Service-capacity allocation in an open migration process (${\varphi }_j-{\alpha }_j\propto \sqrt{{\alpha }_j}$)

For positive traffic rates in single-server colonies, a total service budget $F>{\sum }_j{\alpha }_j$ minimizes the stationary mean population when the spare rates ${\varphi }_j-{\alpha }_j$ are proportional to $\sqrt{{\alpha }_j}$. The Cauchy-Schwarz inequality proves the optimum directly.