We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres.

A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.