The distribution of communication cost for a mobile service scenario

We consider the following mathematical model for a mobile service scenario. Consider a planar stationary Poisson process, with its points radially ordered with respect to the origin (the anchor); these points may correspond to locations of e.g. restaurants. A user, with a location different from the origin, asks for the location of the first Poisson point and keeps asking for the location of the next Poisson point until the first time that he can be completely certain that he knows which Poisson point is his nearest neighbour. The distribution of this waiting time, called the communication cost, is analysed in detail. In particular the expected communication cost and the asymptotic behaviour as the distance between the user and the anchor increases are of interest.