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\title{\bf EE 290Q Topics in Communication Networks\\ Lecture 14: Measurement-Based Admission Control}
\author{Neil Bernstein}
\date{February 29, 1996}
\maketitle
\subsection*{Lecture Outline}
\singlespacing
\begin{itemize}
\item Model
\item Call Admission Schemes
\begin{itemize}
\item Certainty Equivalence Approach
\item Minimax Approach
\item Decision-Theoretic Approach
\end{itemize}
\item Performance
\item Outstanding Issues
\end{itemize}
\oneandahalfspacing
\section{Introduction}
In the last lecture, we completed our examination of admission
control schemes for systems with known statistics. We then turned
our attention to measurement-based schemes, useful when the
statistics of sources are not known but are measureable.
In this lecture we will review three such schemes, examine the
performance of one of them, and discuss some outstanding
issues. A fluid model, whose desciption was begun in the
previous lecture, is used. The majority of today's results
are found in \cite{gibbens}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Model}
Consider a bufferless network resource of capacity $C$. Incoming calls arrive
with Poisson distribution at rate $\nu$, with lengths exponentially
distributed with mean $1$.
Let $X_{i}{(t)}$ represent the amount of fluid being
emitted by the source of call $i$ at time $t$. These sources
are on with probability $p$ and are off otherwise. When on,
they emit fluid at rate 1, i.e. \[\Pr(X_{i}{(t)}\,=\,1)\;=\;p\] Thus,
$p$ can be considered the long-term average rate for each source.
Note that $p$ is unknown at the time of admission decision. Also, recall from the
previous lecture that $X_{i}{(t)}$ and $X_{i}{(s)}$ are independent for $t \neq s$.
Let \[S_{n}{(t)}\,=\,\sum_{i=1}^{n}{X_{i}{(t)}}\]
where $n$ is the number of calls and is a random variable. Thus,
$S_{n}{(t)}$ represents the load of the system at time $t$. We will focus
on call admission schemes that are based solely on the number of calls in the
system and the current load
at the time of decision. If that load exceeds a precalculated threshhold, $s(n)$, new
calls are rejected. Otherwise, they are admitted. The vector \({\bf s}=(s(n),n=0,1,\ldots)\)
defines the call admission scheme.
\subsection{Characteristics}
One can show that $n$, the number of calls in the system, is a Markov chain
whose state is incremented when a new call enters the system and is decremented
when one leaves. Further, if Poisson arrivals and exponential holding times are
assumed, n is a birth-death process with transition rates
\[q(n,n-1)=n;\>q(n,n+1)=\nu a(n)\] where $a(n)$ is the acceptance probability
when n calls are in the system:
\[a(n)=\sum_{i=0}^{s(n)-1}{\left(\begin{array}{c}n\\i\end{array}\right)p^{i}
(1-p)^{n-i}}\]
Looking at the balance equations for the birth-death process, we can find
the stationary distribution of n:
\[\pi (n)\propto \frac{\nu ^n}{n!}\prod_{r=0}^{n-1}a(r)\]
Because of the dependence of $\pi$ on $p$ and $\lambda =\nu p$, the
offered load, we will write $\pi (n)=\pi (n;\lambda,p)$.
Some further definitions:
\begin{itemize}
\item The {\em overall cell loss rate}
\[M(p,\lambda)=\sum_{n}\pi(n;\lambda,p)M(n;p)\]
This is the long term average cell loss rate. (Recall the definition
of $M(n;p)$ from Lecture 13 as the expected cell loss rate with
$n$ calls in the system.)
\item The {\em cell loss ratio} is thus
\[L(p,\lambda)=\frac{M(p,\lambda)}{\sum_{n}\pi(n;\lambda,p)np}\]
\item Making use of the Poisson arrival property and averaging
over all possible states, we define the {\em call blocking probability}:
\[E(p,\lambda)=\sum_{n}\pi(n;\lambda,p)(1-a(n))\]
\item The {\em utilization} is the long term aggregate traffic
rate delivered per unit time:
\[U(p, \lambda)=\lambda(1-E(p,\lambda))-M(p,\lambda)\]
\end{itemize}
We wish to design call admission schemes that meet a target loss rate ($L$) while
minimizing blocking ($E$). Since we have a homogenous situation, this is
equivalent to maximizing the number of calls in the system (thus
maximizing revenue).
\section{Examples of Call Admission Schemes}
\subsection{Certainty Equivalent Approach}
This call admission scheme simply replaces $p$ with an unbiased estimator
\[\hat{p}=\frac{1}{n}S_{n}{(t)}\]
For a given target loss rate, $P_{qos}$, we can accept a new call if
\[L(n+1;\hat{p}) \leq P_{qos}\]
where \(L(n,p)=\frac{M(n;p)}{np}\). That is, the call can be accepted
if the resulting loss rate will still be below the target loss rate.
For binomial sources (as in our model), this can be computed exactly. In
this case,
\[s(n) = \max \left\{ s:L(n+1; \frac{s}{n}) \leq P_{qos} \right\}\]
This scheme is problematic in that it does not take into account any
estimation error on $p$. The estimate $\hat{p}$ will be more accurate
for larger n; fortunately, for small $n$ (relative to system capacity)
the system is operating far
below threshhold, making errors in $\hat{p}$ more tolerable.
\subsection{Minimax Approach}
Under this scheme, we choose $s(n)$ such that
\[\max_{p\in [0,1]}P[S_{n}(t)>C\; and\; S_{n}(0)~~C|p] \leq \exp^{-n\Lambda^{\ast}{(\frac{C}{n};p)}}\]
where $\Lambda^{\ast}$ is the rate function for the binomial random variable.
This is also computable explicitly, with
\[\Lambda^{\ast}{(\mu ;p)}\,=\,\mu\log{\frac{\mu}{p}}+(1-\mu)\log{\frac{1-\mu}{1-p}}\]
Similarly, for the other tail,
\[P[S_{n}(0)>s] \leq \exp^{-n\Lambda^{\ast}{(\frac{s}{n};p)}}\]
Substituting, we get
\[\max_{p} \exp(-n\Lambda^{\ast}(\frac{C}{n};p)+\Lambda^{\ast}(\frac{s}{n};p))\leq P_{qos}\]
The maximum $p$ is achieved at $p^{\ast}=\frac{C+s}{2n}$, the average of the unbiased
estimator $\frac{s}{n}$ and the value at which overflow occurs, $\frac{C}{n}$. So,
for a chosen $s(n)$, cell loss is the result of a combination of two
rare events, namely low $S_{n}(0)$ and overflow (i.e. $S_{n}(t)$ high). The
maximum value $p^{\ast}$ will be between the two.
\subsection{Decision-Theoretic Approach}
This approach relies on some real-world assumptions:
\begin{itemize}
\item Luck won't always be maximally bad
\item We have some prior knowledge of the distributions of $p$ and $\lambda$, and
these will usually continue to hold
\item Costs can be assigned so that we can study the tradeoff between loss and utilization
\end{itemize}
Given a prior distribution $f(p, \lambda)$, choose $s(n)$ using the rule:
\[\max_{s(.)}\int\int[U(p,\lambda )-yM(p,\lambda)]f(p,\lambda )dp d\lambda\]
This scheme directly illustrates the tradeoff between utilization and loss.
The system is operated at a given value of $y$, and $s(n)$ is chosen so as to
maximize the expression. In this case, $y$ is a Lagrangian parameter controlling
the tradeoff, that is, the relative weights of utilization and loss.
\section{Performance}
The authors of \cite{gibbens} found promising results for the decision-theoretic
approach. When compared with an optimal scheme -- one which has pefect
prior knowledge of $p$ and $\lambda$ -- all results were found to be
above 97\% of optimal. For a given $p$, results were better for a small
$\lambda$. For high values of $p$, results were very near 100\% of optimal.
The authors also show that:
\begin{itemize}
\item For heterogeneous on-off sources, the scheme still does reasonably well,
but behaves more conservatively. That is, utilization is lower due to a less
agressive admission policy.
\item Smaller values of $y$ move the loss curve up, resulting in larger loss
probability.
\item In comparison to the decision-theoretic approach, the first two schemes
presented perform more poorly.
\end{itemize}
The scheme as presented depends on the stationarity of the call stream, but since it is memoryless,
it responds well to non-stationarity. The authors altered the value of $p$ with time in order
to investigate how quickly the system responds. They found that response occurred with a
time constant of one to two call durations.
\section{Outstanding Issues}
In setting their model, the authors make a very strong separation of timescales
assumption. They assume that the states of the network at successive call
arrivals are independent. If this independence assumption is not valid, it is
possible that admission mistakes might occur in bursts due to the correlation
of successive states. The impact of this would have on performance is unknown.
Also unknown and left to be explored is the performance of the proposed call admission
schemes with more complex sources, such as {\em Renegotiated Constant Bitrate} sources.
Another open question is whether using historical state information could
improve performance.
\section{Conclusions}
In this lecture we have examined simple threshhold-based call admission schemes
as explored in \cite{gibbens}. The authors of that paper have shown that a
decision-theoretic approach performs nearly as well as an optimal scheme for a
simple model. Further, they claim that the scheme responds well to
nonstationary sources and performs well with heterogeneous on-off
sources.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{1}
\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}
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