Reflected UAS Routing: Stability and Drift Calculation in Queues

A recent study published on arXiv (2607.28688) investigates Reflected UAS routing within heterogeneous multi-server queues under subcritical load conditions. The analysis reveals that the deterministic surrogate is a reflected ordinary differential equation (ODE) situated in the nonnegative orthant, differing from the unconstrained drift equation. This reflected ODE possesses a unique boundary equilibrium defined by a scalar consistency equation and a convex-potential representation, with all trajectories ultimately converging to it. The previously established method of extending deterministic Lyapunov descent to continuous-time Markov chain (CTMC) stability fails, as the precise generator applied to the deterministic potential introduces a boundary term absent in the reflected-ODE descent identity. The authors present a direct Foster-Lyapunov drift inequality for the CTMC utilizing a weighted-quadratic function, avoiding the unsuccessful lift. At the benchmark parameter point, the boundary equilibrium aligns with the numerical attractor to machine precision, and the standard Reflected UAS policy demonstrates a lower mean queue length compared to UAS and JSSQ in independent simulations. This paper is classified as a cross announcement and is available on arXiv.