Security systems and renewal processes

An interesting feature of successive inter-event times in a Poisson renewal process, when observed through a superimposed (possibly random) grid, can be interpreted as an extended form of the ‘inspection paradox’. Probabilistic measures are determined and an estimation and testing procedure is outlined which could be used to examine possible departure from randomness (Poisson form). The problem arose from study of security systems and the results have important applications in that field.