The distribution of the maximum of the unsigned scaled Brownian excursion process and of a modification of that process are derived. These distributions are related to the one-dimensional Brownian bridge.

An asymptotic formula for the mean number of maxima above a level of an n-dimensional stationary Gaussian field has been given by Nosko without proof. In this note a short general proof of this formula is given.

Suppose (X1, Y1), (X2, Y2), …, (Xn, Yn) are independent random vectors such that a ≦ Xi ≦ b and a ≦ Yi ≦ b, i = 1, 2, …, n. An upper bound which exponentially converges to zero is derived for the probability Pr{Sx – nμx ≧ nt1;SY – nμY ≧ nt2} where Sx = Σ Xi, SY = Σ Yi,EYi = μY, EXi = μx and t1 > 0, t2 > 0. The bound is a function of the difference b — a, the correlation between Xi and Yi, μx and μY and t1 and t2.

The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form , assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by , for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.

A generalized occupancy distribution obtained by Uppuluri and Carpenter (1971) is derived by elementary methods, and extended to a multivariate situation.

A finite-state semi-Markov process (SMP) with penalties is considered. A property which is similar to an increasing-hazard-rate property for a Markov chain is defined for an SMP. The SMP is controlled by shifts from the state Ei to immediately after a transition has occurred. Conditions are given which guarantee that the optimal stationary Markovian policy belongs to a subclass of control-limit policies.

A parallel reliability system is subject to shocks which are generated by a renewal point process. The shocks cause the components to fail with certain probabilities. First we study the case when the components fail independently of each other with the same probability. For this case simple closed expressions for the Laplace-Stieltjes transform and expectation of time to system failure are derived. Then a case of dependent components is studied: the probability of failure of a component depends on the number of functioning components. Some extensions are mentioned.

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

The average number of vehicles being able to enter an intersection per time unit from a minor road with a stop or yield sign — the capacity of the intersection — depends on the density of the traffic stream on the major road. In case the time-process of the major road traffic at the intersection is a non-homogeneous Poisson process with a periodic intensity function the capacity is calculated and compared with the capacity in the homogeneous case.

For a single-server queue with one waiting place and increasing arrival rate some necessary and sufficient conditions for infinitely many returns to emptiness with probability one are given.