Jackson's theorem (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
  • \forall i,1\leq i\leq M:\rho_i = utilization of the service at queue i = \frac {\lambda_i}{\mu_i} < 1
  • ni(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\to\infty}P(k_1,k_2,\dots,k_M,t)
  • Arrivals from the outside world are Poisson. All queues have exponential service time distributions.

[edit] 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}\right)^{k_i}\left(1-\frac{\lambda_i}{\mu_i}\right)\right] = \prod_{i=1}^{M}[(1-\rho_i)\rho_i^{k_i}]

(where \rho_i=\frac{\lambda_i}{\mu_i})

[edit] See also

[edit] External links