Let be the sequence of partial sums of independent N(μ, 1) random variables. The boundary in the {(n, Sn)} plane which minimizes the expected number of times n that Sn will be below a boundary when μ = θ > 0 subject to a given expected number of visits that Sn will be above the same boundary when μ = 0 is shown to be linear and is also characterized. Similar results are shown for Brownian motion.

For a single-type Galton—Watson branching process with mean less than one and finite second moment, we establish an upper bound for the mean of the associated Yaglom limit. This bound is attained if and only if the generating function of the process is linear.

A two-step procedure is described to approximate first come first served delay distributions in M/G/1 and GI/G/1 systems. The procedure consists in formulating a coarse initial approximation for the distribution sought, and then correcting this approximation by means of a suitable integral transformation. Examples are displayed.