Propositions 1 and 2, pp. 13–14 — with exogenous demand, dedicated has the shorter wait exactly when λr is below one threshold

Let $C_B\in[0.58,0.59]$. There is a constant $a_1$ with $0<a_1<\tfrac34$ such that for every service radius $r>0$ and every exogenous demand rate $\lambda>0$ with $\lambda r<\tfrac34$ (that is, $\lambda<\mu_D$):

1. $W_D(\lambda,\mu_D,C_D)<W_B(\lambda,\mu_B,C_B)$ if and only if $\lambda r<a_1$;
2. $W_B(\lambda,\mu_B,C_B)<W_D(\lambda,\mu_D,C_D)$ if and only if $\lambda r>a_1$;
3. for every inverse valuation c.d.f. $F^{-1}$, every potential-arrival parameter $\Lambda>0$ with $\lambda\le\Lambda r^2$, and every wait-cost rate $c>0$, with

the dedicated revenue $V(\lambda,W_D)$ strictly exceeds the batch revenue $V(\lambda,W_B)$ iff $\lambda r<a_1$, and the batch revenue strictly exceeds the dedicated one iff $\lambda r>a_1$.

Here $\mu_D=\frac3{4r}$, $C_D=\frac98$, $\mu_B=\frac{45\pi}{4r(32+15\pi)}$, and $W_D,W_B$ are the expected waits (5), (12). For $C_B=0.583$ the paper reports $a_1\approx 0.5689$. Proposition 1 is the threshold $\lambda_{ex}=a_1/r$ in the demand rate at fixed radius, and Proposition 2 the threshold $r_{ex}=a_1/\lambda$ in the radius at fixed demand rate.

Formalization Note lam is $\lambda$. One constant $a_1$ serves all $r$ and $\lambda$, as in the paper's proof. The range $\lambda r<3/4$ is the stability range of dedicated delivery (batch is then stable as well); beyond it only batch is defined and the paper makes no comparison. The revenue clauses include the model's $0<\Lambda$ and $\lambda\le\Lambda r^2$ from p. 8. $F^{-1}$ is represented by an arbitrary function Q; its properties cancel from equal-demand revenue comparisons. $C_B$ ranges over the bracket $[0.58,0.59]$ instead of the paper's numerical value $0.583$.