Proposition 3, p. 21 — ε-equitable district error bound with corrected η

Consider a compact convex service region with normalized Lipschitz demand density, $m$ identical vehicles, a depot, capacity $q$, speed $v$, mean service time $\bar s$, and geometric constant $\beta$. Let $R_1,\ldots,R_m$ partition the region and let their $\mu_1$ and $\mu_2$ shares differ from one $m$th of the corresponding whole-region shares by at most $\varepsilon$. For $\lambda>0$ in the stable regime, $\varepsilon\ge0$, and corrected $\eta<1$, write $L_i$ for the single-vehicle Theorem 3 leading term using district $i$'s own arrival rate and conditional density and $L^*$ for the full-region term. Then

The bound quantifies the loss in each district from approximate equitability.

Formalization Note The source prints $\eta=\varepsilon/\max\{\mu_1(R)/m,(vq/\lambda)(1-\lambda\mu_2(R)/(mvq))\}$, which makes the proposition false; the formal statement uses $\min$ as the proof requires. Its “for small $\varepsilon$” is expressed as $\eta<1$. $L_i$ and $L^*$ are Theorem 3 leading expressions at fixed $\lambda$, rather than the underlying stochastic delivery-time functions.