Stability conditions for a two-resource linear flow network (${\rho }_1+{\rho }_0<1,{\rho }_2+{\rho }_0<1$)

The stationary law of a two-resource linear flow network is normalizable exactly whenThese are the strict offered-load constraints on the two unit-capacity resources. Summing first over the through-route count reduces normalization to two geometric series, which diverge at equality. With positive arrival rates, the nonexplosive continuous-time Markov chain is then positive recurrent.