Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.XY=X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forXY under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenXY is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofXY. Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system.