Testing a Poisson renewal process in the context of security alarm maintenance policies

The results of Barnett and Kenward (1996) on alarm and inspection calls for security systems are extended. Some further theoretical properties of the relevant superimposed renewal process model are obtained for the case of a Poisson alarm process both when the inter-inspection interval is constant and when it takes the form of another independent Poisson process. These properties are used to motivate the choice of inference procedures for examining the basic nature of the underlying processes. The original data of Barnett and Kenward (1996) are analysed using such procedures as well as extensive further data. It is seen that the conventional simple assumptions for the inter-inspection intervals do not hold in practice, and consequently a conditional approach to the problem using the Dirichlet distribution is developed and applied.