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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>
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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]]

Latest revision as of 11:46, 15 August 2014

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