Two Variability Orders

In this paper we study a new variability order that is denoted by ≤ st:icx . This order has important advantages over previous variability orders that have been introduced and studied in the literature. In particular, X ≤ st:icx Y implies that Var[ h ( X )] ≤ Var[ h ( Y )] for all increasing convex functions h . The new order is also closed under formations of increasing directionally convex functions; thus it follows that it is closed, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X ≤ st:icx Y are described. For this purpose, a new order, called the excess wealth order, is introduced and studied. This new order is based on the excess wealth transform which, in turn, is related to the Lorenz curve and to the TTT (total time on test) transform. The relationships to these transforms are also studied in this paper. The main closure properties of the order ≤ st:icx are derived, and some typical applications in queueing theory are described.