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\begin{document}

\title[a LP Algorithm for S-S Distribution of Heavy Traffic Approximations]
{A Linear Programming Algorithm for Stationary Distribution of Heavy Traffic Approximations}

\date{January, 2008}
\begin{abstract}
This paper proposes a linear programming algorithm for computing the stationary distribution of multidimensional diffusions that arise as approximate
models for queueing systems. Our algorithm is based on the necessity and sufficiency of an Adjoint Basic Relationship (BAR): We approximate the state
space considering a finite grid of points and use a finite set of test functions to approximate an infinite dimensional functional space. With these
approximations, the BAR reduces to a set of linear equations that can be solved using standard linear programming techniques. We start by setting the
algorithm for diffusions arising under the classical heavy-traffic regime, namely the Semi-martingale Reflected Brownian Motions (SRBM), and then we
extend the setting for diffusions arising under the Halfin-Whitt heavy-traffic regime. Extensive computational experiences are reported. There, the
algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments.
\end{abstract}

 \maketitle

\section{Introduction}

%\input{Intro_attemp}
\begin{itemize}
\item System of queues and its complexity.
\item Use of heavy Traffic approximations: approximate performance measures and justification for such approximation.
\item No close form solution of diffusion limits, not even for steady state.
\item No ``good'' algorithm to compute s-s distribution. Necessity of s-s distribution.
\item Algorithm idea: Bar characterization, discretization, linear programming. Convergence.
\item Closely related papers.
\item remainder of paper.
\item notation if necessary.
\end{itemize}

\section{The basic adjoint relationship}
\subsection{S-S alternative characterization}
\begin{itemize}
\item Heavy Traffic: diffusion limits, relevant examples. Intractability of solutions, even for steady state.
\item Bar characterization for the stationary distribution.
\end{itemize}
\subsection{Sufficiency of BAR}
\begin{itemize}
\item conditions for sufficiency
\end{itemize}
\subsection{Necessity of BAR}
\begin{itemize}
\item conditions for necessity
\end{itemize}
\section{Proposed algorithm}
\subsection{Bar approximation}
\begin{itemize}
\item State space discretization idea
\item Functional space approximation idea
\end{itemize}
\subsection{LP formulation}
\begin{itemize}
\item LP formulation
\item intuition for tightness constraint
\end{itemize}
\subsection{Convergence}
\begin{itemize}
\item conditions for convergence, and proof if possible.
\end{itemize}
\section{Applications : Semimartingale Reflected Brownian Motion}

\input{Applications-SRBM}
\section{Applications : Two Class Queue}
\section{Concluding remarks}
\section{References}
\end{document}
