The no-aging property and the ℓ1-isotropic model it implies have been introduced to overcome certain shortcomings of the exponential model. However, its definition is abstract and not very useful for practitioners. This paper presents several additional characterizations of the no-aging property. Included are (1) characterizations that appropriately generalize the memoryless property and the constant-failure-rate property of the exponential, (2) behavioral characterizations based on fair bets, and (3) geometric characterizations of the survival and density function and differential-geometric characterizations based on tensor methods.

Consider two exponential single-server queues in tandem and suppose that service rates of customer n are λ n and μ n respectively. In this note, a simple and direct proof is given of the fact that the departure process from the tandem queue is statistically unaffected when the service rates are interchanged if λ n – μn is independent of n. The proof is based only on the memoryless property of exponential distributions.

The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form , assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by , for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.