A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y > 0, given that T >x, the conditional probability of T >x +y is the same as the unconditional probability of T >y; in other words, the physical process operates as if it has no memory concerning the successive occurrences of a certain event. The paper investigates the consequences of defining the property of lack of memory on more general time-domains than the positive reals. As a side issue, there is discussion of a stochastic variation of Cauchy's functional equation.