\documentclass[pdf]{prosper}
\title{Modeling Performance of Mant-to-One Collective Communication Operations in Heterogeneous Cluster}
\author{En-ran Zhou}

\begin{document}
\maketitle

\begin{slide}{Problem and Environment}
\begin{itemize}
\item Accurate model for Many-to-One communication
\item Switched-enabled Ethernet network
\end{itemize}
\end{slide}

\begin{slide}{Communication Models}
\begin{itemize}
\item Point-to-Point
\item One-to-Many
\item Mutiple Point-to-Point
\item Many-to-One
\end{itemize}
\end{slide}

\begin{slide}{Point-to-Point}
\begin{equation}
T_{p2p} = C_i + t_iM + C_j + t_jM + M/\beta_{ij}
\end{equation}

\begin{itemize}
\item $M$: message size
\item $C_i$: fixed processing delays at node $i$
\item $t_i$: times to process a byte at node $i$
\item $\beta_{ij}$: transmission rate between node $i$ and $j$.
\end{itemize}
\end{slide}

\begin{slide}{One-to-Many}
\begin{equation}
C_0 + t_0nM + \left\lbrace\begin{array}{cc}
\max\lbrace c_j + t_jM + M/\beta_{0j}\rbrace & M \leq S \\
\sum_{j=1}^n(C_j + t_jM + M/\beta_{0j}) & M > S
\end{array}\right.
\end{equation}
Due to buffer.
\end{slide}

\begin{slide}{Mutiple Point-to-Point}
\begin{equation}
T_{mp2p} = \max_K\{T_{ij}\}
\end{equation}
\end{slide}

\begin{slide}{Many-to-One}
\begin{itemize}
\item Small message sizes ($0 \leq M \leq M_1$)
\item Congestion region ($M_1 < M \leq M_2$)
\item Large message sizes ($M > M_2$)
\end{itemize}

\vskip 0.2in
Vadhiyar et al noticed the non-linearity behaviour of Many-to-One communication type.
\end{slide}

\begin{slide}{Small message sizes}
Source nodes are sorted in ascending order of the sending overhead.

\vskip 0.1in
Linear Model
\begin{equation}
\Delta_1(C_0 + t_0M + M/\beta_{0j} + \sum_{i=1}^{n-1}(C_i + t_iM))
\end{equation}
$\Delta_1$ is and adjustment factor to improve the accuracy

\vskip 0.1in
Consider the total idle time $T_{idle}$
\begin{equation}
\Delta_1(C_0 + t_0M + M/\beta_{0j} + T_{idle} + \sum_{i=1}^{n-1}(C_i + t_iM))
\end{equation}
\end{slide}

\begin{slide}{Congestion region}
Predict excution time with a constant $C$
\begin{equation}
T_{mto}(M, N) = C
\end{equation}
\end{slide}

\begin{slide}{Large message sizes}
Similar to small message sizes, linear model
\begin{equation}
\Delta_2(C_0 + t_0M + M/\beta_{0j} + \sum_{i=1}^{n-1}(C_i + t_iM))
\end{equation}
$\Delta2$ represent the increased overhead for synchronous sending in reservation mode for larger messages.
\end{slide}

\begin{slide}{Model Parameters}
\begin{itemize}
\item $t_i, t_j, C_i, C_j$ were obtained by a simple ping pong communication
\item $t_i, C_i$ for each node define the characteristics of a particular node
\item $M_1, M_2, C$ are additional parameters found experimentally which are the characteristic of a given network.
\item $\Delta_1, \Delta_2$ ???
\end{itemize}
\end{slide}

\begin{slide}{My Conclusion}
\begin{itemize}
\item Not describe why this behavior appear (because of ethernet properties?)
\item Not describe when this behavior appear ($N > ?$)
\item Not describe how to determine the model parameter
\item This behavior was not discoverd by author
\item The most important parameter of linear model --- sloop, were determied by experimants.
\end{itemize}
\end{slide}

\end{document}