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
* $n\_i(t)$ =# 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