Inequalities between time and customer averages for HNB(W)UE arrival processes

We show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.