  In localization, an outage occurs if the positioning error exceeds a
pre-defined threshold, $\epsilon_{\rm th}$. For time-of-arrival based
localization, a key factor affecting the positioning error is the relative
positions of the anchors, with respect to the target location. Specifically,
the positioning error is a function of (a) the distance-dependent
signal-to-noise ratios (SNRs) of the anchor-target links, and (b) the pairwise
angles subtended by the anchors at the target location. From a design
perspective, characterizing the distribution of the positioning error over an
ensemble of target and anchor locations is essential for providing
probabilistic performance guarantees against outage. To solve this difficult
problem, previous works have assumed all links to have the same SNR (i.e., SNR
homogeneity), which neglects the impact of distance variation among the anchors
on the positioning error. In this paper, we model SNR heterogeneity among
anchors using a distance-dependent pathloss model and derive an accurate
approximation for the error complementary cumulative distribution function
(ccdf). By highlighting the accuracy of our results, relative to previous ones
that ignore SNR heterogeneity, we concretely demonstrate that SNR heterogeneity
has a considerable impact on the error distribution.
