\documentstyle[11pt,amssymbols]{article}
\textwidth 6.5in
\textheight 8.750in
\oddsidemargin 0.1in
\topmargin -0.5in
\input{psfig}
\makeatletter
% These allow switching interline spacing; the change takes effect immediately:
\newcommand{\singlespacing}{\let\CS=\@currsize\renewcommand{\baselinestretch}{1}\tiny\CS}
\newcommand{\oneandahalfspacing}{\let\CS=\@currsize\renewcommand{\baselinestretch}{1.25}\tiny\CS}
\newcommand{\doublespacing}{\let\CS=\@currsize\renewcommand{\baselinestretch}{1.5}\tiny\CS}
\newcommand{\proverflow}{\Pr(\sum_{i=1}^{N}\,{X_{0}^{(i)}} > C)}
\newcommand{\momentfx}{E[\exp^{r\sum_{i=1}^{N}\,{X_{0}^{(i)}}}]\,\exp^{-rC}}
\newcommand{\halfplane}{rC-\;\sum_{j=1}^{J}{n_{j}\;\Lambda_{j}{(r)}}\; \geq \; \gamma}
\newcommand{\adregion}{\frac{\gamma}{r^{\ast}}+\sum_{j=1}^{J}{n_{j}^{\ast}\;\frac{\Lambda_{j}{(r^{\ast})}}{r^{\ast}}}}
% Start with double spacing:
% \oneandahalfspacing
\newtheorem{lemma}{Lemma}
\newtheorem{exmp}{Example}%[chapter]
\newtheorem{dfn}{Definition}%[chapter]
%\newtheorem{asm}{Assumption}%[chapter]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{\bf EE 290Q Topics in Communication Networks\\ Lecture Notes: 12}
\author{Tze-Yau William Ng Huang}
\date{February 22, 1996}
\begin{document}
\maketitle
\bibliographystyle{alpha}
\pagestyle{plain}
\pagenumbering{arabic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Lecture Outline}
\singlespacing
\begin{enumerate}
\item Multiplexing of Markov Streams
\item Renegotiated CBR
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\oneandahalfspacing
In the last lecture, we covered the material on the effective
bandwidth for multiplexed multiple time-scale traffic (see
\cite{Tse}). In this lecture summary, we examine the relationship
between such effective bandwidth and the statistical multiplexing
gain (SMG). We then define the Renegotiated Constant Bit Rate Service
(RCBR see \cite{Gross}). Following this definition, we proceed
to show that the RCBR is able to extract most of SMG.
\section{Multiplexing of Markov Streams}
From last lecture, we learned that an arrival process $X_t$
from a single stream can be modulated by a Markov Chain with
subchains \(S_1, \ldots ,S_K\)to capture the multiple time-scale.
Given $\overline{\mu}_k$ and ${\pi}_k$, the average rate and the
probability of being in the subchain $S_k$ respectively, we can find
${\Lambda}_k(r)$, the log spectral radius in $S_k$.
\subsection*{Effective Bandwidth for Multiple Time-Scale Streams}
The notion of effective bandwidth is given by
\[e({\delta}) \equiv \frac{{\Lambda}(\delta)}{\delta} \]
where
\[{\Lambda}(\delta)=\max_{k \in [1,K]} {\Lambda}_k(\delta)\]
\subsection*{Multiplexing of Large Number of Streams}
We will now look at the situation where many such sources are multiplexed.
\begin{figure}[hbt]
\centerline{\ \psfig{figure=fig1.eps}}
\caption{\em N traffic sources to the node with buffer size $N B$
and link capacity $N c$}
\label{f:graph1}
\end{figure}
Let $W_t$ be the queue length at time t. We are interested in the
probability $P(W_0>NB)$. Last time we found $P(W_0 > 0)$ for a bufferless
system. We will see that in this case, too, the probability of overflow
is of the same nature.
Define
\[L(r)=\log \sum_{k=1}^{N} {\pi}_k e^{r \overline{\mu}_k}\]
The {\em Legendre} transform is
\[L^*(\mu) = \sup_{r > 0} (\mu r - L(r)) \]
\underline{Result}:
For large $N$, large $B$,
such that $P_{ij}(\alpha)B \approx 0$ for $(i,j) \in R$, we have
\[ P(W_0 > NB) \simeq \exp(- N L^* (c)) \]
This is essentially the same as a bufferless system where we replace
the traffic stream with the slow time-scale component. Formally,
\begin{equation}
\lim_{\begin{array}{c} N \rightarrow \infty , B \rightarrow \infty , \\
\rho_{ij}(\alpha) B \rightarrow 0 , (i,j) \in R \end{array}}
\frac{1}{N} \log P(W_0 > NB) = - L^*(c)
\label{eq:lgPrOvf}
\end{equation}
\subsection*{Statistical Multiplexing Gain}
Equation~\ref{eq:lgPrOvf} can be thought as a separation of time-scale
result, i.e. the accommodation of the fast time-scales is made by the
buffer, whereas the multiplexing of the independent sources is accommodated
by a bufferless system.
\begin{figure}[hbt]
\centerline{\ \psfig{figure=fig2.eps}}
\caption{\em large buffer asymptotics}
\label{f:graph2}
\end{figure}
See Figure~\ref{f:graph2} for illustration.
Note: $\hat{\mu} = \max_{k} \overline{\mu}_k$, $\overline{\mu} =
\sum_{k = 1}^{N} {\pi}_k \overline{\mu}_k$
We can identify two components in the gain:
\begin{itemize}
\item smoothing by buffers: effective if the correlation
time-scale is shorter than the buffer/delay time-scale.
\item averaging among sources: effective for slow time-scale
dynamics of the sources.
\end{itemize}
\section{Renegotiated CBR}
We now want to apply the Multiple Time-Scale Markov Models in
designing a service. The issue at hand is how we can provide
robust performance and \underline{yet} be able to take advantage
of the multiplexing gain.
\subsection*{Problems with the Leaky-Bucket Approach}
\begin{figure}[hbt]
\centerline{\ \psfig{figure=fig3.eps}}
\caption{\em leaky bucket $(\sigma , \rho )$}
\label{f:graph3}
\end{figure}
We first examine services with leaky-bucket contract. Under this
type of services, source characterization is done via $(\sigma,\rho)$
parameters. The problems with this approach are:
\begin{itemize}
\item Due to delay constraints, $\sigma$ and the size of the
buffer is limited.
\item We must have $\rho \simeq \hat{\mu} = \max_{k} \overline
{\mu}_{k}$ since otherwise there would be a big probability of
overflow at the buffer in the leaky-bucket.
\end{itemize}
This is a static approach. A solution might be to renegotiate
parameters during the session (e.g. as the Markov chain jumps from
one subchain to another). Next, let us discuss the Renegotiated
Constant Bit Rate (RCBR) service---the simpliest service when there
is renegotiation.
\subsection*{RCBR and its Advantages}
The architecture of the RCBR system is:
\begin{figure}[hbt]
\centerline{\ \psfig{figure=fig4.eps}}
\caption{\em RCBR: smoothing buffer = $B$, network buffer = $0$}
\label{f:graph4}
\end{figure}
In RCBR, users renegotiate CBR rates to accommodate \underline{slow}
time-scale variation. Now, compare this approach with the one where there
is buffering inside the network and allows users to send traffic at will.
\begin{itemize}
\item Disadvantages of RCBR:
\begin{enumerate}
\item sacrifice fast time-scale multiplexing gain
\item increase signaling load
\end{enumerate}
\item Advantages of RCBR:
\begin{enumerate}
\item controlled degradation when failure occurs
(Failure in RCBR is due to unsuccessful renegotiation.)
\item some amount of protection against misbehaving
users
\item simplicity in implementation
\end{enumerate}
\end{itemize}
\subsection*{Applications with Compressed Video Traffic}
We now look at two classes of RCBR application with compressed
video traffic:
{\bf 1). Off-line Application (Stored Video)}:
First, we need algorithms to find good renegotiation
schedules. A schedule is a piecewise constant function
$s(t), t \in [0,T]$. To evaluate a renegotiating schedule,
we need a metric. Given a schedule $s$, we can compute
\[ u(s) = \int_{0}^{T} s(t) dt \]
and
\[ v(s) = \mbox{total \# of renegotiations} \]
Hence, there is a trade-off between $u(s)$ and $v(s)$. We can
formulate this problem on a {\em Lagrangian} optimization problem:
\[ \min_{s} (u(s) + \lambda v(s)) \]
subject to buffer constraint. $\lambda$ is the relative price that
the network charges on renegotiations vs. bandwidth. It turns
out that the problem can be formulated as a dynamic programming
problem and can be solved using a Viterbi-like algorithm.
{\bf 2). On-line Application}: We essentially monitor the buffer
state at the edge of the network and use this information to
renegotiate.
\section{Concluding Remarks}
At the end of the lecture, a few view graphs on experiment
results were shown. They demonstrate that the RCBR are able to
extract most of the statistical multiplexing gain. Those
figures can be found in the reader~\cite{Gross}.
\begin{thebibliography}{2}
\bibitem{Tse}
D. Tse, R. Gallager and J. Tsitsiklis,
``Statistical Multiplexing of Multiple Time-Scale Markov Streams''
{\em IEEE JSAC}, Vol. 13, No. 6, pp. 1028-38, Aug. 1995.
\bibitem{Gross}
M. Grossglauser, S. Keshav, D. Tse,
``RCBR: A Simple and Efficient Service for Multiple Time-Scale Traffic''
{\em Proc. ACM SIGCOMM 95}.
\end{thebibliography}
\end{document}