Properties of the Process of Level Crossings During a Busy Cycle of the M/G/1 Queueing System

The variables d(v), v ∈ (0, ∞) with d(v) the number of downcrossings of level v of the virtual waiting time process during a busy cycle of a queueing system constitute a stochastic process {d(v), v > 0}. For a stable M/G/1 queueing system this process is investigated in the present paper. Under certain conditions on the density of the service time distribution it is shown that this process is a birth and death process with constant birth rate and time-dependent death rate; however, for the M/M/1 system this process {d(v), v > 0} has also constant death rate. A number of properties of the d(v)-process are studied, yielding also some new results for the M/G/1 queueing system. In particular an explicit expression is found for the variance of the area underneath the sample function of the virtual waiting time process during a busy cycle, thus solving a question posed by Iglehart.