Traffic equations for a multiclass queueing network
= Traffic equations for a multiclass queueing network
{title2=$\lambda=\nu+P^T\lambda$}
For queue-class pairs $a$, external arrival rates $\nu_a$, and routing probabilities $p_{ab}$, the effective throughputs satisfy $\lambda_a=\nu_a+\sum_b\lambda_b p_{ba}$. In an open network with routing <spectral radius> less than one, $\lambda=(I-P^T)^{-1}\nu$ is the unique nonnegative solution. Summing gives $\sum_a\lambda_a p_{a0}=\sum_a\nu_a$, balancing total entrances and exits.