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\title{\bf EE 290Q Topics in Communication Networks\\ Lecture Notes: 16}
\author{Tuan Le}
\date{March 7, 1996}
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\section{Introduction}
In this lecture, we focus on understanding how the Erlang Fixed Point (EFP) equations provide a good
approximation to the blocking probabilities of a network. In section 2, we review the main results
about the EFP approximations from Lecture 15. Section 3 states the questions we must answer.
These are the basic questions of existence and uniqueness of solutions and
accuracy of the approximations. Section 4 attempts to answer the existence and accuracy questions,
and it shows that the EFP approximations are accurate in networks where both link capacities and arrival
rates are high. Section 5 concludes these lecture notes with a few summarizing statements.
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\section{A Review of Definitions and Results from Lecture 15}
\label{sec-review}
\begin{itemize}
\item \(R\) = set of all possible routes in network
\item \(\nu_{r}\) = arrival rate of calls on route \(r\), where \(r \in R\)
\item \(n_{r}\) = the number of active calls using route \(r\)
\item \(\vec{n} = (n_{r}, r \in R)\)
\item \(A\) = the matrix whose \(jr^{th}\)-entry, \(A_{jr},\) is the number of circuits used by route
\(r\) on link \(j\)
\item \(C_{j}\) = capacity of link \(j\), \(j = 1, \ldots, J\)
\item \(\vec{C} = (C_{1}, \ldots, C_{J})\)
\item Erlang Fixed Point approximations:
\begin{itemize}
\item The \(A\) matrix is a (0,1) matrix, i.e. each active call uses up 1 circuit from each link
on its route.
\item \(E_{j} = E(\rho_{j},C_{j})\) = approximate blocking probability on link \(j\);
this is the Erlang loss formula for a single link with parameters \((\rho_{j},C_{j})\)
\item By the assumption of independent blocking at each link, \(\rho_{j,r}\) = \(\nu_{r}
\prod_{i \in r-\{j\}} (1 - E_{i})\) = rate of thinned Poisson arrival stream on link \(j\) on
route \(r\). Then \(\rho_{j}\) = \(\sum_{r: j \in r} \rho_{j,r}\) = superposition of
the arrival rates from each route \(r\) that link \(j\) is a part of.
\item The blocking probability of route \(r\) is \(L_{r} \approx 1 - \prod_{j \in r} (1 - E_{j})\).
\end{itemize}
\end{itemize}
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\section{Questions}
The EFP approximations provide a relatively easy method of analyzing the performance of a
circuit-switched network. The independent blocking assumption enables us to calculate
blocking probabilities that are useful in simulations or field work.
However, there are a few questions about the EFP approximations that we have to answer. First, does
there exist a solution \((E_{1}, \ldots, E_{J})\) to the equations? If there is a solution, is
it unique? Finally, how accurate are the equations in approximating the actual probabilities?
We will address these questions in this lecture and in the following lecture.
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\section{Analysis}
We will first show that a solution \((E_{1}, \ldots, E_{J})\) to the EFP approximations does exist.
When we combine \(\{E_{j}\}\) and \(\{\rho_{j}\}\), it is clear that
\beq
\vec{E} = f(\vec{E}) = (E(\rho_{1}(\vec{E}),C_{1}), \ldots, E(\rho_{J}(\vec{E}),C_{J})),
\label{eq:EFP}
\eeq
where
\[f: [0,1]^{J} \rightarrow [0,1]^{J}.\]
\noi By the Brouwer fixed-point theorem, this guarantees at least one solution \((E_{1}, \ldots, E_{J})\)
to (\ref{eq:EFP}).
\subsection{Mode of Distribution of a Single Link}
\label{sec-single}
To answer the question of accuracy, we will analyze the case of a single link first. We
have a single link with capacity \(NC\) and arrival rate \(N\nu\). From the last lecture, we
know that the stationary distribution of this single link ``network'' is
\[\pi(n) \propto \frac{(N\nu)^{n}}{n!} \hspace{.5in} n \leq NC.\]
\noi For large \(N\), if \(\nu \geq C\), then the network is full most of the time because the number of arrivals
are, on the average, greater than the capacity. This leads us to conclude that \(\frac{n}{N}\)
converges to \(C\). The convergence here is similar to Weak Law of Large Numbers convergence.
If \(\nu < C\), then the network is not full most of the time. Since
call holding times are exponentially distributed with a mean of 1, we conclude that \(\frac{n}{N}
\rightarrow \nu.\)
\noi Figure \ref{f:pi} shows how \(\pi(n)\) typically looks. Observe that, for large \(N\), most of the probability
mass is concentrated at the mode of the distribution \(N\nu.\) This idea also applies to arbitrary networks.
\pagebreak
\begin{figure}[ht]
\centerline{\ \psfig{figure=pi_n.eps,width=4in}}
\caption{\em Typical \(\pi(n)\) distribution}
\label{f:pi}
\end{figure}
\subsection{Extension to a General Network}
For a general network, we have
\beq
\pi(\vec{n}) \propto \prod_{r \in R} \frac{\nu_{r}^{n_{r}}}{n_{r}!} \hspace{.5in}
\vec{n} \in F,
\label{eq:pi_n}
\eeq
where
\[F = \{\vec{n}: \vec{n} \geq 0, A\vec{n} \leq \vec{C}\}.\]
\noi \(F\) is the feasible space under the capacity bound. Our problem then is to find
the most probable \(\vec{n}\) that satisfies (\ref{eq:pi_n}). In other words,
we want to find \(\max_{\vec{n} \in F} \prod_{r \in R} \frac{\nu_{r}^{n_{r}}}{n_{r}!}\), which is an integer
optimization problem. For large \(n_{r}\), we can replace the factorial by using Stirling's
approximation: \(n_{r}! \approx (\frac{n_{r}}{e})^{n_{r}}\). We also replace the
integer vector \(\vec{n}\) by a real-valued vector \(\vec{x}\). Our integer
optimization problem has now turned into a convex optimization problem:
\beq
\max \sum_{r} (x_{r} \log \nu_{r} - x_{r} \log x_{r} + x_{r}),
\label{eq:max_xr}
\eeq
subject to the feasible space constraint
\[\vec{x} \geq 0, A\vec{x} \leq \vec{C}.\]
\noi We solve (\ref{eq:max_xr}) by using the following theorem.
\subsection{A Unique Optimum Theorem}
\label{sec-optimum}
\begin{theorem}
There exists a unique optimum \(\tilde{x} = (\tilde{x}_{r}, r \in R)\) that solves (\ref{eq:max_xr}). It can
be expressed in the form
\[\tilde{x}_{r} = \nu_{r} \prod_{j \in r} (1 - B_{j}) \hspace{.5in} r \in R,\]
where \((B_{1}, \ldots, B_{J})\) is any solution to
\[\sum_{r: j \in r} \nu_{r} \prod_{i \in r} (1 - B_{i}) \left\{ \begin{array}{ll}
= C_{j} & \mbox{if \(B_{j} > 0\)} \\
\leq C_{j} & \mbox{if \(B_{j}\) = 0}
\end{array}
\right.,\]
with
\[B_{1}, \ldots, B_{J} \in [0,1).\]
\end{theorem}
The proof is as in Theorem 2.1 in \cite{Kelly}. We will sketch the proof here.
\noi Because (\ref{eq:max_xr}) is a convex optimization problem, we can find a solution
by using the method of Lagrange multipliers. We introduce a vector \(\vec{y}\)
which is a \(J\)-dimensional vector of multipliers. Then, our Lagrangian equation is
\beq
\max \sum_{r} (x_{r} \log \nu_{r} - x_{r} \log x_{r} + x_{r}) -
\sum_{j} y_{j}(C_{j} - \sum_{r} A_{jr}x_{r}).
\label{eq:lag_form}
\eeq
\noi Our solution must also satisfy the complementary slackness condition: \(\vec{y} \geq 0\).
The \(x_{r}\) that maximizes (\ref{eq:lag_form}) is
\[\tilde{x}_{r} = \nu_{r} \exp (- \sum_{j} y_{j}A_{jr}) \hspace{.5in} r \in R.\]
By the duality property of convex optimization, we have a dual problem to (\ref{eq:max_xr}):
\beq
\min_{\vec{y}} \sum_{r} \nu_{r} \exp(-\sum_{j \in r} y_{j}) + \sum_{j} y_{j}C_{j} \hspace{.5in} \vec{y} \geq 0.
\label{eq:min_yj}
\eeq
Because they are dual, solving (\ref{eq:min_yj}) is equivalent to solving (\ref{eq:max_xr}).
\noi Choose \(\vec{y}\) to satisfy the slackness condition and choose \(\vec{x}\) such that
\(A\vec{x} \leq \vec{C}\). Let \(e^{-y_{j}} = 1 - B_{j}\). Then,
\[\tilde{x}_{r} = \nu_{r} \prod_{j \in r} (1 - B_{j}) \hspace{.5in} r \in R,\]
such that \((B_{1},\ldots, B_{J})\) satisfy
\beq
\sum_{r: j \in r} \nu_{r} \prod_{i \in r} (1 - B_{i}) \left\{ \begin{array}{ll}
= C_{j} & \mbox{if \(B_{j} > 0\)} \\
\leq C_{j} & \mbox{if \(B_{j}\) = 0}
\end{array}
\right..
\label{eq:tot_fluid}
\eeq
\subsection{Fluid Flow Approximation}
\label{sec-fluid}
By the form of Theorem 1, we observe that we have replaced our discrete, probabilistic model with a
continuous, deterministic one. Instead of having discrete Poisson arrivals, we now interpret the
arrivals as a fluid flow. \(\nu_{r}\) is thought of as the offered flow of route \(r\). \(B_{j}\) is the fraction
of fluid overflow of link \(j\). The left hand side of (\ref{eq:tot_fluid}) is the total amount of
fluid that can be offered on link \(j\) after thinning by all the links on all the routes that link \(j\)
is a part of. For any link \(j\) with \(B_{j} > 0\), the capacity of that link, \(C_{j}\), is
completely filled, which makes sense intuitively in this fluid flow model.
\subsection{The Limiting Regime}
The results reached in Section \ref{sec-optimum} are appropriate for networks where the capacities and
arrival rates are high. With that in mind, we see that the network approaches a {\em limiting
regime} where the link blocking probabilities, \(\{B_{j}\}\), are independent, and the network traffic becomes
fluid-like.
\noi Consider a sequence of \(N\) networks where
\[\nu_{r}(N) = \nu_{r}N \hspace{.5in} r \in R,\]
and
\[C_{j}(N) = C_{j}N \hspace{.5in} j = 1, \ldots, J.\]
\noi For large \(N\), we have a Law of Large Numbers effect similar to the one characterized in Section \ref{sec-single}.
The mode of distribution is approximated by \(\tilde{x}_{r}\). Then Corollary 3.3 in \cite{Kelly} states
\beq
\lim_{N \rightarrow \infty} E(\frac{n_{r}(N)}{N}) = \tilde{x}_{r} \hspace{.5in} r \in R,
\label{eq:expected}
\eeq
where \(n_{r}(N)\) is the number of route \(r\) calls on the \(N^{th}\) network.
\noi By Little's law,
\beq
(1 - L_{r}(N))\nu_{r}(N) = E(n_{r}(N)).
\label{eq:little}
\eeq
\(L_{r}(N)\) is the blocking probability of route \(r\) calls on the \(N^{th}\) network.
The quantity on the left hand side of (\ref{eq:little}) is the offered load per unit time. The right hand side
of (\ref{eq:little}) is the average number of route \(r\) calls.
\noi By using (\ref{eq:expected}) and (\ref{eq:little}), the result that we achieve is
\[L_{r}(N) \rightarrow 1 - \prod_{j \in r}(1 - B_{j}).\]
Therefore, in the limiting regime, the blocking probabilities of different links are approximately independent.
\noi From Section 3 in \cite{Kelly}, we find that
\[B_{j} = E_{f}(\rho_{j}, C_{j}),\]
and
\[\rho_{j} = \sum_{r: j \in r} \nu_{r}\prod_{i \in r - \{j\}}(1 - B_{i}),\]
where
\beq
E_{f}(\nu,C) = \max(1 - C/\nu, 0).
\label{eq:fluid_loss}
\eeq
The above three equations are equivalent to (\ref{eq:tot_fluid}). They comprise the fluid flow
approximations of the actual blocking probabilities. When \(\nu \leq C\), there is no loss.
When \(\nu > C\), a deterministic \(\frac{\nu - C}{\nu}\) amount of fluid is lost.
\noi In comparing these equations with the EFP approximations in Section \ref{sec-review}, they seem remarkably similar.
Both sets of equations satisfy the same structure, except that \(E_{f}\) is deterministic and \(E_{j}\)
is probabilistic. Since a network is not deterministic in reality, (\ref{eq:fluid_loss}) is a crude approximation of the
loss probabilities. The validity of the EFP approximations is now clear because they replace a deterministic
loss with a more accurate probabilistic measure.
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\section{Conclusion}
The EFP approximations have been used to analyze networks with much success. It is important that we can ensure
the uniqueness and accuracy of these equations. In this lecture, the existence of a solution
\((E_{1}, \ldots, E_{J})\) to the EFP approximations was established. We also briefly showed why the equations
provide good accuracy given a network with high capacities and arrival rates. The question of uniqueness of
solution will be answered in the following lecture.
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\begin{thebibliography}{1}
\bibitem{Kelly}
Kelly, F.P. (1986).
Blocking Probabilities in Large Circuit-Switched Networks.
{\em Adv. Appl. Prob.} 18, pp. 473-505.
\end{thebibliography}
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