On finite exponential moments for branching processes and busy periods for queues

Using a known fact that a Galton-Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.