Open Jackson Network
In an open network, jobs arrive from outside following a Poisson process with rate . Each arrival is independently routed to node j with probability and . Upon service completion at node i, a job may go to another node j with probability or leave the network with probability .
Hence we have the overall arrival rate to node i, including both external arrivals and internal transitions:
Define, then we can solve .
All jobs leave each node also following Poisson process, and define as the service rate of node i when there are jobs at node i.
Let denote the number of jobs at node i at time t, and . Then the equilibrium distribution of, is determined by the following system of balance equations:
where denote the unit vector.
Read more about this topic: Jackson's Theorem (queueing Theory)
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