Nondecreasing Relative Imbalance Along Optimal Two-Server Fluid Paths

Prove that, in the two-server fluid dispatching model with size-aware routing to parallel FCFS servers (each with service rate 1/2), total arrival rate λ<1, and job-size distribution with density f(x), the optimal path that minimizes the total latency has nondecreasing relative imbalance \hat{y} = (u_1 − u_2)/(u_1 + u_2); equivalently, along the optimal path the derivative of the relative imbalance with respect to the total backlog, \hat{y}', satisfies \hat{y}' ≥ 0.

Background

In the two-server case, the authors represent the system state by total backlog x = u1 + u2 and imbalance y = u1 − u2, and analyze optimal paths characterized by a single job-size threshold. They establish a monotonicity result showing that optimal paths are monotone in the relative imbalance \hat{y} = y/x, but leave open whether the optimal path always increases the relative imbalance.

Numerical evidence and analytical observations suggest that unbalancing the queues (increasing \hat{y}) reduces total cost, motivating the conjecture that \hat{y}' ≥ 0 along the optimal path. A rigorous proof would clarify the structural behavior of optimal controls and strengthen theoretical support for heuristics that intentionally unbalance loads.