Extremal property of waiting times in $GI|G|1|\infty$ model

In the present paper stationary distribution functions $W$ and $W^*$ of waiting times, which are limits for actual and virtual waiting times across the time axis, in the $GI|G|1|\infty$ model under $FIFO$ discipline are examined.
The following extremal property is proved. For all $x\in(0,+\infty)$ in the case of non-Poissonian entering stream of demands the strict inequalities $W(x)>W^*(x)>\hat{W}(x)$ are valid, where $\hat{W}$ is the waiting times’ stationary distribution function in the case of the Poissonian entering stream.