one) each Voronoi tile is a convex and bounded polygon, and the graph $\calD :=(\calD_v,\calD_e)$ is a triangulation of the plane [@M91].

![The Delaunay triangulation and the Voronoi tessellation[]{data-label="f1"}](vor2){width="30.00000%"}

The *Voronoi tessellation* $\calV:=(\calV_v,\calV_e)$ is defined by taking the vertex set $\calV_v$ equal to the set of vertices of the Voronoi tiles and the edge set $\calV_e$ equals to the set of edges of the Voronoi tiles.

Each edge $\e\in\calD_e$ is independently assigned a nonnegative random variable $\tau_\e$ from a common distribution $\bF$ (the passage time distribution) that is independent of the Poisson process $\calD_v$. We assume throughout that $\bF$ is continuous and that $$\label{a1}
\int e^{ax}\bF(dx)<\infty\,\mbox{ for some }\,a\in(0,\infty)\,.$$ We denote by $(\Omega,\calF,\bP)$ our underline probability space, i.e. from each realization $\omega\in\Omega$ one can determine the Poisson point process as well the passage time configuration. This model inherits the euclidean (translation and rotational) invariance of the Poisson point process.

The passage time $t(\gamma)$ of a path $\gamma$ in $\calD$ is the sum of the passage times of the edges in $\gamma$: $$t(\gamma):=\sum_{\e\in\gamma}\tau_\e \,.$$ The first-passage time between two vertices $\v$ and $\v'$ in $\calD_v$ is defined by $$T(\v,\v'):=\inf\big\{t(\gamma);\,\gamma\in\calC(\v,\v')\big\} ,$$ where $\calC(\v,\v')$ the set of all paths connecting $\v$ to $\v'$. 