How to use the poisson process for BLE neighbor discovery?

Question

I am trying to use the Poisson process for BLE(Blueetooth Low Energy) neighbor discovery. In this context, there are two devices called Advertiser and Scanner. The advertiser and scanner emit and listen to beacons in a duty-cycled fashion. This leads to longer discovery latency sometimes and I want to use the poisson process for the scanning and advertising patterns and prove the discovery is going to happen. Basically I want to merge two poisson processes and make sure there are two events within a small interval delta.