  In this paper, we mathematically characterize the improvement in device
localizability achieved by allowing collaboration among devices. Depending on
the detection sensitivity of the receivers in the devices, it is not unusual
for a device to be localized to lack a sufficient number of detectable
positioning signals from localized devices to determine its location without
ambiguity (i.e., to be uniquely localizable). This occurrence is well-known to
be a limiting factor in localization performance, especially in communications
systems. In cellular positioning, for example, cellular network designers call
this the hearability problem. We study the conditions required for unique
localizability and use tools from stochastic geometry to derive accurate
analytic expressions for the probabilities of meeting these conditions in the
noncollaborative and collaborative cases. We consider the scenario without
shadowing, the scenario with shadowing and universal frequency reuse, and,
finally, the shadowing scenario with random frequency reuse. The results from
the latter scenario, which apply particularly to cellular networks, reveal that
collaboration between two devices separated by only a short distance yields
drastic improvements in both devices' abilities to uniquely determine their
positions. The results from this analysis are very promising and motivate
delving further into techniques which enhance cellular positioning with
small-scale collaborative ranging observations among nearby devices.
