It is possible to join up all pairs of planar points by using paths based on a variant of a Poisson line pattern which is dense everywhere, but with speed limits on the lines such that lines with speeds exceeding $v$ form a Poisson line process with density depending on $v$. Remarkably, this can be done in such a way that (a) any point can be reached from any other point in finite mean time, with unique fastest connection, (b) the pattern formed by shortest-time connections between a finite set of points is scale-invariant, (c) the pattern of shortest-time connections between points of an independent Poisson point process has finite mean length intensity. In short, it is a SIRSN [3, 4]. I will survey the work which establishes all this, and discuss relevance to evaluation of spatial networks, traffic in spatial networks [1, 2], and qualitative behaviour of shortest-time paths.
1. WSK. Geodesics and flows in a Poissonian city. Ann. Appl. Prob., 21 (2011), no. 3, 801-842.
2. WSK. Return to the Poissonian City. J. Appl. Prob (2014), 15A, 297-309.
3. WSK (2016). From Random Lines to Metric Spaces. Ann. Prob. (to appear), 46pp.
4. Kahn, J. (2016). Improper poisson line process as SIRSN in any dimension. Ann. Prob. (to appear).