Directional Derivatives of Oblique Reflection Maps

Given an oblique reflection map and functions , Dlim (the space of functions that have left and right limits at every point), the directional derivative () of along , evaluated at , is defined to be the pointwise limit (as 0) of the family of functions () . = 1 [( + ) - ()] . Directional derivatives are shown to exist and lie in Dlim for oblique reflection maps of the socalled Harrison-Reiman class. When and are continuous, the convergence of () to () is shown to be uniform on compact subsets of continuity points of the limit () and the derivative () is shown to have an autonomous characterization as the unique fixed point of an associated map. Motivation for the study of directional derivatives arises from the fact that they characterize functional central limits of non-stationary queueing networks. This work also shows how the various types of discontinuities of the derivative () are related to the topology of the network as well as to the states (of underloading, overload...

• Type: Journal
• Year: 2010
• Where: MOR
• Authors: Avi Mandelbaum, Kavita Ramanan
• Keywords: Directional Derivatives, MOR 2010, Oblique Reflection Maps, Reflection Map, Security Privacy