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\title{\bf EE 290Q Topics in Communication Networks\\
Lecture 19: State-Dependent Routing in Circuit Switched Networks}
\author{Michael Hu}
\date{March 21, 1996}
\maketitle
\section{Introduction}
In past lectures we examined the static routing strategy, in which calls
are routed along pre-specified routes in circuit switched networks. In
this strategy, calls are blocked if there are no free circuits in any of
the links on the pre-specified route. One improvement to this strategy
is to use adaptive routing, in which the pre-specified route may change in
adaptation to changes in the call arrival statistics. However, this
adaptive routing operates in the slow time-scale and may not allow calls to be
admitted into the network although there may be free circuits.
In this lecture, we will consider state-dependent routing, which admits
and routes
calls depending on the current state of the network on a per call basis.
Thus, the call admission and routing depends on the instantaneous
unused capacity of the links in the network. We will introduce
a model for state-dependent routing and apply Erlang Fixed Point (EFP)
approximation to analyze network performance. Then we will
apply trunk reservation to a single link and to a fully connected network.
\section{State-Dependent (Dynamic) Routing}
The state-dependent routing strategy that we will consider is
{\it alternate routing}. In this strategy, each call has a primary route
(the direct link between the source and destination nodes). Instead
of blocking the call if there are no available circuits on this route, as in
static and adaptive routing, a random alternate route is attempted.
By allowing the
call to be re-routed, we can potentially reduce the blocking probability
of the network and can increase the amount of admitted calls.
This strategy provides a straight-forward implementation. However, the
consequence is that under certain circumstances,
the performance may be worse than static routing (to be shown).
The potential problem is that calls that are routed on an alternate
routes use
more network resources than those that are routed on direct routes. Thus,
the implied cost of routing calls on an alternate routes may be greater
than the direct revenue obtained from those calls. If this is the case,
it is better to block calls rather than re-route them, which reduces
the strategy to static routing.
We will consider dynamic routing under a specific topology:
one in which the network is fully
connected and symmetric such that the statistics of all links are identical.
Furthermore, we will assume that calls arrive at rate $\nu$ between each
pair of nodes and that each link has capacity $C$. Note that in a network of
$M$ nodes, there is one direct route between any two nodes and there are
$M-2$ two-link (alternate) routes. Furthermore, we will consider this
dynamic routing strategy:
\begin{itemize}
\item Try to route the call on a direct route (single link). If there is a
free circuit, then admit the call on this route.
\item If the direct route fails, then randomly try a alternate route. If
both links in this route have at least one free circuit, then accept the
call.
\item If the alternate route also fails, then block the call.
\end{itemize}
\noindent{Note} that this represents just one strategy. Others can be
formulated
to choose the alternate route by some objective function.
\subsection{Analysis of dynamic routing using EFP}
We will now evaluate the blocking probability of a call using the EFP
for this model. Recall the assumptions made under EFP:
\begin{itemize}
\item links block independently of each other
\item arrival of thinned streams is Poisson
\end{itemize}
\noindent{Definitions:}
\begin{itemize}
\item $p\stackrel{\bigtriangleup}{=}$ blocking probability of each link
in steady state
\item $\lambda \stackrel{\bigtriangleup}{=}$ total arrival rate of calls
to each link
\item $\nu \stackrel{\bigtriangleup}{=}$ arrival rate of direct calls
to each link
\item $C \stackrel{\bigtriangleup}{=}$ capacity of each link
\item $M \stackrel{\bigtriangleup}{=}$ number of nodes in network
\end{itemize}
\noindent{Note} that the above definitions apply to all links because
the network is
defined to be fully connected and symmetric.
By applying the EFP assumption that links block independently of each
other, the probability that a call is accepted into the network can
approximated by:
\begin{equation}
Pr(call\ is\ accepted) = (1-p) + p(1-p)^{2}
\label{1}
\end{equation}
\noindent{regardless} of the node from which the call is requested.
The first term $(1-p)$ in equation (\ref{1}) is the probability that
the call is accepted on the direct link. The second term $p(1-p)^{2}$
is the probability that the call is
accepted on an alternate two-link route. Thus, to fully characterize the
performance of the network under alternate routing, we need to determine $p$,
which is approximated using the EFP equations. Recall from
previous lectures that the blocking probability of a single link
is approximated by:
\begin{equation}
p = E(\lambda, C)
\label{2}
\end{equation}
\noindent{Thus,} we will focus on a single link in our model.
The total call arrival rate on any link is the sum of the arrival rate
of direct calls and the arrival rate of calls in which the link is a part
of a alternate route. The arrival rate of direct calls is merely $\nu$.
For each direct route, the overflow rate is $\nu p$. Thus, the total overflow
rate in the entire network is given by
$(\stackrel{M}{2}) \nu p $,
which is obtained by multiplying the overflow rate per link with the number of
direct links $(\stackrel{M}{2})$.
Since each overflow call results in two link requests, the total number of
link requests in the network is given by
$2(\stackrel{M}{2}) \nu p $.
The number of requests for this link due to overflow calls is
$2 \nu p$ because link requests are uniformly distributed across the
$(\stackrel{M}{2})$
links. Furthermore, the actual arrival rate of overflow calls will be
thinned by
the other link in the alternate route. As a result, the total arrival rate
due to overflow calls is given by:
$2\nu p (1-p)$
\noindent{and} the overall arrival rate of calls on a single link is:
\begin{eqnarray}
\lambda = \nu+2\nu p(1-p) \nonumber \\
= \nu(1+2p(1-p))
\label{3}
\end{eqnarray}
\noindent{Applying} the call arrival rate into equation (\ref{2}) results in
\begin{equation}
p = E(\nu(1+2p(1-p)), C)
\label{4}
\end{equation}
\noindent{Solving} for p (not done in this lecture) and plotting the result
as a function of the offered load
normalized by the link capacity $C$ for several values of $C$
yields the graph in figure \ref{5}.
\begin{figure}[hbt]
\centerline{\psfig{figure=dyn_block.eps,width=120mm}}
\caption{Link blocking probability vs. arrival rate for several values
of $C$.}
\label{5}
\end{figure}
\noindent{Note} that for certain fixed load $\frac{\nu}{C}$, there
may be multiple
solutions for $p$, while in static routing, EFP yields an unique solution.
Recall that EFP solves
for the mode (or maximum point) of the stationary distribution. In static
routing, there is a unique mode, while in dynamic routing, the non-unique
EFP yields multiple modes. Of the three solutions at $\frac{{\nu}^{'}}{C}$
for $C = 1000$,
the top and bottom solutions for $p$ are locally stable modes, while the
middle solution doesn't correspond to a modal peak. Thus, the system will
exhibit a hyteresis effect by staying near one of the two stable modes
for a length of time and then jumping to the other mode. This bi-stability
is can be illustrated by the following example.
Suppose the average normalized call arrival rate is $\frac{{\nu}^{'}}{C}$
where $C$ is 1000 and that the solution to $p$ is initially the lower solution.
This implies that there is little blocking such that most calls are
routed on the direct link. If a large burst of calls arrive, the direct
link will quickly reach capacity and force the system to route
calls on alternate routes. Each call such routed uses an extra circuit,
which would otherwise be used by calls routed on direct routes,
and thus causes future calls to be blocked on the direct path. This positive
feedback causes the system to switch states by causing $p$ to jump to the
higher solution such that there is a higher blocking probability on the
link than before the burst. The system will remain in this state
until there is an unusually low arrival of calls. When this occurs, circuits
will become free as calls terminate, which no longer block future direct calls.
Thus, the blocking probability jumps down to the lower solution.
This bi-stability is undesirable because performance under the higher value
of p is worse than merely performing static routing (to be shown),
due the high implied cost of admitting calls only on alternate routes.
\section{Trunk Reservation}
To alleviate the negative effects of bistability, trunk reservation (TR) can
be used as an admission control strategy to reserve link capacity for the
sole use of direct calls. In this strategy, all calls that are routed on
a direct route are admitted as long as there is a free circuit. Calls that are
routed on an alternate route are admitted only if it
leaves a predetermined number
of circuits for future direct calls. Thus, this strategy prevents the
alternate routed calls from blocking future direct routed calls.
\subsection{TR in a single link network}
To analyze TR, we will first consider it in a single link network with
capacity $C$. Consider the following two types of calls
arriving into the network:
\begin{itemize}
\item A high priority class denoted class H where \\
\ \ \ \ \ \ \ - arrival rate $\stackrel{\bigtriangleup}{=} \nu$ \\
\ \ \ \ \ \ \ - revenue per call $\stackrel{\bigtriangleup}{=} W_{H}$ \\
\ \ \ \ \ \ \ - number of class H calls $\stackrel{\bigtriangleup}{=} N_{H}$.
\item A low priority class denoted class L where\\
\ \ \ \ \ \ \ - arrival rate $\stackrel{\bigtriangleup}{=} \sigma$ \\
\ \ \ \ \ \ \ - revenue per call $\stackrel{\bigtriangleup}{=} W_{L}$ \\
\ \ \ \ \ \ \ - number of class L calls $\stackrel{\bigtriangleup}{=} N_{L}$.
\end{itemize}
\noindent{In} the context of our dynamic routing strategy, class H is
synonymous with
calls using the link as a direct route and class L is synonymous with calls
using the link as a part of an alternate route.
The goal is to maximize the long term average revenue by
making admission decisions based on the current state of
the network $(N_{H}, N_{L})$. It can be shown that the optimal
strategy in the single link scenario is trunk reservation.
The proof of this claim is omitted in this lecture.
We can compute the blocking probability of both types of calls by modelling
the state of the link as a modified birth-death process with $C$ states.
\begin{figure}[hbt]
\centerline{\psfig{figure=network.eps,width=120mm}}
\caption{Link state as a modified birth-death process.}
\end{figure}
\noindent{Note} that there are two different arrival rates.
When the link is in a state in $[0,C-s-1]$, then there are enough circuits
such that a low priority call can
be admitted. Thus, in any
of these states, the arrival rate is $\nu + \sigma$.
When the link is in a state in $[C-s, C]$, then the remaining free circuits are
reserved for calls using this link as a direct route. Since all low
priority calls are rejected, the arrival
rate in any of these states is $\nu$.
To obtain the blocking probability, the stationary distribution
must be computed. Although this will not be done in this lecture, it
can easily be done by solving the detailed balance equations. Instead, we
will denote the blocking probability of classes H and L as the following:
\begin{displaymath}
F_{H}(\nu,\sigma,C,s)
\end{displaymath}
\begin{displaymath}
F_{L}(\nu,\sigma,C,s)
\end{displaymath}
\subsection{TR in a fully connected network}
When considering networks, TR is generally not the optimal admission control
strategy. However, it is widely used due to the simplicity of implementation.
We will analyze the call blocking probability using a fixed point model.
Definitions:
\begin{itemize}
\item $p_{1} \stackrel{\bigtriangleup}{=}$ blocking probability of a
direct call on a link
\item $p_{2} \stackrel{\bigtriangleup}{=}$ blocking probability
of an overflow call on a link
\item $L \stackrel{\bigtriangleup}{=}$ blocking probability of an
incoming call
\end{itemize}
\noindent{$L$} is given as follows. Note the similarity with equation
(\ref{1}):
\begin{equation}
L = p_{1} - p_{1} (1 - p_{2})^2
\end{equation}
Again, by solving the modified birth-death
equations, we can obtain the blocking probabilities:
\begin{displaymath}
p_{1} = F_{H}(\nu,\sigma,C,s)
\end{displaymath}
\begin{displaymath}
p_{2} = F_{L}(\nu,\sigma,C,s)
\end{displaymath}
\noindent {where} $\nu$ is the external call arrival rate, $\sigma$
is the overflow
arrival rate $= 2 \nu p_{1}(1-p_{2})$. Since $\nu$ and $C$ are known and
$s$ is a configurable parameter, this yields two unknowns and two equations.
Furthermore, for each fixed set $(\nu,\sigma,C)$, $s$ is chosen such
that $L$ is minimized.
The following graph compares the use of the different call admission strategies.
\begin{figure}[hbt]
\centerline{\psfig{figure=various_block.eps,width=120mm}}
\caption{Link blocking probability vs. arrival rate for several call admission
strategies}
\end{figure}
\noindent{It} is apparent from this graph that alternate routing
without TR performs
very badly under high load and performs much worse than static routing.
As expected, alternate routing with TR blocks less calls than static routing.
However,
it should be noted that the performance is only marginally better under
high call arrival rates.
\section{Conclusion}
In this lecture we have introduced state-dependent routing as a poor
alternative to static routing. However, by coupling state-dependent
routing with the trunk reservation admission control strategy, we
actually improve upon the network performance in terms of number of
calls admitted.
\begin{thebibliography}{6}
\bibitem{kelly1} F.P. Kelly, "Blocking Probabilities in Large
Circuit-Switched Networks", {\em Advanced Applied Probability}, Vol. 18,
pp. 473-505, 1986.
\bibitem{kelly2} F.P. Kelly, "Loss Networks", {\em Annals of Applied
Probability}, pp. 319-378, 1991.
\end{thebibliography}
\end{document}