Traffic equations for a multiclass queueing network ($\lambda =\nu +P^T\lambda$)

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.