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)
Famous quotes containing the words open, jackson and/or network:
“For it is not possible to join serpentine wisdom with columbine innocency, except men know exactly all the conditions of the serpent: his baseness and going upon his belly, his volubility and lubricity, his envy and sting, and the rest; that is, all forms and natures of evil: for without this, virtue lieth open and unfenced.”
—Francis Bacon (1561–1626)
“Oh, write of me, not “Died in bitter pains,”
But “Emigrated to another star!””
—Helen Hunt Jackson (1830–1885)
“Parents need all the help they can get. The strongest as well as the most fragile family requires a vital network of social supports.”
—Bernice Weissbourd (20th century)