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

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'''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:

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 <math>''i</math>''

* <math>\lambda_i</math> = total rate at which jobs arrive at queue <math>''i</math>''

* <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==

*[[Jackson network]]

*[[Little's Lawlaw]]

==External links==