*#REDIRECT[[Jackson network]] ▼'''Jackson's theorem''' is the first significant development in the theory of [[queueing theory|networks of queues]]. It assumes an open queueing network of single-server queues with the following characteristics:
* M = # of queues in the system, not counting queue 0 which represents the outside world
* <math>\mu_i</math> = service rate at queue ''i''
* <math>\lambda_i</math> = total rate at which jobs arrive at queue ''i''
* <math>\forall i,1\leq i\leq M:\rho_i = </math> utilization of the service at queue <math>i = \frac {\lambda_i}{\mu_i} < 1</math>
* <math>n_i(t)</math> =# of jobs in queue ''i'' at time ''t''
* <math>n(t)=(n_1(t), n_2(t), \dots, n_M(t))^T</math>= the system state at time ''t''
* <math>P(k_1, k_2, \dots, k_M, t) = \Pr(n(t)=k_1, k_2, \dots, k_M)^T)</math>
* <math>P(k_1, k_2, \dots, k_M)=\lim_{t\to\infty}P(k_1,k_2,\dots,k_M,t)</math>
* Arrivals from the outside world are Poisson. All queues have exponential service time distributions.
==Production form of Jackson's network==
:<math>P(k_1,k_2,\dots,k_M)=\prod_{i=1\to M}\left[\left(\frac{\lambda_i}{\mu_i}\right)^{k_i}\left(1-\frac{\lambda_i}{\mu_i}\right)\right]=\prod_{i=1\to M}[(1-\rho_i)\rho_i^{k_i}]</math><br>
(where <math>\rho_i=\frac{\lambda_i}{\mu_i}</math>)
==See also==
*[[Little's law]]
==External links==
*Sinclair, B. (2005, June 9). ''[http://cnx.rice.edu/content/m10888/latest/ Jackson's Theorem]''. Connexions
[[Category:Stochastic processes]]
[[Category:Mathematical theorems]]
[[Category:Queueing theory]]
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