  Two classical stochastic processes are considered, the Ehrenfest process,
introduced in 1907 in the kinetic theory of gases to describe the heat exchange
between two bodies and the Engset process, one of the early (1918) stochastic
models of communication networks. This paper investigates the asymptotic
behavior of the distributions of hitting times of these two processes when the
number of particles/sources goes to infinity. Results concerning the hitting
times of boundaries in particular are obtained. The paper relies on martingale
methods, a key ingredient is an important family of simple non-negative
martingales, an analogue, for the Ehrenfest process, of the exponential
martingales used in the study of random walks or of Brownian motion.
