We study the sample path behaviour of χ2 processes in the neighbourhood of their level crossings and extrema via the development of Slepian model processes. The results, aside from being of particular interest in the study of χ2 processes, have a general interest insofar as they indicate which properties of Gaussian processes (which have been heavily researched in this regard) are mirrored or lost when the assumption of normality is not made. We place particular emphasis on the behaviour of χ2 processes at both high and low levels, these being of considerable practical importance. We also extend previous results on the asymptotic Poisson form of the point process of high maxima to include also low minima (which are in a different domain of attraction) thus closing a gap in the theory of χ2 processes.

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

For the sample functions of the stationary virtual waiting-time process vt of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the vt-process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the vt-process of the GI/G/1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.