Local-count independence in a linear flow network ($E{n}_i={\rho }_i/(1-{\rho }_0-{\rho }_i)$)

Summing the stationary law of a linear flow network over its through-flow count gives independent local geometric distributions with ratios $q_i={\rho }_i/(1-{\rho }_0)$. This gives the displayed means. The local counts are independent of one another in this marginal distribution; the through count is generally dependent on them. The independence is a stationary distributional fact, not an assertion that their dynamics evolve independently.