Jackson's theorem (in queueing theory)
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Jackson's theorem is the first significant development in the theory of 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
- μi = service rate at queue i
- λi = total rate at which jobs arrive at queue i
utilization of the service at queue 
- ni(t) =# of jobs in queue i at time t
- n(t) = (n1(t),n2(t),...,nM(t))T= the system state at time t
- P(k1,k2,...,kM,t) = Pr(n(t) = k1,k2,...,kM)T)
- Arrivals from the outside world are Poisson. All queues have exponential service time distributions.
[edit] Production form of Jackson's network
![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/e/f/2/ef2c515d43481610e8e94d318a845fb4.png)
(where 
)
[edit] See also
[edit] External links
- Sinclair, B. (2005, June 9). Jackson's Theorem. Connexions
