 Product-form stationary distribution of a multiclass queueing network

Connect multiclass single-server queues by Markovian routing. Each node must either have common exponential service parameters across its classes or use a symmetric service discipline. If the traffic equations for a multiclass queueing network give loads ${\rho }_j={\sum }_r{\lambda }_{jr}/{\mu }_{jr}<1$, its equilibrium word law is the product of the individual queue laws. Queue populations are independent at a fixed equilibrium time. A proof reverses routing with $p_{ba}^{\ast }={\lambda }_a{p}_{ab}/{\lambda }_b$, external arrivals with ${\nu }_a^{\ast }={\lambda }_a{p}_{a0}$, and exits with $p_{a0}^{\ast }={\nu }_a/{\lambda }_a$, and interchanges the insertion and service position rules. Weight ratios verify each reversed transition, and the traffic equations match total rates. This is a quasireversibility construction. A standard ordered-queue formulation is described in Walton's paper on multiclass queueing networks, Section 2.