Jackson's theorem (queueing theory)

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
* mu_i = service rate at queue "i"
* lambda_i = total rate at which jobs arrive at queue "i"
* forall i,1leq ileq M: ho_i = utilization of the service at queue i = frac {lambda_i}{mu_i} < 1
* n_i(t) =# of jobs in queue "i" at time "t"
* n(t)=(n_1(t), n_2(t), dots, n_M(t))^T= the system state at time "t"
* P(k_1, k_2, dots, k_M, t) = Pr(n(t)=(k_1, k_2, dots, k_M)^T)
* P(k_1, k_2, dots, k_M)=lim_{t oinfty}P(k_1,k_2,dots,k_M,t)
* Arrivals from the outside world are Poisson. All queues have exponential service time distributions.

Product form of Jackson's network

:P(k_1,k_2,dots,k_M)=prod_{i=1}^{M}left [left(frac{lambda_i}{mu_i} ight)^{k_i}left(1-frac{lambda_i}{mu_i} ight) ight] = prod_{i=1}^{M} [(1- ho_i) ho_i^{k_i}]
(where ho_i=frac{lambda_i}{mu_i})

ee also

*Jackson network
*Little's law

External links

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


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