On up- and downcrossings

Abstract

For the sample functions of the stationary virtual waiting-time process v t of the GI / G /1 queueing system some properties of the number of up- and downcrossings of level v by the v t -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 v t -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.