Jackson's theorem (queueing theory): Difference between revisions

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{{moveto|Jackson's Theorem}}

'''Jackson's theoremTheorem''' is the first significant development in the theory of [[queueing theory|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 the service 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>

<br>

==Production form of Jackson's networkNetwork==

<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>

<br>

==See also==

*[[Jackson Network]]

==External links==

*Sinclair, B. (2005, June 9). ''[http://cnx.rice.edu/content/m10888/latest/ Jackson's Theorem]''. Connexions

[[Category:Stochastic processes]]