Queues under preemptive LIFO and ladder height distributions for risk processes: A duality

A sample-path duality is shown between the stationary service time in progress for single-server queues with general stationary input operating under the preemptive last-in-first-out (LIFO) discipline, and the first strictly descending ladder height in risk processes with a general stationary input. In addition to giving new insight, the duality allows one to easily obtain some known probabilistic formulas for risk processes by calculating corresponding elementary queueing quantities. The duality thus ‘explains’ why these risk formulas hold under such generality. For example, under time stationarity, the first descending ladder height distribution for the risk process is shown to be the same as the time-stationary remaining service time in a preemptive LIFO queue. The duality is very simple and based on basic queueing ideas and time reversal of input. An arrival-stationary analog is also given as is an extension covering all descending ladder heights jointly.