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Directional Derivatives of Oblique Reflection Maps\\

Avishai Mandelbaum\\
School of Industrial Engineering and Management\\
Technion, Haifa, Israel\\

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and


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Kavita Ramanan\\
Department of Mathematical Sciences\\
Carnegie Mellon University \\
Pittsburgh, PA 15213\\
kramanan@andrew.cmu.edu
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\noindent {\bf Abstract}.  Given an oblique reflection map $\Gamma$
 and functions $\psi, \chi \in {\cal D}_{\lim}$ (the space of
 $\mathbb{R}^K$-valued functions that have left and right limits at
 every point), the directional derivative $\nabla_\chi \Gamma (\psi)$
 of $\Gamma$ along $\chi$, evaluated at $\psi$, is defined to be the
 pointwise limit (as $\varepsilon \downarrow 0$) of the family of
 functions $\nabla^\varepsilon_\chi \Gamma (\psi) \doteq
 {\varepsilon}^{-1} \left[ \Gamma (\psi + \varepsilon \chi) - \Gamma
 (\psi) \right]$.  Directional derivatives are shown to exist and lie
 in ${\cal D}_{\lim}$ for oblique reflection maps associated with
 reflection matrices of the so-called Harrison-Reiman class.  When
 $\psi$ and $\chi$ are continuous, the convergence of
 $\nabla_{\chi}^{\epsilon} \Gamma (\psi)$ to $\nabla_\chi\Gamma(\psi)$
 is shown to be uniform on compact subsets of continuity points of the
 limit $\nabla_\chi \Gamma (\psi)$ and the derivative $\nabla_\chi
 \Gamma (\psi)$ is shown to have an autonomous characterization as the
 unique fixed point of an associated map.  Motivation for the study of
 directional derivatives stems from the fact that they arise as
 functional central limit approximations to time-inhomogeneous
 queueing networks as well as transient time-homogeneous queueing
 networks.  This work also shows how the various types of
 discontinuities of the derivative $\nabla_{\chi}Gamma(\psi)$ are
 related to the reflection matrix and properties of the function
 $\Gamma (\psi)$.  In the queueing network context, this describes the
 influence of the topology of the network and the states (of
 underloading, overloading or criticality) of the various queues in
 the network on the discontinuities of $\nabla_\chi \Gamma (\psi)$.


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