2 VARIABILITY ORDERS

In this paper we study a new variability order that is denoted by less than or equal to(st:icx). This order has important advantages over pr evious variability orders that have been introduced and studied in the literature. In particular, X less than or equal to(st:icx) Y implies that Var[h(X)] less than or equal to Var[h(Y)] for all increasing conv ex functions h. The new order is also closed under formations of incre asing directionally convex functions; thus it follows that it is close d, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X less than or equal to(st:icx) Y are described. For this purpose, a new order, call ed 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 re lationships to these transforms are also studied in this paper. The ma in closure properties of the order less than or equal to(st:icx) are d erived, and some typical applications in queueing theory are described .