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\title{\bf EE 290Q Topics in Communication Networks\\ Lecture Notes: 9}
\author{Emile Sahouria}
\date{February 13, 1996}
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\maketitle
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\section{Introduction}
Statistical multiplexing is a mechanism for more efficiently utilizing
the bandwidth of packet switched networks. A key element of such a scheme
is determining the probability of overflow in the buffers at the nodes of
a network that uses this discipline. Large deviations theory provides
the analysis of this hopefully rare event. This naturally leads to the
definition of the effective bandwidth of an information source in a
statistically multiplexed network. The following is elaborated on
in Shwartz \cite{Shwartz} and in Tse \cite{Tse}.
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\section{A Review of a Large Deviations Result}
Recall that if \( \{X_i\} \) is a sequence of independent and identically
distributed random variables with expected value $\bar{\mu}$, then for
\( \mu > \bar{\mu} \),
\[ P(\sum_{i=1}^{n} X_{i} > n\mu) \approx \exp(-n\Lambda^{\ast}(\mu)), \]
where
\[ \Lambda^*(\mu) = \max_{\mu>0}( \mu r - \Lambda(r)),\]
and \(\Lambda(r)=\log E[e^{rX_{i}}] \) is the log moment generating function
of the random variables.
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\section {Buffer Overflow Analysis}
\begin{figure}[ht]
\centerline{\ \psfig{figure=qu.ps,width=5in}}
\caption{\em Queue with single arrival process.}
\label{f:q}
\end{figure}
Consider a discrete queue with a single i.i.d. arrival process $\{X_t\}$,
a very large buffer,
and service rate c, as shown in figure \ref{f:q}. The random variables
$X_t$ are the amount of traffic arriving in the interval begginning at time
$t$. It is assumed that the queue has been operating for a long time, and
is in steady state by time zero. The amount of
work remaining in the buffer at time t is $W_t$; for simplicity of notation,
we will consider what happens at time zero. Further, we assume the stability
requirement $ E[X_t] < c$.
The above result predicts that that for very large buffer sizes, the probability
of overflow, i.e., that $W_0 > B$ for some B, is
\[ P(W_0 > B) \approx e^{-r^{*}B}. \]
\begin{figure}[ht]
\centerline{\ \psfig{figure=rstar.ps,width=5in}}
\caption{\em Geometric interpretation of $r^*$. The convex
curve is $\Lambda(r)$.}
\label{f:rstar}
\end{figure}
Figure \ref{f:rstar} illustrates the meaning of $r^*$. It is the single positive
root of the log moment generating function of $X_i - c$. Note that this is
different from the log generating function in the previous section.
An upper bound and a lower bound for the overflow probability will be obtained;
in the limit they will converge to the desired expression.
The derivation of the upper limit relies on two methods of analysis:
decomposition of the problem into a sum of disjoint events
and the large deviations probability above. First, the event
of overflow is decomposed into a number of disjoint events.
Specifically, let
\[S_n = \sum_{t=-n}^{-1}(X_t - c) \]
be the work remaining in the queue starting from time $t$, and define $E_n$
as the event the system was last empty at time $-n$. Then
\begin{equation} P(W_0 > B) = \sum_{n = 1}^{\infty} P((W_0 > B) \cap E_n) \leq
\sum_{n = 1}^{\infty} P(S_n > B). \label{eq:sum}
\end{equation}
The inequality follows from the fact that $P((W_0 > B) \cap E_n)
\Rightarrow P(S_n > B)$.
But now large deviations theory tells us that
\[P(S_n > B) \approx \exp[-n\Lambda^{*}(\frac{B}{n})]. \]
This can also be seen independently just from the Chernoff bound:
\[P(S_n > B) \leq E[e^{rS_n}]e^{-rB}
=\exp[-rB + n\Lambda(r)]
=\exp[-B(r-\frac{n}{B}\Lambda(r))].\]
When evaluated at the $r$ that minimizes the exponent,
this produces
\[P(S_n > B) \leq \exp[-BT\Lambda^*(\frac{1}{T})] \mbox{ where $T = \frac{n}{B}$ }, \]
or
\[P(S_n > B) \leq e^{ -n \Lambda^{*}(\frac{B}{n}) }.\]
Now the sum (\ref{eq:sum}) must be bounded. A logical candidate uses the
approximation $e^{-aB} + e^{-bB} \approx e^{-\min(a,b)}$. However, the
infinite sum complicates matters; in particular, the behavior of the
exponent for large n is uncertain. This behavior can be more carefully
\begin{figure}[ht]
\centerline{\ \psfig{figure=expbehv.ps,width=5in}}
\caption{\em Determining the minimum exponent.}
\label{f:expbhv}
\end{figure}
studied with the help of figure \ref{f:expbhv}. The r-intercept of the
straight line is $T\Lambda^*(\frac{1}{T})$. As n decreases, the slope
of this line increases, eventually intercepting the r-axis at r*. This,
then, gives the minimum value of the exponent. The figure also shows that
for large $T$(large $n$), $T\Lambda^*(\frac{1}{T}) \approx -T\min_{r}
(\Lambda(r))$. Thus, pick a $T_0 >0$, and a $\beta >0$ such that
\( T\Lambda^*(\frac{1}{T}) > r^* + \beta T\) for $ T>T_0 $.
This allows the decomposition of (\ref{eq:sum}):
\begin{eqnarray}
P(W_0 > B) & \leq &\sum_{n = 1}^{\infty} P(S_n > B)
\leq \sum_{T>0}\exp[-BT\Lambda^*(\frac{1}{T})] \nonumber \\
& = & \sum_{TT_0} \exp[-BT\Lambda^*(\frac{1}{T})]. \nonumber
\end{eqnarray}
The first term on the right hand side is a finite sum, and can thus be
approximated by $e^{-Br^*}$. The second term is bounded using the $T_0$
and $\beta$ described above to give
\[ \sum_{T>T_0} \exp[-BT\Lambda^*(\frac{1}{T})]
\leq \sum_{T>T_0} e^{-B(r^* + \beta T)}
= e^{-Br^*} \sum_{T>T_0} e^{-B\beta T}
\leq Ce^{-r^*B} \]
by virtue of the fact that the second to last term is a convergent
geometric series. As B becomes large, the upper bound is established:
\[ \lim_{B \inf} \frac{1}{B}\log P(W_0 > B) \leq -r^*B. \]
The lower bound is a particular event in the sum (\ref{eq:sum}):
\[ P(W_0 > B) \geq P(S_{T^*B} > B) \]
where $T^*$ achieves $ r^* = \min_{T}(T\Lambda^*(\frac{1}{T}) )$.
As B grows, figure \ref{f:expbhv} shows that the required exponent in
the large deviations estimate of the probability of this event
is $r^*$.
So $P(W_0 > B) \geq e^{-r^*B}$, and we have shown
\[ P(W_0 > B) \approx e^{-r^*B}. \]
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\section {Effective Bandwidths}
The analysis for a single traffic stream in the previous section
can be easily extended to multiple traffic streams.
\begin{figure}[ht]
\centerline{\ \psfig{figure=multq.ps,width=5in}}
\caption{\em A multiple input stream queue.}
\label{f:multq}
\end{figure}
Figure \ref{f:multq} shows the details of a node in
such a network:
\begin{itemize}
\item The N incoming streams are each independent of each
other.
\item Each stream, $\{X_t^{(i)}\}$ consists of i.i.d random
variables specifying the number of data packets arriving
at time $t$. The streams are independent of each other,
but do not necessarily have the same statistics.
\item The streams are combined into a single stream
\[ Y_t = \sum_{k = 1}^{N}X_t^{(k)} \]
prior to entering the queue.
\item The queue is served at a constant maximum rate c, and is
work conserving.
\end {itemize}
Applying the result found above, one obtains $P(W_0 > B) \approx e^{-r^*B}$,
where $r^*$ is the unique positive root of
\[ \Lambda(r) = \log E[e^{-r(Y_0 - c)}]. \]
In terms of the indvidual log moment generating functions
\[ \Lambda_k(r) = \log E[e^{-rX_0^{(k)}}], \]
the aggregate log moment generating function is
\[ \Lambda(r) = \log E[\exp(-r(\sum_{k=1}^{N}X_0 - c))]
= \sum_{k = 1}^{N}\Lambda_k(r) - cr. \]
Finally it is apparent that one can specify a limited quality
of service for such a statistically multiplexed node. The QOS is
stated in terms of worst loss for any traffic stream; this can be
translated into an exponent, $\delta$, in the large deviations loss
probability: $P(W_0 > B) \approx e^{-\delta B}$; a smaller
\begin{figure}[ht]
\centerline{\ \psfig{figure=qos.ps,width=5in}}
\caption{\em Demonstrating the QOS parameter.}
\label{f:qos}
\end{figure}
$\delta$ produces a larger probability of loss. This means $\delta
\leq r^*$ in figure \ref{f:qos}. Since
\[ \Lambda(r^*) = \sum_{k}\Lambda_k(r^*) - r^*c = 0, \]
an equivalent condition is
\[ \Lambda(\delta) = \sum_{k}\Lambda_k(\delta) - c\delta \leq 0 \]
or
\[ \sum_{k}\frac{\Lambda_k(\delta)}{\delta} \leq c.\]
One then defines the the {\em effective bandwidth} of source $k$ as
\mbox{ $ \frac{\Lambda_k(\delta)}{\delta} $ }. The effective
bandwidth is fixed by the statistics of a stream; the requested
QOS can be guaranteed if the sum of the effective bandwidths for all
sessions is less that the outgoing bandwidth of the node.
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\section{Summary}
Large deviations theory provides a mechanism for analyzing
loss probabilities in statistically multiplexed networks, and
introduces a mechanism for providing quality of service using
effective bandwidths. However, this analysis only considers
i.i.d. traffic streams, and treats all streams alike. A
more refined analyisis is thus in order, and will be provided
in the sequel.
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\begin{thebibliography}{2}
\bibitem{Shwartz}
Shwartz, Adam and Weiss, Alan (1995).
Large Deviations for Performace Analysis: Queues, communications, and
computing. New York: Chapman and Hall, 1995, ch. 1.
\bibitem{Tse}
Tse, N. C., Gallager, Robert G. and Tsitsiklis, Hohn N. (1995).
Statistical Multiplexing of Multiple Time-Scale Markov Streams.
{\em IEEE Journal on Selected Areas in Communications}, August 1995,
vol. 13, no. 6, pp. 1028-1038.
\end{thebibliography}
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\end{document}