Let ℒ be the set of probability measures concentrated on { − 1, 0, 1, 2, …}. Then, if μ0 is in ℒ, there exists at most one μ1 in ℒ, with μ0≠ μ1, such that for all n = 1, 2, ….

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 the identical distribution of the random variables nX1,n and (n − i + 1)(X1,n −; Xi–1,n) for one i and one n with 2 ≦ i ≦ n.

This paper contains a simple proof of Steck's recurrence equation for determining the power of the standard one-sided Kolmogorov test for a distribution function.

We study a Markov quality control model in which we may either reset the machine or keep the machine producing items. Both discounted and average costs are considered. We shall show that a monotone policy is optimal and show how to compute the optimal critical point. Two special models are studied along with some numerical results.

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.