Letter to the Editor—Bruce's Spider and Renewal Theory

The theory of an alternating renewal process with absorption at the renewal points is examined. The problem is discussed in the context of the behavior of Bruce's spider that attempted to climb a smooth wall but repeatedly slipped back and had to start again; but it provides a model for many other processes, of which two typical examples are given. Results are obtained under fairly general distributional assumptions for the distribution of the time to absorption and for the probabilities, conditional upon no absorption having taken place, of the process being in one phase or the other of the alternating process.