Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. In this paper, we introduce, by taking into account the age of systems, a time-dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t)=P(X
t
>
Y
t
) which enables one to obtain, at time t, the probability that the residual lifetime X
t
is greater than the residual lifetime Y
t
. It is mentioned, in Brown and Rutemiller (1973) that the probability of type R(t) is important for designing as long-lived a product as possible. Several properties of R(t) and its connection with well-known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.
