= Product-form stationary distribution of a multiclass queueing network
{title2=$\pi((x_j)_j)=\prod_j(1-\rho_j)\prod_i\lambda_{j,c_{ji}}/\mu_{j,c_{ji}}$}
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 $p_{ba}^*=\lambda_a p_{ab}/\lambda_b$, external arrivals with $\nu_a^*=\lambda_a p_{a0}$, and exits with $p_{a0}^*=\nu_a/\lambda_a$, 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 https://arxiv.org/pdf/0809.2697[Walton's paper on multiclass queueing networks, Section 2].
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