A recent study on the GI/G/1 queue derives the Maclaurin series for the moments of the waiting time and the delay to respect to some parameters. By the same approach, we obtain an identity on the moments of the transient delay of the M/G/1 queue. This identity allows us to understand the transient behavior of the process better. We apply the identity with other established results to study convergence rate and stochastic concavity of the transient delay process, and to derive bounds and approximations of the moments. Our approximation and bound both have simple closed forms and are asymptotically exact as either the traffic intensity goes to zero or the process approaches stationarity. Performance of the approximation of several M/G/1 queues is illustrated by numerical experiments. It is interesting to note that our results can also help to gain variance reduction in simulation.

We study the structure of genealogical trees of reduced subcritical Galton-Watson processes in a random environment assuming that all (randomly varying in time) offspring generating functions are fractional linear. We show that this structure may differ significantly from that of the ‘classical’ reduced subcritical Galton-Watson processes. In particular, it may look like a complex ‘hybrid’ of classical reduced super and subcritical processes. Some relations with random walks in a random environment are discussed.

An algorithm for calculating the probability distribution of ladder epochs is derived. Two theorems are given for bounds on the distribution function of ladder epoch probabilities.