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
  
• ${\displaystyle \mu _{i}}$ = service rate at queue ${\displaystyle i}$
  
• ${\displaystyle \lambda _{i}}$ = total rate at which jobs arrive at queue ${\displaystyle j}$
  
• ${\displaystyle \forall i,1\leq i\leq M:\rho _{i}=}$ utilization of at queue ${\displaystyle i={\frac {\lambda _{i}}{\mu _{i}}}<1}$
  
• ${\displaystyle n_{i}(t)}$ =# of jobs in queue i at time t.
  
• ${\displaystyle n(t)=(n_{1}(t),n_{2}(t),...,n_{M}(t))^{T}}$= the system state at time t
  
• ${\displaystyle P(k_{1},k_{2},...,k_{M},t)=Pr(n(t)=k_{1},k_{2},...,k_{M})^{T})}$
  
• ${\displaystyle P(k_{1},k_{2},...,k_{M})=\lim _{t\to \infty }P(k_{1},k_{2},...,k_{M},t)}$
  
• Arrivals from the outside world are Poisson. All queues have exponential service time distributions.
  

Production form of Jackson's Network

${\displaystyle P(k_{1},k_{2},...,k_{M})=\prod _{i=1\to M}[({\frac {\lambda _{i}}{\mu _{i}}})^{k_{i}}(1-{\frac {\lambda _{i}}{\mu _{i}}})]=\prod _{i=1\to M}[(1-\rho _{i})\rho _{i}^{k_{i}}]}$ 
(where ${\displaystyle \rho _{i}={\frac {\lambda _{i}}{\mu _{i}}}}$ )

See also

Jackson Network