#REDIRECT[[Jackson Networknetwork]] ▼
Jackson's Theorem is the first significant development in the theory of networks of queues. It assumes an open queueing<br>
network of single-server queues with the following characteristics:<br>
* M = # of queues in the system, not counting queue 0 which represents the outside world<br>
* <math>\mu_i</math> = service rate at queue <math>i</math><br>
* <math>\lambda_i</math> = total rate at which jobs arrive at queue <math>j</math><br>
* <math>\forall i,1\leq i\leq M:\rho_i = </math> utilization of at queue <math>i = \frac {\lambda_i}{\mu_i} < 1</math><br>
* <math>n_i(t)</math> =# of jobs in queue i at time t. <br>
* <math>n(t)=(n_1(t), n_2(t), ..., n_M(t))^T</math>= the system state at time t<br>
* <math>P(k_1, k_2, ..., k_M, t)=Pr(n(t)=k_1, k_2, ..., k_M)^T)</math><br>
* <math>P(k_1, k_2, ..., k_M)=\lim_{t\to\infty}P(k_1,k_2,...,k_M,t)</math><br>
* Arrivals from the outside world are Poisson. All queues have exponential service time distributions. <br>
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==Production form of Jackson's Network==
<math>P(k_1,k_2,...,k_M)=\prod_{i=1\to M}[(\frac{\lambda_i}{\mu_i})^{k_i}(1-\frac{\lambda_i}{\mu_i})]=\prod_{i=1\to M}[(1-\rho_i)\rho_i^{k_i}]</math><br>
(where <math>\rho_i=\frac{\lambda_i}{\mu_i}</math>)<br>
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==See also==
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