A method is presented for constructing a continuous, stationary process with a given rate of upcrossings at all levels and for which it is possible to find the exact distribution of the number of upcrossings, in an interval, at all levels.

A sequence of independent, identically distributed random variables is observed. After a sample of m has been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied.

The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.

Consider a queue for a special performance or sale. A person with arrival number i will decide to join the queue with probability pi. Here the pi denote fixed constants. Let X denote the number of arrivals, X″ the number of customers leaving the queue and X′ = X – X″ the number of customers who stay with the queue. For the case that pi+1 ≦ pi and X has a Poisson distribution, it is shown that Var (Xʺ) ≧ E(Xʺ) and Var (X′) ≦ E(X′). There are also results for the case where {pi} and X are rather arbitrary.

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.