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\title{\bf EE 290Q Topics in Communication Networks\\ Lecture Notes: 13}
\author{Yuan-Chi Chang}
\date{February 27, 1996}
\maketitle
\subsection*{Lecture Outline}
\singlespacing
\begin{enumerate}
\item Effective bandwidth for bufferless multiplexing
\item Measurement-based admission control
\end{enumerate}
\oneandahalfspacing
\section{Introduction}
In the last lecture, we covered the material on statistical multiplexing of
multiple time-scale traffic (see \cite{tse_1})
as well as the {\bf Renegotiated Constant Bit Rate
(RCBR)} scheme (see \cite{tse_2}). Today, we are going to
look at the multiplexing gain obtained by sending a large
number of slow time-scale traffic sources into a bufferless node.
In the second half of the lecture, we will begin another topic which is
{\bf measurement-based admission control}.
For further references, one may find the material covered today in the
papers by Tse \cite{tse_2}\ and Gibbens \cite{gibbens}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bufferless Multiplexing}
\begin{figure}[hbt]
\centerline{\ \psfig{figure=node.eps}}
\caption{\em N traffic sources to the bufferless node with link capacity $C$}
\label{f:graph1}
\end{figure}
Consider a bufferless network node with link capacity $C$. At time $t$, there
are total $N$ stationary traffic sources arriving this node.
See Figure~\ref{f:graph1}
for an illustration. Without loss of generality,
set $t = 0$. The quantity we are
interested in is the probability that the total arrival traffic
exceeds $C$, i.e.\ \(\proverflow\)
We can bound this probability by the {\em Chernoff} approximation.
\[\proverflow\; \leq \; \momentfx\]
Further, we assume that these traffic streams are independent of each other.
The traffic streams belong to $J$ traffic classes. Traffic sources in each
class are identical. In class $j$, source $i$, denote its log moment
generating function by $\Lambda_{j}{(r)}$\ , i.e.
\[\Lambda_{j}{(r)} = \log\,E[\exp^{r\,X_{0}^{(i)}}]\]
Assume there are $n_{j}$ sources in class $j$, and \(\sum_{j=1}^{J}{n_{j}} = N \).
The {\em Chernoff} approximation above can be rewritten as
\[\proverflow\; \leq \; \exp^{-rC+\;\sum_{j=1}^{J}{n_{j}\;\Lambda_{j}{(r)}}}\]
By choosing the tightest $r$, we have
\[\proverflow\; \leq \; \exp^{-{\max_{r}{[rC-\;\sum_{j=1}^{J}{n_{j}\;\Lambda_{j}{(r)}}]}}}\]
Suppose the {\em QoS}\ requires the loss probability to be less than
$\exp^{-\gamma}$, then \(\proverflow\; \leq \; \exp^{-\gamma}\). This gives us:
\[\max_{r}{[rC-\;\sum_{j=1}^{J}{n_{j}\;\Lambda_{j}{(r)}}]}\; \geq \; \gamma\]
Define the admissable region \(A = \{(n_{1},\ldots,n_{j})\;|\;\max_{r}
{[rC-\;\sum_{j=1}^{J}{n_{j}\;\Lambda_{j}{(r)}}]}\, \geq \, \gamma \,\}\).
So for each $r$, there is a halfplane constraint given like:
\[\halfplane\]
In other words,
\[A = \bigcup_{r}{\{(n_{1},\ldots,n_{j})\;|\; \halfplane\}}\]
The admissable region can be thought of as a union of halfplanes.
The complement of $A$, $A^{c}$, is a convex set. See Figure~\ref{f:graph2}
for an illustration. In this figure, only two classes are used and
a point \((n_{1}^{\ast},n_{2}^{\ast})\) at the boundary of $A$ is selected.
The region bounded by the halfplane is shown with shades.
\begin{figure}[hbt]
\centerline{\ \psfig{figure=region.eps}}
\caption{\em Using a half-plane to approximate $A$}
\label{f:graph2}
\end{figure}
Find \((n_{1}^{\ast},\ldots,n_{j}^{\ast})\) at the boundary of $A$. One
can show that, at this point
\[\max_{r}{[rC-\;\sum_{j=1}^{J}{n_{j}^{\ast}\;\Lambda_{j}{(r)}}]}\,=\,\gamma\]
Let $r^{\ast}$ be this $r$, then
\[r^{\ast}C-\;\sum_{j=1}^{J}{n_{j}^{\ast}\;\Lambda_{j}{(r^{\ast})}}\,=\,\gamma\]
Thus,
\[\adregion\;=\;C\]
Therefore, $A$ is now lower-bounded by
\[\{(n_1,\ldots,n_j)| \adregion\;\leq\;C\}\]
It is a conservative approximation of the admissable region in the sense that
it uses one half-plane to linearly approximate the region.
>From the above derivations, a natural definition of the effective bandwidth of a source
in class $j$ given the above derivation would be
$\frac{\Lambda_{j}{(r^{\ast})}}{r^{\ast}}$.
Note that even for the bufferless case, the effective bandwidth of stochastic
traffic sources can be defined. Although it's not a tight bound, it gives a
reasonable well estimation when the number of traffic sources is large.
\section{Measurement-based Admission Control}
If the statistics of traffic sources are not known in advance, the analysis
approaches we did in the past several lectures will not apply. Therefore,
measurement based admission control schemes are proposed.
It's a relatively young field and there are still a lot of on-going research
work. We will focus on one approach described in \cite{gibbens}.
And still assume the bufferless setup.
The main goal of the measurement-based approach is to make good admission
control decisions rather than trying to estimate some stochastic properties
of the source. Several issues relating to this approach are:
\begin{enumerate}
\item estimation errors---how would the estimation errors impact its
effectiveness?
\item prior knowledge---how to make use of prior knowledge about the
traffic sources to improve on-line decisions?
\item stationarity of statistics---in general, all approaches have to
assume certain levels of stationarity. But at what time scale and which
statistics have to be stationary?
\end{enumerate}
We focus on models that incorporate statistics of sources as well as those of call
activities. We assume a simple model in which calls are homogeneous and can
be described as on-off sources. When it is in the on state, it emits traffic
at rate $1$ and when it is in the off state, it emits nothing. The output link
of this bufferless switch has a rate $C$. Calls arrive at a {\em Poisson} rate $\nu$.
The length of calls are exponentially distributed with a mean value of $1$.
For a call $i$ in the system, define $X_{i}{(t)}$ as the rate at which the
source is emitting fluid. \[\Pr(X_{i}{(t)}\,=\,1)\;=\;p\] where $p$ is
unknown at the time of the decision.
\begin{quote}
Sidenote: if $p$ is known in advance, then the effective bandwidth of this simple
on-off source can be calculated, which makes the admission control problem trivial.
\end{quote}
We can further assume that $X_{i}{(t)}$ and $X_{i}{(s)}$ are independent of each other
when $t \not= s$. This is a seperation of time scale assumption. It essentially says that
the burst scale process is much shorter than the length of a call; since time is normalized by the average length of a call.
\begin{quote}
Sidenote: for traffic sources having long time-scale correlations like compressed
video, the above assumption may not be valid.
\end{quote}
Define
\[S_{n}{(t)}\,=\,\sum_{i=1}^{n}{X_{i}{(t)}}\]
where $n$ is the number of calls and is a random variable.
Given $n$ calls in the system, define
\[L(n;p)\,=\,\frac{1}{np}\,M(n;p) \;where\; M(n;p)\,=\,E[(S_{n}-C)^{+}]\]
Here $L$ represents the proportion of cells (or load) lost and $M$ measures the
mean number of cells lost given $n$ traffic sources in the system.
Note that these quantities can be computed explicitly. Use {\em Chernoff's} approximation,
\[L(n;p)\;\cong\;\Pr(S_{n}{(t)} > C)\;\cong\;\exp^{-n\Lambda^{\ast}{(\frac{C}{n})}}\]
One can show that
\[\Lambda^{\ast}{(\mu)}\,=\,\mu\log{\frac{\mu}{p}}+(1-\mu)\log{\frac{1-\mu}{1-p}}\]
\subsection*{Call admission schemes}
The scheme checks the load of the system. The admission decision of the incoming calls is
based on the current load of the system i.e. \ $S_{n}{(t)}$ at time $t$. The control
rule is described by $s(n)$, which is a set of predetermined decision thresholds. When a
new call is requested, it is accepted when $S_{n}{(t)} < s(n)$. Otherwise, it is rejected.
\section{Concluding Remarks}
Today we derived the effective bandwidth for bufferless statistical multiplexing. It
characterizes the multiplexing gain one can get from mixing a large number of slow
time-scale traffic sources. We also started introducing a new topic which is
measurement-based admission control. Next time we are going to continue the topic.
\begin{thebibliography}{3}
\bibitem{tse_1} 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{tse_2} M. Grossglauser, S. Keshav, D. Tse, ``RCBR: A simple and efficient service for multiple time-scale traffic,'', {\em Proc. ACM SIGCOMM 95}.
\bibitem{gibbens} R. Gibbens, F. Kelly, P. Key, ``A decision-theoretic approach to call admission control in ATM networks,'', {\em IEEE JSAC}, vol.13, no.6, pp.1101-14, Aug. 1995.
\end{thebibliography}
\end{document}