Jackson's theorem

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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
$μ i$ = service rate at queue $i$
$λ i$ = total rate at which jobs arrive at queue $i$
$\forall i,1\leq i\leq M:\rho_i =$ utilization of the service at queue $i = \frac {\lambda_i}{\mu_i} < 1$
$n i (t)$ =# of jobs in queue i at time t
$n (t) = (n 1 (t), n 2 (t),..., n M (t)) T$ = the system state at time t
$P (k 1, k 2,..., k M, t) = P r (n (t) = k 1, k 2,..., k M) T)$
$P(k_1, k_2, ..., k_M)=\lim_{t\to\infty}P(k_1,k_2,...,k_M,t)$
Arrivals from the outside world are Poisson. All queues have exponential service time distributions.