An application of the theory of regenerative phenomena

Abstract

One of the classical models of applied probability is that which may be described as the covering of a line with a random collection of intervals of random length. It appears as such in ((9), pp. 23–25), but also as the problem of Type II counters with random dead time (1, 4, 10), as a model for a pedestrian crossing a busy road(5), and as the queue M/G/∞. If an excuse is required for returning to a problem which has been the object of so much research in the past, it is to be found in the fact, noted by Kendall (5), that the model gives rise to the general stable infinitely divisible p-function, and that the theory of the semigroup P of standard p-functions (6) requires deeper information than previous, practically motivated, studies have provided.