General explicit Lyapunov function for transient balanced Jackson networks

Construct an explicit Lyapunov function proving transience for every open Jackson network with at least three balanced nodes, including the general case in which the oblique-derivative subsolution problem on the associated spherical simplex must be solved.

Background

The paper proves transience for networks with $k_0\geq 3$ balanced nodes using capacity comparisons and symmetrisation, but this proof does not provide an explicit state-space Lyapunov function. Remark 3.2 discusses the natural candidate based on the inverse power of the elliptic radius of the corrected balanced-node process, together with angular corrections and stable-node correctors.

For general routing among balanced nodes, the boundary derivatives of the natural candidate have no fixed sign. The authors explain that seeking a function of the form $\varrho^{-\alpha}\psi$, with an angular factor $\psi$, leads to an oblique-derivative subsolution problem on a spherical simplex whose corners are difficult. Although the problem can be solved in specific examples, its general solution for all networks with $k_0\geq 3$ remains unresolved.