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\title{\bf EE 290Q Topics in Communication Networks\\ Lecture Notes: 17}
\author{Mark Soper}
\date{March 12, 1996}
\begin{document}
\maketitle
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{\footnotesize
\subsection*{Introduction to Today's Lecture}
In today's lecture we continue our study of call blocking in
large circuit switched networks, and begin to discuss how this
analysis can be applied to the development of optimal routing
schemes for such networks. So the lecture is divided into
two sections:
\begin{itemize}
\item[1] We finish the discussion of approximating call blocking
probabilities using the Erlang Fixed Point Approximation and the
Fluid Flow Approximation (please refer to the
preceeding two lectures for the entire development).
Last time, we analyzed the EFP to gain insight into the accuracy
of its approximation and when it is appropriate to use EFP. We
found that as the scale of the network becomes large (in other
words, as the link capacity and the offered traffic increase
simultaneously) the loss probabilities are {\em as if} each link
blocks independently, and we may use the Erlang Fixed Point
for an accurate approximation of actual call blocking
probabilities. In today's lecture we introduce the Fluid Flow
Approximation, also valid for large scale networks, which allows
us to characterize the previously introduced, discrete-arrival,
stochastic network model with a continuous-flow, deterministic
model that is much easier to analyze. We show how Erlang's
formula for individual link blocking probabilities can be used
to improve the accuracy of the Fluid Flow Approximation, and
we compare the Erlang Fixed Point and Fluid Flow models as
network scale becomes large. This material is based on first
Kelly paper. \cite{Kelly_1}
\item[2] Now that we have an understanding of call blocking
probabilities in an arbitrary, large, circuit switched network,
we move on to discussion of how we might use this information
to devise optimal routing schemes for the network. We begin
by outlining a basic routing model, with a few assumptions about
the routing disciplines and the optimization criteria, and we
formulate a global optimization problem. The analysis of this
problem is begun today, to be continued in subsequent lectures. This
material is based on the second Kelly paper. \cite{Kelly_2}
\end{itemize}
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\section{Approximating the Call Blocking Probability }
\begin{figure}[hb]
\centerline{\ \psfig{figure=net.eps,height=1.75in}}
\caption{An Arbitrary Circuit Switched Network}
\end{figure}
\pagebreak
\subsection*{The Fluid Flow Approximation}
% I might want to add the exact stat. distrib. (1.2)
% and the optimization problem (2.1) as well as a diagram
% of an basic nodal topology (Fig. 1)
Recall from previous lectures that we derived the exact stationary
distribution for blocking probability in an arbitrary network, such
as the one illustrated in the figure above, but
evaluation of this distribution requires calculation of a complicated
normalizing constant. So we sought an easier way to approximate
the stationary distribution. We observed that for large scale
networks (capacity and offered traffic increase simultaneously),
the majority of the stationary distribution lies under the mode,
or most likely state, of the distribution. Hence we formulated
the following optimization problem, which is equivalent to finding
the mode of the stationary distribution:
\noindent{maximize} $\sum_{r}( x_{r}log \nu_{r} -
x_{r}log x_{r} + x_{r}$
\noindent{subject} to $x \ge 0 , Ax \le C$
\noindent{We} begin our discussion today by arguing that for a
large scale network, we may model the stochastic arrival and
existence of a discrete number of calls as a continuous,
deterministic fluid flowing in the network. Under this assumption
we present the Fluid Flow Approximation, an approximated solution
to the above optimization problem:
\noindent{There} exists a unique optimum:
\begin{displaymath}
\tilde{x}_{r} = \nu_{r}\prod_{j \in {r}} ( 1 - B_{j} )
\end{displaymath}
\noindent{where} $B_{1}, ... , B_{J}$ satisfy:
\begin{displaymath}
\sum_{r:j\in{r}} \nu_{r} \prod_{i \in {r}} (1 - B_{i}) \;\;\;
\left\{ \begin{array}{l}
= C_{j} \: if \, B_{j} > 0 \\
\le C_{j} \: if \, B_{j} = 0
\end{array} \right.
\end{displaymath}
\noindent{Where:}
\begin{itemize}
\item $\nu_{r}$ is the offered flow on route $r$
\item $B_{j}$ is the fraction of fluid which overflows on link $j$
\item $C_{j}$ is the capacity of link $j$
\item $\tilde{x}_{r}$ is the remaining flow actually carried in the
network for route $r$
\end{itemize}
\begin{figure}[h]
\centerline{\ \psfig{figure=pipe.eps,height=2in}}
\caption{Pipe Analogy for Fluid Flow Model}
\end{figure}
\noindent{As} done previously, we consider the limiting regime where
offered traffic and capacities are increased in line with one another.
\noindent{As} $N \to \infty$, then $\nu_{j}(N) = \nu_{j}N$, and $C_{j}(N) = C_{j}N$
\noindent{So} as $N$ tends to infinity:
\begin{displaymath}
1 - L_{r}(N) \to \prod_{j \in r} (1 - B_{j})
\end{displaymath}
\noindent{where} $L_{r}(N)$ is the blocking probability for route $r$.
\noindent{What} the above equation tells us is that to find the overall
probability of blocking across route $r$, simply find individual
link blocking probabilites and treat links as if they are
independent.
\subsection*{Relating the Fluid Flow Model to the Erlang Fixed Point
Approximation}
It can be easily shown that the Fluid Flow Approximation given
above can be equivalently represented by stating that $B_{1}, ... ,
B_{J} $ satisfy:
\begin{displaymath}
B_{j} = E_{f} \left( \sum_{r:j \in r} \nu_{r} \prod_{i \in r - \{j\}}
(1 - B_{i}) \;\;\; , \;\;\; C_{j} \right)
\end{displaymath}
\noindent{where} $E_{f}(\nu , C)$ is defined as:
\begin{displaymath}
E_{f}(\nu , C) = max(\frac{\nu-C}{\nu}, 0)
\end{displaymath}
\noindent{with} $\nu$ the offered flow on the link and $C$ the capacity of the link.
\noindent{Notice} that with this simplified link loss probability function:
\begin{itemize}
\item if $\nu < C$, loss on the link = $0$
\item if $\nu > C$, loss on the link = $\frac{\nu-C}{C}$
\end{itemize}
\noindent{What} we have done here is put the Fluid Flow solution in
a form similar to that of the Erlang Fixed Point solution. If
we desire more accuracy, we could find the individual link blocking
probabilities using Erlang's formula ( $E(\nu,C)$ ) in place of
$E_{f}(\nu,C)$ in evaluating $B_{1}, ... , B_{J}$. Recall that
$E(\nu,C)$ is given by:
\begin{displaymath}
E(\nu , C) = \frac{\nu^C}{C!}\left(\sum_{n=0}^{C} \frac{\nu^n}{n!}\right)^{-1}
\end{displaymath}
\noindent{Indeed} we can show that:
\begin{displaymath}
E(\nu N, CN) \to E_{f}(\nu, C) \; as \; N \to \infty
\end{displaymath}
\noindent{What} this tells us is that as the scale of the network
becomes large (as $ N \to \infty $), the individual link probabilities
given by the Erlang formula approach those of the deterministic
Fluid Flow model ( $E_{f}$ ). In other words, where
$E_{1}(N), ... , E_{J}(N)$ are the EFP link blocking probabilities
of the $N^{\underline{th}}$ scaled system, as $N /to /infty$, then:
\begin{displaymath}
\prod_{j \in r} \left( 1 - E_{j}(N) \right) \to \prod_{j \in r} ( 1 - B_{j} )
\end{displaymath}
\noindent{We} conclude our discussion of Blocking Probability
Approximation with an intuitive argument about the existence
and uniqueness of a fixed point solution. Although we don't make
this explicit, it is possible to
write a modified version of our previous optimization problem. This
modified optimization problem is stricly convex (hence has a unique
optimal solution). Also, we can show that there is a 1-1 correspondence
between the stationary points of the optimization problem and the fixed
points of the EFP equations. Therefore, since the optimization problem
is strictly convex, there is a unique stationary point, and this implies
that the EFP equations have a unique fixed point.
\section{Developing an Optimal Routing Scheme}
\noindent{At} this point we shift our focus from estimating call
blocking probabilities in an arbitrary circuit switched
network to using these estimates to develop optimal routing
schemes.
\subsection*{Basic Assumptions about Routing}
\noindent{We} start with the basics: What is routing? In an arbitrary
network, there may be many possible paths, or routes, between
two nodes A and B. For our purposes we will define routing
as the discipline through which we answer the question of
how to best split the overall A $\to$ B traffic among the
different routes. Here we will also assume Poisson arrivals
of calls at A for the A $\to$ B connection.
\noindent{Routing} disciplines can be categorized based on their criteria
for routing decisions:
\begin{itemize}
\item {\em fixed} - when a call arrives, it follows a route
that has already been pre-determined
\item {\em adaptive} - routing decisions are congestion
sensitive, based on periodic network
load measurements
\item {\em dynamic} - routing decisions are made to
instantaneously exploit un-used network capacity on a
per call basis
\end{itemize}
\noindent{Initially} we will consider a {\em fixed} routing scheme.
\noindent{We} wish for our routing scheme to be optimal, in other words
that our schemes {\em best} splits the traffic from A to B
among the different routes, and there are many ways in which
the word {\em best} can be interpreted, such as minimizing
average blocking probability, minimizing blocking probability
for a given subset of calls, maximizing network revenue, etc.
For our purposes we formulate the optimization problem in
terms of {\bf Revenue Maximization}.
\subsection*{The Problem Formulation: Revenue Maximization}
\noindent{We} fix:
\noindent{$\nu_{r}$} = offered traffic for route $r$ \\
$C_{j}$ = the capacity of the link $j$
\noindent{We} assume that a call on route r will, if accepted, bring
in a revenue $w_{r}$.
\noindent{The} optimization problem is to maximize this revenue,
which will be in terms of the blocking probabilites, and we
will use the EFP approx. to do this. As done previously,
we define $E_{1}, ... , E_{J}$ to be the EFP individual link
blocking probabilities for links $1, ..., J$. Earlier we
have shown that we can readily solve for these parameters.
\noindent{We} define $\lambda_{r}$ to be the thinned traffic rate
on route $r$, or the rate of accepted traffic, (Ie: traffic
which isn't blocked.)
\begin{displaymath}
\lambda_{r} = \nu_{r} \prod_{j \in r} ( 1 - E_{j})
\end{displaymath}
\noindent{We} define the objective funciton, which may be interpreted
as the total revenue per unit time, which is defined in
terms of $\lambda$ instead of $\nu$ (because we gain revenue
from accepted, not offered calls) :
\begin{displaymath}
W(\vec{\lambda}, \vec{C}) = \sum_{r} \lambda_{r}w_{r}
\end{displaymath}
\noindent{It} is important to note that this total network revenue
per unit time function $W$ is not necessarily concave, so may have
several local maxima. We are interested in finding the overall maximum
revenue, so we formulate our {\bf Global Optimization Problem},
defined as follows:
\\
\noindent{For} all A, B, let $\Re_{AB}$ be the set of all routes from
A to B.
\noindent{Let} $\nu_{AB}$ be the Poisson arrival rate of calls requesting
connection from A to B.
\noindent{Our} global problem is to maximize:
$$W(\vec{\lambda}, \vec{C})$$
\noindent{subject} to:
$$\sum_{r \in \Re_{AB}} \nu_{r} = \nu_{AB} \;\;\;\;\;\; \forall A, B$$
\noindent{Our} initial strategy is to find a solution
to the global problem using hill climbing techniques (Ie:
at each point, find the direction of the steepest ascent, or
gradient, and travel in that direction). In subsequent lectures we will
consider both decentralization and adaptive routing, where
we will measure or predict what $\nu_{AB}$ is and make
decisions accordingly.
\noindent{So} before we may climb the hill, we need to know which
direction is up, so we find the gradient. Here is the
result:
\begin{displaymath}
\frac{\partial}{\partial\nu_{r}} W(\vec{\lambda}, \vec{c}) =
( 1 - L_{r} )(w_{r} - \sum_{j \in r} \alpha_{j})
\end{displaymath}
\noindent{where} $L_{r}$ = blocking prob of route $r$, and:
$$\alpha_{j} = W(\vec{\lambda}, \vec{c}) -
W(\vec{\lambda}, \vec{c} - \vec{\delta_{j}})$$
\noindent{Here $\vec{\delta_{j}}$} is the 1 x J impulse vector, with zeros
in all entries except the jth entry, which is one.
\noindent{$\alpha_{j}$} is the so-called {\em implied cost} on link
$j$, which can be interpreted as the revenue loss incurred
by removing one circuit from link $j$ for a unit time.
\noindent{This} result is fairly subtle, and we will be trying to
clarify in the forthcoming discussion. Keep in mind that
$\alpha$ measures the knock-on cost, or effect of re-allocating
resources away from link $j$ for a unit time, and that these applied costs
superimpose in an additive manner.
\subsection*{Calculating the Implied Cost $\alpha$}
\noindent{To} calculate the gradient, we need to solve
for the implied cost $\alpha$.
\noindent{So} we wish to find [ $\alpha_{1}, ... , \alpha_{J}$ ], the
implied costs for each link.
\noindent{We} state that [ $\alpha_{1}, ... , \alpha_{J}$ ] satisfy the
following system of equations:
\begin{displaymath}
\alpha_{j} = \eta_{j} \sum_{r: j \in r} \rho_{j,r}
\left(w_{r} - \sum_{i \in r - \{j\}} \alpha_{i}\right)
\end{displaymath}
\noindent{where:}
\begin{displaymath}
\rho_{j,r} = \nu_{r} \prod_{i \in r - \{j\}} ( 1 - B_{i} )
\end{displaymath}
\noindent{This} $\rho_{j,r}$ should be interpreted as the thinned
traffic on route $r$, not yet thinned by route $j$.
\begin{displaymath}
\rho_{j} = \sum_{r: j \in r} \rho_{j,r}
\end{displaymath}
\noindent{This} $\rho_{j}$ should be interpreted as the overall
thinned traffic of all routes that go through link
$j$, before this traffic is thinned by $j$.
\begin{displaymath}
\eta_{j} = E(\rho_{j} , C_{j} - 1) - E(\rho_{j}, C_{j})
\end{displaymath}
\noindent{This} $\eta_{j}$ should be interpreted as the difference
between two Erlang loss probabilities, one with capacity
$C_{j} - 1$, and one with capacity $C_{j}$.
\subsection*{Single Link Model}
To better understand how these $\alpha_{j}$ equations
are derived, we consider a simplified model of a single
link, with capacity $C$ and arrival rate $\rho$.
\noindent{If} you remove a circuit for a one time unit, what does
this do to the probability of loss ?
\noindent{The} expected number of lost calls per unit time in the
original system was:
\begin{displaymath}
\rho E(\rho,C)
\end{displaymath}
\noindent{The} expected number of lost calls per unit time in the
new system (with one circuit removed) is:
\begin{displaymath}
\rho E(\rho, C-1)
\end{displaymath}
\noindent{So} the change in expected number of lost calls is:
\begin{displaymath}
\rho [ E(\rho , C-1) - E(\rho , C) ]
\end{displaymath}
\subsection*{Arbitrary Network Model}
\noindent{We} now extend the single node model by focusing
on a certain link $j$ in an arbitrary network model.
Although we have run out of time in this class, we
present one point as a precursor to the next lecture:
\noindent{If} we remove a single circuit from link j for unit
time in the arbitrary network, the expected additional number
of route $r$ calls that will be blocked is:
\begin{displaymath}
\rho_{j,r} [ E(\rho_{j}, C_{j} - 1) - E(\rho_{j} , C_{j}) ]
\end{displaymath}
\noindent{Important} note: the {\em aggregate} arrival rate to link
$j$ ($\rho_{j}$) must be used as the first argument to
the Erlang loss function.
\section*{Next Lecture}
Next time we continue our analysis of the revenue maximization
problem in looking at the concept of the implied cost of
removing a circuit for unit time in the arbitrary network model.
\begin{thebibliography}{2}
\bibitem{Kelly_1}
F.P. Kelly (1985).
Blocking Probabilities in Large, Circuit-Switched Networks.
{\em Adv. Appl. Prob.} {\bf 18}, 473-505 (1986).
\bibitem{Kelly_2}
F.P. Kelly (1986).
Routing in Circuit-Switched Networks: Optimization, Shadow
Prices and Decentralization. {\em Adv. Appl. Prob.} {\bf 20},
112 - 144 (1988).
\end{thebibliography}
\end{document}