Note on the GI/GI/1 queue with LCFS-PR observed at arbitrary times

Consider the GI/GI/1 queue with the Last-Come First-Served Preemptive-Resume service discipline. We give intuitive explanations for (i) the geometric nature of the stationary queue length distribution and (ii) the mutual independence of the residual service requirements of the customers in the queue, both considered at arbitrary time points. These distributions have previously been established in the literature by either first considering the system at arrival instants or using balance equations. Our direct arguments provide further understanding of (i) and (ii).