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.

This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

Let X1, X2, ···, Xn be a random sample of size n from a population with probability density function f(x), x > 0, and let X1, n < X2, n < ··· < Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − Xi−1, n) and (n − i)(Xi+1, n − Xi, n) for one i and one n with 2 ≦ i ˂ n.

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.

For a distribution function F on (0,∞), regular variation of its tail is known to imply that F is subexponential. Let be merely of dominated variation. This note shows that F need not be subexponential, and investigates which of the known necessary conditions for subexponentiality become sufficient when insisted upon for such an F.

We study the asymptotic behavior as α → 0 of the number of independent random arcs of length a needed to cover at least 1 − Kα (K ≧ 1) of a circle of unit circumference.

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.

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

Using Laplace transforms an explicit analytical expression is obtained for the autocorrelation function of the number in the system for the two-server Poisson queue. The method employed may be extended to Poisson queues with more than two servers.

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.