordpress backup], you could call us at our own site. You may discover this probably the most time-consuming part of building a Word - Press MLM website. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. After activating, you will find their website link and get the activation code from their website. <br><br>Creating a website from scratch can be such a pain. If a newbie missed a certain part of the video then they could always rewind. This plugin is a must have for anyone who is serious about using Word - Press. Now, I want to anxiety that not every single query will be answered. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. After age 35, 18% of pregnancies will end in miscarriage. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Websites using this content based strategy are always given top scores by Google. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. I don't want that for my visitors and I'm quite sure they don't either. The days of spending a lot of time and money to have a website built are long gone. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.

	~~The arrival theorem always holds in open [[product-form solution\|product-form networks~~]~~] with unbounded queues~~ at ~~each node, but it also holds in more general networks~~. ~~A necessary and sufficient condition for~~ the ~~arrival theorem to be satisfied in product~~-~~form networks is given in terms~~ of ~~[[Palm probabilities]] in Boucherie & Dijk~~, ~~1997~~.~~<ref name=boucherie~~-~~dijk /> A similar result also holds~~ in ~~some closed networks~~. ~~Examples of product-form networks where~~ the ~~arrival theorem does not hold include reversible Kingman networks<ref name=boucherie-dijk>{{cite doi\|10~~.~~1016/S0166-5316(96)00045-4}}~~<~~/ref~~><~~ref~~>~~{{cite jstor\|3212273}}</ref> and networks with~~ a ~~delay protocol~~.~~<ref name="dijk" />~~

	~~Mitrani offers the intuition that "The state~~ of ~~node ''i'' as seen by an incoming job has a different distribution from~~ the ~~state seen by~~ a ~~random observer~~. ~~For instance~~, ~~an incoming job can never see all '''k'' jobs present at node ''i'', because it itself cannot~~ be ~~among the jobs already present~~.~~"<ref>{{cite book \| page = 114 \| title = Modelling~~ of ~~Computer and Communication Systems \| first = Isi \| last = Mitrani \| publisher = CUP \| year = 1987 \| isbn = 0521314224 }}</ref>~~

	~~==Theorem for arrivals governed by~~ a ~~Poisson process==~~

	~~For [[Poisson processes]] the property~~ is ~~often referred to as~~ the ~~'''PASTA property''' (Poisson Arrivals See Time Averages)~~ and ~~states that the probability of the state as seen by an outside random observer is the same as the probability~~ of ~~the state seen by an arriving customer~~.<~~ref~~>~~{{cite doi\|10.1287/opre.30.2.223}}~~<~~/ref~~> ~~The property~~ also ~~holds for the case of~~ a ~~[[doubly stochastic Poisson process]] where the rate parameter is allowed to vary depending on the state~~.~~<ref>{{cite doi\|10.1016/0167-6377(88)90036~~-~~3}}</ref>~~

	~~==Theorem for Jackson networks==~~

	~~In an open [[Jackson network]] with ''m'' queues~~, ~~write <math>\scriptstyle{\mathbf{n} = (n_1, n_2, \ldots, n_m)}</math> for~~ the ~~state~~ of ~~the network. Suppose <math>\scriptstyle{\pi(\mathbf{n})}</math> is the equilibrium probability that the network is~~ in ~~state <math>\scriptstyle{\mathbf{n}}</math>. Then~~ the ~~probability that the network is in state <math>\scriptstyle{\mathbf{n}}</math> immediately before an arrival~~ to ~~any node~~ is ~~also <math>\scriptstyle{\pi(\mathbf{n})}</math>~~.

	~~Note that this theorem does not follow from [[Jackson's theorem (queueing theory)\|Jackson's theorem]]~~, ~~where the steady state~~ in ~~continuous time is considered~~. ~~Here we~~ are ~~concerned with particular points in time, namely arrival times~~.<~~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> This theorem first published by Sevcik and Mitrani in 1981.<ref name="sevcik" />~~

	~~==Theorem~~ for ~~Gordon–Newell networks==~~

	In a ~~closed [[Gordon–Newell network]] with ''m'' queues, write <math>\scriptstyle{\mathbf{n} = (n_1, n_2, \ldots, n_m)}</math>~~ for the ~~state of~~ the ~~network~~. ~~For a customer in transit~~ to ~~state ''i'', let <math>\scriptstyle{\alpha_i(\mathbf{n}-\mathbf{e}_i)}</math> denote the probability that immediately before arrival the customer 'sees' the state of the~~ system ~~to be~~

	~~:<math>\mathbf{n}-\mathbf{e}_i = (n_1~~,~~n_2~~,~~\ldots~~,~~n_i - 1~~,~~\ldots,n_m)~~.\,<~~/math~~>

	~~This probability,~~ <~~math~~>~~\scriptstyle{\alpha_i(\mathbf{n}-\mathbf{e}_i)}</math>,~~ is ~~the same as the steady state probability~~ for ~~state <math>\scriptstyle{\mathbf{n}-\mathbf{e}_i}</math>~~ for ~~a network of the same type with ''one customer less~~''.~~<ref>{{cite doi\|10~~.~~1007/1~~-~~4020~~-~~3631~~-~~0_5}}</ref> It was~~ published ~~independently by Sevcik and Mitrani~~,~~<ref name="sevcik">{{cite doi\|10.1145/322248.322257}}</ref>~~ and ~~Reiser and Lavenberg~~,~~<ref>{{cite doi\|~~10~~.1145/322186.322195}}</ref> where the result was used~~ to ~~develop [[mean value analysis]]~~.

	~~==Notes==~~
	~~{{Reflist}}~~

	~~{{Queueing theory}}~~

	~~[[Category:Queueing theory]]~~
	~~[[Category:Stochastic processes]]~~
	~~[[Category:Probability theorems]]~~

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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], the '''arrival theorem'''<ref>{{cite doi|10.1007/0-387-21525-5_4}}</ref> (also referred to as the '''random observer property''', '''ROP''' or '''job observer property'''<ref>{{cite book|title=Sample-path Analysis of Queueing Systems|first=Muhammad|last=El-Taha|publisher=Springer|year=1999|isbn=0-7923-8210-2|page=94}}</ref>) states that "upon arrival at a station, a job observes the system as if in steady state at an arbitrary instant for the system without that job."<ref name="dijk">{{cite doi|10.1016/0169-7552(93)90073-D}}</ref>

The arrival theorem always holds in open [[product-form solution|product-form networks]] with unbounded queues at each node, but it also holds in more general networks. A necessary and sufficient condition for the arrival theorem to be satisfied in product-form networks is given in terms of [[Palm probabilities]] in Boucherie & Dijk, 1997.<ref name=boucherie-dijk /> A similar result also holds in some closed networks. Examples of product-form networks where the arrival theorem does not hold include reversible Kingman networks<ref name=boucherie-dijk>{{cite doi|10.1016/S0166-5316(96)00045-4}}</ref><ref>{{cite jstor|3212273}}</ref> and networks with a delay protocol.<ref name="dijk" />

Mitrani offers the intuition that "The state of node ''i'' as seen by an incoming job has a different distribution from the state seen by a random observer. For instance, an incoming job can never see all '''k'' jobs present at node ''i'', because it itself cannot be among the jobs already present."<ref>{{cite book | page = 114 | title = Modelling of Computer and Communication Systems | first = Isi | last = Mitrani | publisher = CUP | year = 1987 | isbn = 0521314224 }}</ref>

==Theorem for arrivals governed by a Poisson process==

For [[Poisson processes]] the property is often referred to as the '''PASTA property''' (Poisson Arrivals See Time Averages) and states that the probability of the state as seen by an outside random observer is the same as the probability of the state seen by an arriving customer.<ref>{{cite doi|10.1287/opre.30.2.223}}</ref> The property also holds for the case of a [[doubly stochastic Poisson process]] where the rate parameter is allowed to vary depending on the state.<ref>{{cite doi|10.1016/0167-6377(88)90036-3}}</ref>

==Theorem for Jackson networks==

In an open [[Jackson network]] with ''m'' queues, write <math>\scriptstyle{\mathbf{n} = (n_1, n_2, \ldots, n_m)}</math> for the state of the network. Suppose <math>\scriptstyle{\pi(\mathbf{n})}</math> is the equilibrium probability that the network is in state <math>\scriptstyle{\mathbf{n}}</math>. Then the probability that the network is in state <math>\scriptstyle{\mathbf{n}}</math> immediately before an arrival to any node is also <math>\scriptstyle{\pi(\mathbf{n})}</math>.

Note that this theorem does not follow from [[Jackson's theorem (queueing theory)|Jackson's theorem]], where the steady state in continuous time is considered. Here we are concerned with particular points in time, namely arrival times.<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> This theorem first published by Sevcik and Mitrani in 1981.<ref name="sevcik" />

==Theorem for Gordon–Newell networks==

In a closed [[Gordon–Newell network]] with ''m'' queues, write <math>\scriptstyle{\mathbf{n} = (n_1, n_2, \ldots, n_m)}</math> for the state of the network. For a customer in transit to state ''i'', let <math>\scriptstyle{\alpha_i(\mathbf{n}-\mathbf{e}_i)}</math> denote the probability that immediately before arrival the customer 'sees' the state of the system to be

:<math>\mathbf{n}-\mathbf{e}_i = (n_1,n_2,\ldots,n_i - 1,\ldots,n_m).\,</math>

This probability, <math>\scriptstyle{\alpha_i(\mathbf{n}-\mathbf{e}_i)}</math>, is the same as the steady state probability for state <math>\scriptstyle{\mathbf{n}-\mathbf{e}_i}</math> for a network of the same type with ''one customer less''.<ref>{{cite doi|10.1007/1-4020-3631-0_5}}</ref> It was published independently by Sevcik and Mitrani,<ref name="sevcik">{{cite doi|10.1145/322248.322257}}</ref> and Reiser and Lavenberg,<ref>{{cite doi|10.1145/322186.322195}}</ref> where the result was used to develop [[mean value analysis]].

==Notes==
{{Reflist}}

{{Queueing theory}}

[[Category:Queueing theory]]
[[Category:Stochastic processes]]
[[Category:Probability theorems]]