In queueing theory, a discipline within the mathematical theory of probability, the arrival theorem[1] (also referred to as the random observer property, ROP or job observer property[2]) 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."[3]
The arrival theorem always holds in open 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.[4] 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[4][5] and networks with a delay protocol.[3]
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."[6]