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ALFRED
P.
"Models
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
for
Planning Capacity Expansion in
Local Access Telecommunication Networks"
A. Balakrishnan, T. L. Magnanti
A. Shulman, R. T. Wong
MIT
Sloan School Working Paper #3048-89-MS
Revised: August 1989
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
UCh.
13
"Models
for
Planning Capacity Expansion in
Local Access Telecommunication Networks"
A. Balakrishnan, T. L. Magnanti
A. Shulman, R. T. Wong
MIT
Sloan School Working Paper #3048-89-MS
Revised: August 1989
\
OCT
1
3 1989
,
Models for Planning Capacity Expansion in
Local Access Telecommunication Networks ^
A. Balakrishnan
**
T. L. Magnanti
Sloan School of Management
M. I. T.
Cambridge, Massachusetts
A.
GTE
Shulman
Laboratories, Incorporated
Waltham, Massachusetts
R. T.
Wong
"*
Krannert Graduate School of Management
Purdue University
West Lafayette, Indiana
May
1989
Revised: August 1989
This research
Supported
* *
was
in part
initiated
by an
through a grant from
AT&T
research
GTE
Laboratories, Incorporated
award
Supported in part by Grant # ECS-8316224 from the Systems Theory and Op)eralions
Research Program of the National Science Foundation
Supported
Research
in part
by
ONR Contract No. NOOOO-14-86-0689
from the Office
of
Naval
Abstract
The rapid progress
of communications technology has created
new
opportunities for modeling and optimizing the design of local telecommunication
The complexity,
systems.
diversity,
and continuous evolution
pose several modeling challenges. In
this paper,
we
of these networks
present an overview of the local
telephone network environment, and discuss possible modeling approaches.
particular,
we
(i)
discuss the engineering characteristics of the network, and
introduce terminology that
literature;
(ii)
In
is
commonly used
in the
communications industry and
describe a general local access network planning
model and
framework, and motivate different possible modeling assumptions;
(iii)
various existing planning models in the context of this framework; and,
summarize
(iv)
describe
some new modeling approaches.
The discussion
modeling
local
in this
is
directed both to researchers interested in
telecommunications systems and
such models. Our goal
environment
paper
is to
to
planners interested in using
present relevant aspects of the engineering
for the local access
telecommunication network, and
to discuss the
relationship of the engineering issues to the formulation of economic decision
models.
We
indicate
how changes
in the
underlying switching and transmission
technology affect the modeling of the local telephone network.
We
also review
various planning issues and discuss possible optimization approaches for treating
them.
Introduction
1.
Over
the last three decades,
problem don\ain
for
developing and applying optimization models.
these modeling efforts:
$60 billion in 1980 in
over $100 billion in
improvements
communication network planning and routing has been a
the
(i)
enormous investments
System transmission
Bell
total for the
in the design
U.
communication
in
facilities
S.) offer significant
Two main driving
alone
(AT&T
facilities
fertile
forces underlie
(estimated at around
Bell Laboratories (1986)),
and
opportunities for cost savings with even modest
and operation of communication networks, and
(ii)
rapid technological and
regulatory changes provide novel design alternatives and operating environments. This paper reviews
and develops
in
alternative
modeling approaches
for
addressing contemporary design problems that arise
one major component of a telecommunication system: the
paper
first sets
a
backdrop
for
local access
network.
As a
starting point, the
our discussion by reviewing relevant technological developments as well
as the evolution of the local access network.
In the next
few years, the nature of services and the volume
communications industry should change
radically. Several
era in commurucations: replacement of analog transmission
increasing
bandwidth of
increasing competition
fiber optic transmission
among
Integrated Services Digital
of
demand
in the tele-
developments mark the emergence of a new
by
equipment
digital technology,
decreasing cost and
relative to conventional
copper cables,
providers of telecommunication services, and adoption of international
Network (ISDN) standards.
As ISDN becomes
fully operational,
and
telephone companies complete the transition to digital switching and fiber optic transmission, users
will
have access
to a
broad range of new services combirung voice, data, graphics, and video.
applications include telemetry, database access, videophone
access to packet networks,
facilities,
New
improved networking
services,
and customer controlled network management. Telephone companies are
already planning for an even more ambitious expansion of services and capabilities (the so-called
broadband ISDN network) when
fiber optics will
way
homes (Kostas
to the individual
and Yates
customers'
permeate the entire communication system,
(1984),
(1985), The Economist (1987), Fortune (1988)).
and transmission technologies
To accomodate
modeling and planning
is
expected
the anticipated
efforts to
Dettmer
Thus,
to greatly stimulate
demand
(1985), Toth,
-1
the
Colombini, McClaren,
ISDN combined with
the
new
switching
network usage.
increase, telephone
expand and upgrade
all
companies have
their switching
initiated extensive
and transmission
facilities.
Network modernization and expansion
communication system, both
carriers
is
for strategic
particularly critical in the local access
and economic reasons.
have almost completed the transition
In the last
component
of the
few years, the long-distance
to digital switching technologies
and
fiber optic
transmission. In contrast, the technological changes in the local telephone network, which accounts for
approximately 60% of the
total
investment in communication
For instance, in 1987, only
20%
of
all
facilities,
have been much more modest.
the local access networks in the U. S.
is
limited
by the current
capabilities of local networks,
and
digital switching
ISDN telecommunication
(The Economist, 1987). Thus, the ability to offer the proposed advanced
services
employed
local telephone
companies
face
competitive pressures to upgrade their networks rapidly.
Because modernizing and expanding switching and transmission
facilities
investments, telephone compianies typically priortize expansion projects based on
potential
and emphasize cost effectiveness
in
implementing the selected
network planners face complex choices concerning where and when
technology in order
to
meet the increasing demand
deploying concentrators and multiplexers in the
method
alternatives
that did not arise in the traditional analog
local access
new decision support models
demand growth
For each project,
The emergence
of
new
and tradeoffs and, hence, new
and copper environment. For
instance,
network now provides an alternative
(instead of cable expansion) for increasing network capacity.
planners require
enormous
expand capacity or replace current
for different types of services.
communication technologies has created additional decision
modeling challenges
to
projects.
requires
As
to identify cost effective
a consequence,
network
expansion and modernization
strategies.
This paper focuses on contemporary expansion planning models for the local access component
(from the customer premises to the serving sv^tching center) of public telephone networks.
We do not
address design issues, such as the blocking of potential transmissions or network vulnerability, that are
more
relevant for long-distance networks. Similarly, our models might not apply directly to data
networks or rural networks since these
rural
latter
network types employ different technologies
(for
example,
networks use radio transmission, and data networks employ packet switching) and different
design
criteria (e.g.,
reducing packet delay in data networks).
Our purpose
methodology
in this papjer
for local access
is to
discuss alternative modeling approaches rather than a specific
network planning. The various models
-2-
that
we
consider differ in their
underlying assumptions, complexity and computational
tractability.
We focus on economic models for
aggregate planning (also called fundamental planning in the industry) rather than on detailed
engineering models of different technologies. Thus,
pattern of network evolution, specified
switching and transmission resources.
by
we are concerned
the capacity, location,
with identifying the broad
and timing
of investment in different
We review some of the underlying telecommunications
technology, and contrast the traditional network planning methods developed for the copper and
analog environment with the requirements imposed by the newer technologies.
solution
methods
The
for the different
rest of this
paper
is
orgar\ized as follows. Section 2 describes the evolution
communications industry and
briefly describe
models.
characteristics of local telecommunication networks,
the
We also
literature.
and engineering
and introduces some terminology commonly used
This discussion has two purposes:
(i)
to highlight
technological issues that are important in formulating appropriate optimization models, and
introduce analysts
who might
not be familiar with the telecommunication industry to
prevailing and expected technology.
network representation
that
Section 3 develops a general
some
framework based on
encompasses a vdde range of single-period
local access
in
(ii)
to
of the
a layered
network planning
models, and motivates different possible modeling assumptions. Section 4 discusses several planning
models
in the context of
literature,
our modeling framework.
We first review some models propx)sed in the
and then describe two new models - one using
a fixed -charge
network design formulation, and
another based on tree covering concepts. Section 5 offers concluding remarks.
The Local Telecommunication Network
2.
This section describes the
Icxral
access network, traces
and introduces some communications terminology. CXir
features so that
we can
its
evolution over the last few decades,
intent is to describe
some important
represent them adequately in economic planning models.
The Communication Network Hierarchy
2.1
Most rudonal telecommunications networks can be broadly divided
shown
(i)
technological
in Figure 1,
into the three
main
levels
namely,
the long-distance,
toll
or inter-city network that typically connects city pairs through gateway
nodes (also called point-of-presence nodes, Lavin (1987));
the inter-office or switching center
(ii)
network within each
city that interconnects switching centers
(also called local exchanges, or central offices) in different subdivisions (clusters of customers),
and
provides access to the gateway node(s); and,
(iii)
the local access network that connects individual subscribers belonging to a cluster to the
corresponding switching center.
These three levels of the communication system hierarchy
processing capabilities and
services they perform,
tree configuration
and
amount
differ in several respects: the
of intelligence they contain, the technologies they employ, the
their design criteria.
For instance, the
local access
network typically has a
and contains a dedicated communication channel connecting each customer
to the
switching center. Currently, most (approximately
80%
transmission on copp>er cables, and
electronic devices. In contrast, thelong-distance
do not contain
in the U.S.) local access
networks use analog
network has a relatively dense topology providing multiple communication paths between each origindestination pair.
The gateway nodes on
this
network contain
switching, traffic compression (concentration), and
The long-distance networks
in the U.S. are
some
intelligent
hardware
to
perform
service functions (such as directory assistance).
almost completely digitized, and employ high frequency
transmission using fiber optics, microwave (radio), and
satellite
communications. The
inter-office
the switching centers within a restricted geographical region (for example, within
network links
all
each
high speed transmission lines and possibly through tandem switches;
city) via
access to the nearest gateway
lin-iited
node of the long-distance network. The
intelligence for routing
incoming messages
to the
inter-office
it
also provides
network contains
appropriate downstream switching centers or
gateway nodes.
Ideally, the design of a
levels of the
telecommunication network should simultaneously account for
network hierarchy since the capacity requirements
interdependent. For instance, the
respective
number of customers assigned
communication requirements
(i.e.,
determine the desired switching capacity
on the
inter-office
task, analysts
to
all
three
at the different levels are
to
each switching center, and their
whom they communicate and how often) would
at the
switching center as well as the transmission capacity
network. However, because of differences in ownership and to simplify the planning
decompose
the overall planning
example, Dawson, Murphy, and
Wolman
problem by considering each
(1984)). In this paj^er
level separately (see, for
we focus on decision models for
designing and expanding the local access component.
2.2
Evolution of the Local Access Network
The
local access
links individual
network
customers
to a
(also called the outside plant, local loop, or local exchange network)
switching center that interconnects them for local communications, and
also serves as an interface to the higher levels in the
communication system,
feeder networks,
A
and
this
network
distribution
is
network hierarchy. Like the overall
also hierarchical;
consists of three levels referred to as routes,
networks.
route is a portion of the local access network;
with the switching center via the same link incident
local access
it
it
contains
to the center.
all
customer nodes that commurucate
Figure 2 shows a typical route of a
network. Each switching office might serve as the termination point for 3 to 5 routes
(Koontz (1980)); the
Bell
System contains around
40,(X)0
such routes (Ciesielka and Douglas
purpKDses of capacity planning in the local network, each route
•5-
is
indep>endent of the others.
(1980)).
For
Each route
is in
turn divided into two segments: the feeder network connecting the switching
center to intermediate nodes called distribution points (or control points), and distribution networks
connecting each distribution
pxjint to the
customer premises. The feeder network consists of cable groups
of varying gauges that are either buried, installed in ducts, or
mounted on
poles,
and are
accessible at
intermediate points. The segment of cables between two adjacent distribution points along the route
often called a feeder section.
The feeder network has
a tapered structure,
we move away from
each feeder section decreases as
the switching center.
taps into the feeder network via lateral cables at the distribution points.
distribution network,
sometimes called an
diameter of a few thousand
feet,
i.e.,
allocation area
the
The
number
is
of cables in
distribution network
The area served by a
(Gibson and Luber (1980)), typically has a
and may include as many as
5(X)
customers. The
number of distribution
points assigned to a switching center varies from 20 to 200. Most feeder and distribution networks have
a tree structure
and thus provide
(See Griffiths (1986) for a
a unique transmission path
more comprehensive
Traditionally, the distribution
On
cables.
to the switching center.
description of local telecommunication networks.)
network
small) in order to exploit economies of scale,
from each customer
and
is
designed for ultimate
to avoid
demand (which
In this paper,
we
focus on the
evolution of technologies and planning practices in the feeder network.
classify the technological
Sta ge
The basic feeder network
1
:
developments into three stages
The basic feeder network employs analog transmission
copper cables (twisted wire
pairs).
center. Physically, the line for a
It
demand growth
Elken (1980), Friedenfelds and McLauglin (1979)).
medium-term feeder capacity planning problem.
we might
new
medium-term demand;
telephone companies periodically review and increase feeder capacity to accomodate
(1980),
relatively
subsequent disruption of service for laying
the other hand, feeder networks are designed to meet only the
and customer movement (Ciesielka and Long
is
From
(see also
at the voice
a
We next
trace the
modeling
Dawson
{perspective,
et al. (1984)).
frequency of 4 Khz over
uses a dedicated line to connect each customer to the switching
customer might consist of wire segments (possibly with different
gauges) belonging to each downstream feeder section, with sections joined at the intermediate
distribution points.
-6-
In this setting, a principal design concern
is to
provide acceptable transmission quality by
ensuring that the circuit connecting each customer to the switching center
permissible wire resistance (around 1300 ohms, increasing to 2500
the
network engineering
task,
satisfies the
ohms with
maximum
range extenders). Thus,
sometimes called Resistance Design (Ciesielka and Douglas
determine a cost-effective combination of wire gauges
to
(1980)), is to
use in each feeder section that will satisfy
all
maximum resistance requirements.
Observe
that the basic feeder
network can respond
by adding and reassigning cables within each feeder
as physical pair facility
said to
have
customers
exhaust.
move
relief.
Any
or the
number
section. Planners
telecommunication
sometimes
demand
refer to this
exercise considers
of customers increases:
(i)
two
strategies to relieve exhaust
feeder cable reallocation, and
(ii)
only
method
where demand exceeds the available cable capacity
section
The feeder planning
to increased
is
when
feeder cable
expansion.
Given the projected changes
reallocation
methods attempt
in the
medium term demand
to identify a feasible
at
each distribution
f>oint, feeder
reassignment of currently allocated and spare
feeder cables within each section to various upstream distribution points in order to delay cable
expansion. Gibson and Luber (1980) describe a heuristic feeder allocation method; Elken (1980)
formulates the reallocation task as a separable convex progranrvming problem, and proposes an iterative
procedure that solves a sequence of linear programs.
Feeder expansion
models
(e.g.,
Freidenfields and McLaughlin (1979)
determine the number of additional cables
planning horizon in order
cable expansion
is
to install in
to relieve the projected
the only available
method
and Koontz
(1980))
every time period (typically, every year) of the
exhaust at
minimum
for relieving exhaust,
discounted
total
and
if
cost.
the local access
When
network has
a tree structure, each feeder section can be analyzed independently for capacity planning purposes (since
the
number of customers served by each
each section). Thus, physical pair
spatial
distribution point uniquely determines the cable requirements in
facility relief
models used
in this context
do not incorporate any
coupling between sections. Friedenfelds and McLaughlin (1979) formulate a multiperiod
capacity expansion
heuristic solution
model
method
to find the
for this
optimal mix of cable gauges in each section; they describe a
model.
-7-
Stage
Feeder networks with remote electronics
2:
From a modeling p>oint of view,
the next nujor stage in local network evolution occurred
the communication industry developed pair gain or remote electronic devices,
concent -ators
and remote
switches, for use in the local network.
compresses or interleaves signals from several incoming
that
A
lines into a
has a higher frequency but requires only a single line (or a pair of
incoming signal
to a separate 'channel' in the
multiplexer
lines).
i.e.,
is
when
multiplexers,
an electronic device
composite outgoing signal that
The system assigns each
combined outgoing transmission. (Channels correspond
to
preassigned non-overlapping frequency bands in frequency division multiplexing, and to time slots in
We refer
time division multiplexing.)
compression ratio
traffic
(also called the
to the ratio of input to
multiplexing
ratio).
output signal frequencies as the traffic
Like multiplexers, concentrators also perform
compression, transforming multiple incoming signals into a single outgoing high frequency signal.
However, the output
signal
from a concentrator does not have a dedicated channel
for
each input line
(and so signals might be blocked); rather, the output channels are dynamically assigned to the input
lines as the
need
arises.
The
Remote switches perform
ratio of
incoming
to
outgoing channels
local switching functions to interconnect all
the switching center through them; thus, each remote switch
the
main switching
is
center. Typically,
is
called the concentration ratio.
customers
a decentralized
who communicate with
and smaller version of
remote switches also perform multiplexing and /or concentration
functions for traffic destined to the main switching center.
In local access
the
as
same physical
demand
less
line
increases.
network applications, remote electronic devices enable multiple users
on the feeder network, thus providing an
They
alternative
also eliminate circuit resistance restrictions,
method
to
to relieve
share
exhaust
and permit the use of fewer and
expensive wire gauges. Multiplexers, concentrators, and remote switches are available in several
configurations, with varying input capacities
compression ratios ranging from
2:1 to
(i.e.,
number
of input lines)
and
different traffic
as high as 96:1.
While multiplexing, concentration and remote switching reduce the number of cables required
downstream feeder
sections, these cables
must now handle higher frequencies. Conventional copjser
cables (twisted pairs) have a limited effective
(150
Khz
to 2
Mhz)
in
bandwidth (around 150 Khz). Higher frequency signals
require either coaxial cables or conditioned (or groomed) copper cables,
i.e.,
twisted
wire pairs with intermediate repeaters (which are electronic devices to eliminate distortion in the
-8
very high frequency signals (over 2 Mhz) require
signals);
for
fiber optic cables. Often, existing ducts (built
copper cables) can accomodate these enhanced transmission media as well.
For the second generation local access network with electronic devices, the planner must
consider various choices for locating, sizing, and timing the installation of remote electronics
(multiplexers, concentrators, switches) as well as the conventional option of physical pair facility
relief
(i.e.,
increasing cable capacities in different feeder sections). Furthermore, unlike the older
technologies, these
new devices
section in isolation since higher
introduce spatial couplings,
demand
on every intermediate feeder
longer consider each feeder
does not necessarily translate
section.
Fiber in the local access network
:
Many
telephone companies are currently planning
access networks because of
facilities consist of a
cable.
we can no
at up>stream distribution points
into increased cable capacity requirements
Sta ge 3
i.e.,
its
introduce fiber optic technology in local
to
extremely high bandwidth. Fiber optic or lightwave transmission
pair oi fiber optic terminals (or fiber terminating equipment) connected
a fiber
Fiber optic terminals convert electrical (analog or digital) signals into very high frequency
optical signals,
and might perform
optical coupling
and multiplexing functions as
of fiber cables (some with a transmission capability of over
1
well.
Tera bits per second)
unlimited for local network applications; therefore, they can accomodate a large
is
factor limiting the
number
the cost of fiber terminating
equipment
cables are installed in existing
distribution points
and
of channels that can
is
expected
be multiplexed on
to
dominate the
underground ducts) due
fiber.
The bandwidth
effectively
number
high frequency channels. Indeed, the electronics in the fiber terminating equipment
main
by
is
of multiplexed
currently the
For local access networks,
fiber cable costs (especially,
to the relatively short distances
if
fiber
between the
the switching center.
Except for a few experimental networks, telephone companies have not deployed fiber optic
transmission extensively in the local access network. The characteristics, capabilities and deployment
of the technology,
Anderson
Kaplan
(1988),
(1984)).
networks
(e.g.,
and even the network configuration plans are constantly changing
Carse (1986), Ensdorf, Keller, and Kowal (1988), Toth
et al. (1985),
(see, for instance,
Snelling
and
Researchers have proposed several competing topologies for fiber-based local access
Campbell
(1988), Garbanati
and Palladino
-9-
(1988), Sirbu
and Reed
(1988),
White
(1988)).
Some of the
topologies, such as the ring, loop,
vulnerability,
an issue
that
is
architectures, seek to reduce
becoming increasingly important
a single cable can affect service to a large
In this papjer,
and mesh
we assume
for fiber
network
networks since damage
that, for topxslogical
design purposes, fiber optic terminals essentially
we do
not represent the unique
characteristics of fiber optic transmission in great detail, particularly since this technology is
When
deploying
it,
even
number of customers.
act like concentrators with very high traffic compression ratios. Thus,
evolving.
to
the technology develops further arnl telephone
still
companies gain experience with
network plaruiers might require more sophisticated models
that distinguish fiber optic
transmission from conventional electrical transnussion.
Table
The next
I
summarizes the technologies and expansion options we have discussed
section formalizes the feeder
modeling assumptions;
we will
network planning problem, and motivates several possible
subsequently use these assumptions to differentiate various modeling
approaches.
Table
I
Local Access Network Expansion Options
Technology Stages
in this section.
Local Access Network Planning: Problem Definition, Modeling
3.
Framework and Assumptions
This section presents a general conceptual and modeling framework for studying the local access
(feeder)
network planning problem, defines the scope of models
identifies
some key assumptions
that distinguish different
that
we will
subsequently consider, and
modeling approaches
We
for this problem.
present an example to clarify the decisions and tradeoffs in the planning task, and to illustrate the
modeling implications of
traffic
processing and cable expansion options.
We then develop a general
layered network representation that captures the important issues and tradeoffs of the local access
network planning problem.
The feeder capacity planning
exercise begins with a forecast (based
customer movements) of telecommunication demand
demand
planning horizon. The basic unit of
at
each distribution
for voice transmission
is
on new construction and
p)oint for the
a circuit. For
duration of the
analog transmission,
each circuit represents a bandwidth requirement of 4 Khz, and requires one twisted pair of copper wires.
The corresponding
Kbps
digital equivalent in the U.S.
(Kilobits per second).
The demand
is
the
DSO
for data, video,
signal
which has a transmission
and other wideband
services
is
usually
expressed as a multiple (or fraction, for some types of data transmission) of the basic DSO
example, the DSl, DS2, and DS3 rates
are, respectively, 24, 96,
rate of 64
rate; for
and 672 times the DSO transmission
rate.
The
total
objective of the planning exercise might be to satisfy
all
the projected
(investment plus operating) discounted cost, or to selectively satisfy
profit.
In this
paper
we
we
restrict
our discussions
models rather than multi-period models. Multi-period models can account
one year
scale,
i.e.,
in anticipation of higher
harder to solve than a
demand
to
at
minimum
maximize
total
focus on the cost minimization perspective partly because telephone companies
currently use this form most often. Furthermore,
caused by economies of
demand
static
planning horizon. Studying
for the
temporal couplings
the optimal investment strategy might install excess capacity during
demand
model
static
to single-period or static
in
subsequent years. However, multi-period models are
that seeks only to satisfy
models might
demand
fX)ssibly give
much
in the terminal year of the
us insights about the more general
problem. For instance, the single-pseriod solution algorithms could serve as building blocks for multiperiod versions (see, for example, Shulman and Vachani (1988)). Or, as Minoux (1987) has proposed.
11-
model might be used
the static
to first identify the final target
model would then determine the evolution
To meet
appropriate
the
traffic
demand
of the existing
for different services, the
network; a subsequent multi-period
network toward the
target.
network design must provide adequate and
processing (multiplexing, concentration and switching) and transmission
from each distribution point
to the
facilities
switching center. In general, different services and processing
devices require different transmission rates and, hence, different transmission media (twisted wire
pairs,
groomed copper
cables, coaxial cables,
and
fiber optic cables).
For example, video signals cannot
be transmitted over twisted pair copper cables; similarly, remote electronic devices with high
multiplexing ratios require enhanced media that can carry high frequency signals. The local access
network design must account
for these
bandwidth
specifications
and provide compatible communication
resources.
In addition to providing adequate capacity to
network configuration must also
telephone
company might wish
meet the projected demand, the
satisfy various technological
to
and policy
restrictions.
local access
For example, the
provide multiple paths to some preferred large-volume business
customers. Similarly, to ensure adequate transmission quality (particularly for data transmission
applications)
and
to facilitate future
expansion and modernization, designers might
Sf)ecify a
maximum
permissible distance (some companies use a limit of 12 kilofeet) betv\'een each customer and the nearest
electronic device;
we
refer to this type of constraint as a proximity restriction.
Thus, the single-period local access network planning problem has the following ingredients:
Given
(i)
the projected (terminal year)
(ii)
(iii)
demand
for different services at
each distribution point,
the current switching and transmission capacities at each location, and
the costs of installing, expanding and operating
new
switching and transmission
facilities,
find
the cost minimizing expansion plan that meets the
demand and
satisfies all technological
and
policy restrictions.
The optimal expansion plan should specify
(i.e.,
(a) the location
and
size of various
network enhancements
addition or expansion of transmission media and remote electronic devices), and (b) the routing of
traffic
from each distribution point
to the
switching center.
-12-
This definition of the local access network planning problem makes
For instance,
by some
we have
ignored
some
implicit assumptions.
processing steps (such as database queries) required
spjecial information
services. Currently, all information processing occurs at higher levels of the
hierarchy
(i.e.,
in the inter-office or
backbone network); the
local access
communication
networks f>erform only
traffic
compression. Future designs might decentralize certain information processing functions as well to the
local access
network. Also, our single-p>eriod model does not consider tradeoffs between overlay and
replacement strategies for
new
technologies.
When we consider a
multi-p>eriod
modernization, the planner must also decide whether to introduce
new
framework
for
network
by
initially
digital technology
overlaying existing analog technology, or by immediately replacing the analog components.
and Epstein
(1979),
Combot and Mason
(1979),
Combot, Tsui, and Weihmayer
(1981),
Combot
and Hoang and Lau
(1984) consider these issues for inter-office network planning.
Before presenting a general modeling framework for the local access network planning problem
we
discuss a simple example that illustrates the basic decisions and tradeoffs that the network planner
must address.
Example
3.1
Figure 3a illustrates a local access network problem with a single service
network has a
as
shown
tree structure, consisting of a single
in the figure.
heavy shaded
However,
medium
(say,
copper cables) with existing capacities
inadequate for the projected
this capacity is
lines represent feeder sections with projected exhaust
projected exhaust represents the
(i.e.,
section
level.
capacity shortfall).
employ
The
The
expand cable
to
traffic
processing
3b shows a representative cost function for expanding cable capacity along the feeder
between distribution points i and
In this example, the given
j,
consisting of a fixed charge plus a (constant) per unit cost.
network does not have any existing processors. To
the planner can use a processor with a 10:1 traffic conversion ratio:
traffic
demand
amount by which telephone companies would need
capacities in a conventional physical pair relief strategy (that does not
options). Figure
The given
typ>e.
entering on every set of 10 incoming lines onto
-13
1
outgoing
i.e.,
line.
relieve exhaust,
the processor compresses the
We assume for simplicity that the
copper cables can transmit both the base rate signal
compressed output
might arise
signal. Figure 3c
the processor
if
capacities (Y|, Y2,
and
is
shows an
at
which the service originates and the processor's
This cost function
illustrative processor cost function.
available in three
models with differing costs (G^, G2, and G3) and
<»).
Figure 4 shows one possible expansion plan for the network example of Figure
entails installing a processor with a capacity of
along sections
(3,1)
and
the traffic from nodes
node
(4,2)
by 100 and 10
2, 4, 5, 8,
and
9.
Its
400 units
at
node
output signal, shown
the processor performs a tenfold compression of
its
1;
relieved the projected exhaust
would have expanded cable
either direction
traffic
on edges
400 incoming
(2,5), (2,1),
circuits. All the
of the network. For instance,
(from nodes 2 and 4) from distribution point 2
concentrated
traffic
and
capacities in these feeder sections).
on each edge
to the
node 5 compresses
this signal requires
signals at the base (unconcentrated) rate to the switching center.
we have
at
in dotted lines, travels
nodes 2 and
edge
By
from node 5
all
to
only 40 lines since
other nodes transmit
installing the processor at
node
5,
(1,0) (the physical pair relief strategy
Observe that we permit
(2,5) carries
(to the
traffic
flow in
150 uiuts of unconcentrated
processor located at node
flow in the opposite direction from 5 to 2
This plan
and expanding the cable segment
The processor
units, respectively.
(the switching center) via the intermediate
5,
3.
5,
and 40 units of
switching center); thus, the 200
available lines in section (2,5) can accomodate both these flows. Also note that the expansion plan
shown
in Figure 4 involves backfeed,
section
(2,5).
Some of
The example
(i)
the
and
2, 4, 5, 8,
(1,0).
flow that
models we discuss
and
in Section 4
and cable expansion:
is
more economical
if
of expanding cables along the three sections.
exhaust on section
to
directed
do not
two
away from
(3,1).
6's traffic
the switching center,
tradeoffs in local access
Installing a processor at
network planning:
node 5 and assigning
downstream
the cost of the traffic processor
We
on
pjermit backfeed.
9 to this processor relieves the exhaust in the
This strategy
process node
is
of Figures 3 and 4 illustrates the
the tradeoff between processors
nodes
i.e.,
sections (5,2), (2,1),
is
lower than the cost
could follow a similar strategy to relieve the
For instance, locating a processor at node 6 (with a capacity of 120 units) to
would
expand cable capacities
if
relieve the 100 units exhaust
on edge
the cost of a 120-unit processor at
cost (for 100 additional circuits
on
section (3,1)).
-14
node
(3,1).
However, we might prefer
6 exceeds the cable expansion
(ii)
the tradeoff between installing a
few large centrally located processors versus many small
geographically dispersed processors:
For example, should
we
two processors, one
locate
at
node
4
(with a capacity of 150 units, serving nodes 2 and 4) and the other at node 5 (with a capacity of 250
units, serving
nodes
and
5, 8,
9),
instead of a single processor at
avoids cable expansion on section
cost
likely to
is
decreases.
scale, its total
processor
expansion cost might pxsssibly increase as the
On the other hand, installing fewer, but larger, processors reduces
The planning model must address
the total processing cost.
economies of scale
however, because of economies of
In general, the total cable
be higher.
number of processors
(4,2);
node 5? The two-processor solution
in processor costs
this tradeoff
between exploiting
and avoiding transmission capacity expansion by employing a
decentralized processor location strategy.
Our example considered
transport both concentrated
a single service, a single processor type,
and unconcentrated
traffic.
and a
single
medium
Additional complexities arise
that can
when we
consider multiple processor types, multiple processing steps in series, and multiple transmission media.
In the next
two sections we develop
network representation
a layered
for this
more
general problem.
This representation serves as a framework for comparing various modeling approaches for local access
network expansion planning.
It
encompasses
a
wide range of existing and proposed transmission and
processing technologies, cost structures, and network topologies.
Modeling Principles
3.2
This section discusses the modeling elements for representing local access network planning
problenas with multiple transmission rates, service types, processor types, and transmission media. In
the next section
we develop a
conceptual framework based on a multi-layer network representation that
captures the interrelationship between the three main modeling elements:
different services that
traffic
satisfied;
(ii)
transmission
processors (e.g., multiplexers, concentrators,
compress
(i.e.,
must be
signals.
As mentioned
facilities for
remote switches or
(i)
customer demands for
carrying signals; and
(iii)
fiber optic devices) that
previously, the network expansion plan
must match
the
can
bandwidth
transmission rate or frequency) specifications of these three elements. The layered network
representation ensures this compatibility
by
sef>arately identifying the traffic flows at each
-15
transmission rate.
We first discuss how the transmission rate serves as a common link between customer
demands, transmission
facilities,
We assume that
and
traffic
network employs a discrete
the local access
For instance, telephone compjanies in the U.
multiples of
bits
processors.
use four standard
S.
per second or bps) labeled DSO, DSl, DS2,
respectively, to 64 Kbps, 1.536
set of 'standard' transmission rates.
transmission rates (expressed as
digital
and DS3; these
rates correspond,
Mbps, 6.144 Mbps, and 43.008 Mbps.
Customer Demands
Customer demands are forecasted by
point.
Each service type originates
a basic rate of 1.5
service
(the
must be transmitted;
terminal year
basic rate.
demand
The
final
number of higher
that, since
we
Mbps
for
DSl
service type (voice, data, video) at each distribution
For instance, certain types of video services require
at a basic rate.
rate).
This rate represents the
frequency
the signals could be multiplexed to higher frequencies,
each type of service
is
rate channels) for each service typje
we do not consider any
all
at
if
which the
necessary.
The
expressed as the number of required channels at the
design should provide the required
can effectively aggregate
minimum
number
of basic rate channels (or an equivalent
from every node
to the switching center.
Observe
unique information processing requirements for different service types,
services requiring the
same
basic rate into a single service type.
Transmission Facilities
We differentiate
transmission
facilities
according to the
medium
or cable type (such as twisted
wire pairs, copper cables with repeaters, coaxial cables, and fiber optic cables). Each transmission
medium can handle
and DSl
a limited set of transmission rates. For example, coaxial cables can transmit
signals, while the
DS2 and DS3
copper cables) are typically installed
rales require fiber optic cables.
in sections; to
joined at intermediate distribution points.
media
(e.g., fiber
Some
DSO
transmission media
(e.g.,
connect two non-adjacent nodes the cables must be
We refer
to these
media as
sectional
media. Other types of
cables) provide direct point-to-point connections without intermediate connectors;
however, they use the same physical infrastructure, namely, the underground ducts or trenches
intermediate feeder section. Most high frequency
traffic
-16-
requires point-to-pjoint cables.
in
each
Transmission capacities are specified by the number of lines for each
pair of nodes. In general, the total (transmission) cost to
node has two components: a separable
or shared cost that
the
same
different
is
common
cost
component
to several (or all)
infrastructure. For instance, once
expand these
medium between every
capacities
between any pair
medium, and
f>ertaining to each individual
of
a joint
media. Joint costs arise because different media share
we incur the costs
for building
media could share these ducts. Because of economies of
scale,
underground ducts, several
both the separable and
joint cost
functions are likely to be concave; for example, they might consist of a fixed cost and a variable cost
that
depends on the volume of
traffic.
Traffic Processors
As explained
previously, traffic processors combine several incoming (lower frequency) signals
The economic model
into a single outgoing (higher rate) signal.
processor type by three parameters:
conversion
mean
ratio.
the ratio of
differ
from the
for spare
input rate
(i.e.,
We do not model other detailed
number of incoming
lines for contingencies).
(number of input
its
lines
traffic
-
lines (e.g.,
that
we
consider characterize each
frequency of input signals), output rate, and
technological differences.
copper wire pairs)
to
By
outgoing
conversion ratio
number
of output lines) +
compression
ratio,
i.e.,
number of output
lines.)
(including spare
lines
(The industry uses a related measure called the pair gain
we
ratio
defined as
This conversion ratio
may
the ratio of output rate to input rate, because of provisions
outgoing lines and differences between the number of input and output channels (as noted
previously, concentrators require a smaller
number
of output channels since they allocate channels
dynamically). Effectively, the input and output rates determine the type of input and output
transmission media that the processor requires, while the conversion ratio determines the
physical lines of the output
medium
required per incoming
Processor capacities are specified by their
number
of
line.
maximum number of input lines.
Installing
new
processors or expanding existing capacities entails processor costs which nnight vary by location,
processor
tyj)e,
and the required additional
capacity. For instance, this cost function
fixed costs (for acquiring land, constructing buildings, pedestals, cabinets
variable or volume-dependent costs
(e.g.,
for each
module)
that
and other
depend on
might consist of
infrastructure),
the desired capacity.
expect these processor cost functions to be concave (as a function of additional capacity) because of
economies of
scale.
17
and
We
A
(Ciesielka
commerdally available
spjecific
and Douglas
traffic
processing device, namely, the SLC-96 system
The SLC-96
(1980)) illustrates these concepts.
a modular digital
is
carrier/concentrator system introduced in the Bell System around 1979. Each
frequency input
The input
lines.
signals before retransnussion.
signals are analog; the
The input
Khz
rate of 4
module supports 96 voice
SLC-96 system converts them
is
equivalent to a
number
DSO
into digital
digital rate (64 Kbps).
The system performs two-to-one
digital concentration,
number of input lines;
module has 48 output channels. These 48 output channels are
thus, each
i.e.,
the
of output channels
half the
is
transmitted over two standard so-called Tl digital lines, each carrying 24 channels. The system also
requires a spiare Tl line to assure continuity of service
when one of
main Tl
the
Each Tl
lines fails.
might consist of two pairs of copper wires, with intermediate repeaters; the transmission
line is 1.536
Mbp>s (= 24 channels x DSO input rate of 64 Kbps), which corresponds exactly
rate over each
Thus, the SLC-96 has a concentration ratio of (96 input channels + 48 output channels) =
traffic
compression
{3T1 lines x 2pairs/line)) = 16.
(to
An
provide service for up
One
II
summarizes the relationships between transmission
medium
and
-
rate)
1
the
and DS2. In
with a conversion ratio of
DS2
ratio; for instance,
2,
rate.
DSl
DSO
-
up
to ten
modules
and
customer demands for
For ease of illustration,
and assume
this
DSO
12,
Observe
The example
-
whose
basic
example, the designer can use three types of
signals to the
DSl
rate
(i.e.,
with a
a tyf)e 2 processor to compress
from the
signals represent a 24-fold compression of the
requires
1
DSO
rate,
rate
signals to the
traffic
is
more
and a
DS2
rate
compress
compression
while the type
outgoing line for every 12 incoming
signals directly using a typje 3 processor
-18-
DSl
DSO input
ratio of 48 to directly
that the conversion ratios differ
(i.e., it
we
that each rate has a unique
fiber cable, respectively).
and a type 3 processor wnth a conversion
processor has a conversion ratio of only 12
Also, compressing
rates,
processors.
DSO, DSl, and E>S2
processor to compress
with a conversion ratio of
traffic to
+
voice communication, high spjeed data, and video service
rates are, respectively DSO, DSl,
processors: a type
-
traffic
(twisted wire pair, coaxial cable,
considers three service types
DSO
ratio of (96 input pairs
960 customers).
consider only three transmission rates
DSl output
and a conversion
a
version of the SLC-96 system p>ermits stacking of
different service types, transmission facilities,
corresponding
24,
2,
System
Illustrative
Table
to
Mbps + 64 Kbps) =
DSl
to the
rate.
ratio of (1.536
line
effective (in terms of the
1
lines).
numer
of
DS2
in
lines required for the
tandem;
lines,
for, say,
same number of incoming DSO
480 incoming E>SO
whereas the type
1 -
tyf)e 2
lines, the
lines)
than using a type
combination requires 480/(12*2) = 20 DS2
Table
lines.
Service
Transmission
Rate
Type
Medium
shown
Data
section
in Table n.
Traffic Processors
Type
DSl
The next
System
Transmission
Voice
2 processor
II
Illustrative
DSO
and a type
type 3 processor requires only 480/48 = 10 outgoing DS2
presents a layered network representation that models the interrelationships
Layer
1
1
Type 3
Type 2
Twisted pair
U
Coaxial
2:1
DS2
3.3
Video
Layered Network Representation
Let G:(N,E) represent the physical network whose nodes
switching center (node 0) and the distribution points (nodes
edges E contains edge
(i,j) if
model since
processor
typ)es.
it
1
N=
{0,l,2,...,n)
correspond
to n) a^ssigned to that center.
to the
The
set of
the local access network currently contains or can potentially contain a
feeder section between nodes
of the
T
Fiber
i
and
j.
This physical network does not provide a complete representation
does not differentiate the various transmission
To incorporate these modeling
different layer with each transmission rate.
features,
we
rates,
transmission media, and
use a multi-layer network that associates a
Each layer contains a copy of the original network, with
-19-
additional edges that represent direct pxjint-to-point cables. The arc flows within a layer correspond to
transmission at a specific rate, and flows across layers model
To describe
summarized
associated with each transmission rate (DSO, DSl,
media
for
The main simplifying assumption
each transmission
rate. In particular,
transmission rates and media,
rate,
and every
i.e.,
setting,
in Table
we
first
consider a
This example has one service type
II.
and DS2), and processor types corresponding
we
we assume a one-to-one correspondence between
each transmission
will indicate
every
to
example concerns the available transmission
in this
medium can accomodate only one
unique medium. After developing the layered network
rate requires a
problem
simplified
compression.
the layered network representation in greater detail,
representation for the illustrative system
pair of rates.
traffic
network enhancements that
will
transmission
for this
permit us
to
model
transmission media with overlapping frequency ranges.
The layered network, denoted as G^, contains one layer
and hence the network layers and
the transmission rates,
for
each transmission
from / =
service types,
1
to
L
rate.
Let us index
in increasing
order of frequency. Because of our simplifying assumption concerning the correspondence between
transmission rates and media,
medium
that carries rate
(rate 2) as well as
the
G^
the
example of Table
signals. In the
high speed data (service type
Each layer of
We denote
/
we can conveniently use
and
2)
same index
II,
/
the index
coaxial cables
to refer to the
/
transmission
= 2 correspnands
(medium
to
DSl
signals
2).
contains a replica of the nodes of the original (physical) network G:(N,E).
copy of the
node
original
i
in layer
as
/
(i,/).
The layered network contains two types
of
edges: transmission edges contained within each layer, and processor edges connecting the different
layers.
The transmission edge connecting node
medium
/
lines connecting distribution px)ints
transmission between
to
i
and
j.
If
each original feeder section
to-point, then the
medium
/
network
medium
(i,j)
/
i
i
to
and
/
j.
j
in layer
then layer
network G.
contains edges
/,
The flow on
is sectional,
in the physical
in layer
node
(i,j,/)
for
denoted as edge
this
/
(i,j,/),
edge corresfHsnds
contains an edge
On the other
hand,
to rate
(i,j,/)
if
represents
corresponding
medium
every pair of (original) nodes
can connect. Observe that the edges of this point-to-point network represent the
rather than physical, layout of
medium
/
lines; physically, the
20'
medium
/
/
i
/
is
and
point-
j
that
logical,
lines connecting distribution
points
i
and would be routed through
j
the ducts of the intermediate feeder sections. Transnnission edges
are either directed or undirected depending on whether
The edges connecting two
connecting node
(i,/')
in layer
located at distribution point
Table
II,
a processor
distribubon point
i.
i
/'
higher indexed layers since
to
that
edge from
Observe
different layers of
node
(i,/")
/'
/'
unidirectional or bidirectional.
is
traffic
<
/")
processors.
The processor edge
represents a traffic processor
traffic to rate /" traffic.
Thus, for the example of
to (i,3) will represent a type 3 traffic processor installed at
edges are always directed from lower indexed layers
that processor
we
/
G^ represent
in layer /" (with
compresses rate
(i,l)
medium
only permit
traffic
compression
(i.e.,
to
transformations from lower to
higher transmission rates).
Figure 5 shows the equivalent layered network representation for a local access planning
problem
that
is
defined over a (physical) network with a tree topology, and with transmission rates,
shown
transmission media, and processor types as
pairs
and coaxial cables are
in Table n. This figure
assumes
sectional, while fiber cables are point-to-p>oint
optic terminal to the switching center. (Thus, layers
1
and
2
that twisted wire
from each potenfial
fiber
have the same topology as the given
physical network, while layer 3 has a star topology.)
(Dur objective
is
to represent the local access
planning problem as a cost minimizing network flow
formulafion defined over the layered network. To complete the layered network flow formulation,
must
specify:
nodes,
(iii)
(i)
demand and
the units of
flow measurement,
the cost funcrions and capacities of different arcs,
conservation at each node.
(ii)
the
and
We consider each of these issues in
demands and
(iv)
we
supplies at different
the laws governing flow
turn.
Units of measurement
When
different processor types
demands, supplies, and flows and
of
medium / lines
have arbitrary
traffic
required. Thus, for the example of Table
II,
we can use
the
coaxial cable carries a fixed
number
number
of rate
of rate
/
/
the
we measure
ratios,
capacities of transmission edges in each layer
expressed in terms of the number of coaxial lines required for
Equivalently,
conversion
demand
/
in
for high
this service at
terms of the number
speed data
is
each distribution point.
channels as the unit of measurement in layer
channels (without loss of generality,
-21
the
we
/
since each
can assume that
each rate
/
line carries
one
terms of number of input
number
rate
lines; thus, the
incoming coaxial
of
from layer
channel). For processor edges,
/
lines.
we
gains formulation rather than a standard
at
each node. In Section 43,
measurement
for all layers,
flows and caf>acities in
capacity of a type 2 processor (in Table
Observe that
to layer; consequently, as
we measure
this
measurement scheme implies
minimum cost network
special situation in
by the
specified
that the units differ
becomes a network-flow-with-
elaborate later, the problem
we consider a
11) is
flow formulation that conserves flow
which we can use a common unit of flow
and transform the general network-flow-with-gains problem
to a
flow
conserving network flow formulation.
Demands and
supplies
Each node
number
of
(i,/)
medium
/
in layer
we can
corresponding
to a distribution point
lines required to satisfy the
expositional convenience,
Equivalently,
/
we
treat the line
i
has 'supply' djj equal to the
requirement for service type
/
at location
requirements at each node as a supply
also define these requirements to be
at that
node demands rather than
i.
(Note: For
node.
supplies.)
The
switching center node (0,L) in the highest rate layer L serves as the sink (or destination) node for
flows; other switching center nodes (0,0, for
Edge
/
For representing costs and capacities,
modeling existing
facilities, (ii)
new
(iv)
and
we
i
to
j
distinguish between four types of arcs:
those modeling the collection of
in layer
/
transmission and processing
edges representing existing
edges.
merely as transshipment nodes.
those modeling additions to existing capacities,
existing transmission or processing facility, say,
edge from node
L, serve
and capacities
cost functions
facilities,
<
The cost of flow on
traffic at the
B medium
/
lines
between nodes and
i
the expansion
any).
Similarly,
and new
facility
those modeling
j,
we introduce an
We represent additional
by uncapacitated expansion edges
facilities, if
those
switching center. To model an
with zero cost but a flow capacity of B.
facilities
(iii)
(i)
we model new
(parallel to the capacitated
facilities
edges corresponds
with uncapacitated
to the separable cost
we discussed
component
for transmission or processor expansion
Section
flows might incur joint transmission costs that dejTend on the combination of flows on
3.2,
all
and
installation. In addition, as
in
all
transmission edges (in different layers) that use the same physical feeder section(s). Finally, the cost
parameter for the processor edges {0^'J"),
for
/'
<
/",
incident to the switching center nodes
•22
depends on
our assumption regarding permissible transmission rates
particular,
if
for traffic entering the switching center. In
the switching center has the capability to receive multiple signal rates, each of these
edges has zero
cost.
processor edge
(0,/',/")
however,
If,
all
entering
must have the same transmission
traffic
rate,
the
carries the cost corresponding to a /'-to-/" processor located at the switching
center.
Flow conservation law
As mentioned
from
previously, because the unit of
measurement
we
different layers interact via the processor edges,
to relate the
consider
incoming and outgoing flows
some node
in layer 2 for the
(i,2)
at
differs
from layer
and flows
to layer,
require a network-flow-with-gains equation
each node and every layer. To
example of Table
II.
illustrate this constraint,
This node has three types of incident
edges:
transmission edges
(i)
pxjint
(
i i )
j,
2)
representing coaxial cables carrying DSl
distribution point
XI J =
at that location.
which we define
i
to
DSl
2, 3)
(i,
has
representing a type
signals;
= incoming
Yjj2
= outgoing
processor located
at distribution
point
i
representing a type 2 (DSl-to-DS2) processor located at
node
d^ incoming
DSl
(in the
layered network) have associated flow
processor (expressed in terms of the
1
number of incoming DSO
lines)
i;
traffic (or line
traffic
signals corresponding to service type 2 (high
as:
throughput of the type 2 processor located
Yjj2
1
and
Each node's incident edges
throughput of the type
located at
X2j =
and from distribution
i.
In addition, distribution point
speed data)
1, 2)
(i,
compresses DSO signals
an outgoing processor edge
variables
traffic to
i;
an incoming processor edge
that
(iii)
(i,
from
i
requirements) from
to
j
on
at
j
node
to
coaxial cables.
23-
i
i
(in
terms of number of DSl input
on coaxial
cables; and,
lines);
For convenience,
facility
we define all
on
four variables to include flow
existing as well as
edges. Observe that the line requirement variables Yjp and Yjj2 are 'directed' even though
the edge
{i,j,2) is
undirected (assuming coaxial cables are bidirectional). Xlj, X2j, Yjj2, and Yjj2 are the
would determine how much transmission and processor
decision variables in the model. Their values
capacity to
add
at location
i.
The flow conservation law equates
network
to the total
the total inflow of traffic at each
outflow at that node. The throughput Xlj of the type
one component of the node's inflow; since the throughput of type
input
to
(medium
we must
1) lines,
translate this throughput into
maintain consistent units of measurement
at
node
(i,2).
processor enables us to express Xlj in equivalent units of
medium
by the
definition of conversion ratio
Thus, the input-output flow equation
To simplify our
namely, node
more general
i
traffic to
in layer 2 for the
settings in
more lower
discussion,
rate layers
higher rates
I"
/'
>
<
/,
medium 2
at
measured
example shown
in layer
Table
in
via the type
ratio, call
1
it
processor
lines required
node
11.
in
number of
(i,2)
by
2 lines
p j, of the type
lines; in particular, the
is
a type
number
1
of
Xlj/pj (which,
1
processor with
has the following form:
(1)
formulated the flow equation
(1) for a specific
example,
However, the equation readily extends
to
receives the output of several processor types from one
/
and provides inputs
(1) differs
to several processor types that
from the flow conservation equation
network flow formulation because of the
local access
compress layer
/
'loss'
in a standard
minimum
factors p corresponding to traffic processors. Thus, to
networks with remote electronic devices, we require a network-
flow-with-gains formulation. In Section 4.3
traffic
The conversion
(i,2)
is
(i,2) is
I.
Observe that equation
model second generation
processors
of the layered
processor at node
lYij2 + X2i.
=
we have
which the node
1
1
node
an equivalent number of medium
number of output
the
pj, is
lYji2+ Xypi+dj;
cost
node
2 lines of concentrated traffic arriving at
XI: input lines).
or
expansion/new
we show
conversion ratios for different processor types,
equivalent base rate channels instead of
number
that,
we
under
certain
can define
all
assumptions regarding the
decision variables in terms of
of lines for different media. This alternative
24-
definition enables us to use a standard
network flow problem formulation that conserves flow
each
at
node.
Extensions
The layered network representation extends
transmission
in
Table
II),
could either
medium
by associating
representation
for different
is
to
model
between layers and transmission
alternative transmission media, or
a different layer for each transmission
appropriate
distribution pxDints without
when we can
directly interface
interlayer edges that connect
media. In this case,
we
combination for both
these
its
which each
we assumed
rates,
but introduce parallel
decompose the problem
(ii)
rate-medium combination. The
two
different
media
at
first
intermediate
any additional, expensive equipment. For situations requiring interface
equipment, the second representation enables us
on
settings in
media overlap. To represent these overlapping ranges, we
retain the correspondence
edges within each layer
further
more complex problem
can handle a range of transmission rates (rather than the single rate
and the ranges
(i)
to
also
to
model the cost of
two layers corresponding
assume
to the
this
equipment by associating a cost
same transmission
rate but different
that each traffic processing device requires a specific
input and output. Depending on the application context,
two representations. The layered network can
also
model
rate-medium
we might
use a mix of
different processor technologies as
parallel interlayer edges.
Within the framework of
this layered
network representation, the general
local access
network
planning problem becomes:
Minimize
total
separable + joint transmission and processor expansion/installation cost
subject to
network flow with gains constraints
non-negativity constraints, for
and
for all
nodes
(i,/)
of the layered network,
and
transmission and processor edge throughput variables (X
Y).
Observe that
originating at a
all
(1),
this
formulation models multiple processing steps in series,
node might possibly be compressed
at
i.e.,
traffic
two or more downstream nodes before reaching
the switching center. Also, the formulation permits bifurcated routes,
-25-
i.e.,
two customers connected
to
the
same
distribution point might
communicate with the switching center
processing steps. To prevent this bifurcation,
traffic
we
via different routes
and
require a different formulation that distinguishes the
originating at different nodes. This latter formulation can also incorporate various proximity
restrictions.
Depending on the
variables
and /or constraints. For
similar to Figure 3b
(Zm^ =
0)
specific application context, the formulation
we
instance,
we can model
by introducing additional binary
install a
new
type
m processor at node
objective function coefficient for variable Zm^,
i.
might contain additional
a fixed plus variable processor cost structure
variables Zrrij denoting whether (Zmj = 1) or not
The
fixed cost of the processor serves as the
and the formulation contains additional
constraints to relate this location variable to the processor throughput variable
Nemhauser and Wolsey
model
(1988)).
Similarly, the formulation
Xmj
'forcing'
(see, for instance,
might contain additional constraints
to
certain policy restrictions (e.g., proximity rules, contiguity requirements (see Section 4.3)).
The general network flow formulation with gains and with non-separable edge
problem dimensions might be quite large
difficult to solve, especially since the
networks. Therefore, models proposed in the literature
make
that are inherently true for specific application contexts
development. These assumptions lead
exploit. Next,
we
to special
several additional assumptions,
and others
problem structures
new
some
that facilitate algorithmic
that solution algorithms
planning models as special cases
local access
can
we show how
in the general layered
to
view
network
framework.
3.4
Possible Modeling Assumptions
The problem formulation described
access network planning models that
we
in Section 3.3 defines a basic
describe in Section
4.
Within
modeling approaches are possible, each characterized by an additional
assumptions are motivated by three
factors:
26
is
for practical local access
discuss several categories of possible assumptions. In Section 4
various existing and
cost functions
framework
this
for all the local
framework, different
set of
assumptions. These
They make the model computationally
(i)
tractable.
For instance, certain
problems defined over general networks become pwlynomially solvable
They
(ii)
(iii)
They
NP -hard
location
for tree networks.
reflect uncertainties in the technology.
and
arise because of differences in corporate jx)lides
practices.
For instance, some local
operating comjjanies might emphasize non-bifurcated routing to reduce the burden of
managing/rearranging the network.
We have
identified the following six areas that cover
most of the differences
in
modeling assumptions.
Selecting different combinations of assumptions in these areas gives rise to different models.
(a)
New
Models
versus expansion projects:
that
apply only
designing
to
capacities) are typically easier to solve than expansion planning
switching and transmission resources. As
we mentioned
new networks
models
in Section 3.3,
(with no existing
that account for existing
expansion planning models
with existing capacities require an additional set of (capacitated) parallel edges, and associated
decision variables and constraints that
(b)
make
the
model more
difficult to solve.
Network topology: Some models assume that the physical network has a tree structure. For
example,
we show
in Section 4.3
how
this
assumption makes a version of the planning model
solvable in pjolynomial time (because tree networks have a unique path from each distribution
jx)int to the
Many models assume
switching center).
point media connecting the
traffic
that all
compressed
traffic
requires point-to-
processor directly to the switching center. Effectively, this
assumption implies that the network layer corresponding
to
compressed
traffic
has a star
topology, with the svvitching center as the central node.
(c)
Backfeed vs unidirectional flows:
movement away from
toward the switching
Some
tree
network models incorporate
the switching center, while others
center.
upstream distribution points;
assume
Without backfeed, a processor that
that
is
all
backfeed,
i.e.,
traffic
flows must be directed
located at
node
this restriction limits the set of solutions that the
i
can serve only
algorithm must
consider.
(d) Processor
and transmission
cost functions:
In Section 3.3,
we
discussed generic cost functions for
expanding processor and transmission capacity. These cost functions can be specialized
-27-
in various
ways.
First, all
we
the models
discuss in Section 4 ignore joint costs,
two or more transmission media. Further,
functions of capacity,
we
When we include fixed
problem becomes much more
Routing
restrictions:
Some
at
sections, the
may
processor, then
all
to solve the local
enforce a non-bifurcated routing restriction
from each distribution point must follow the same route
This policy
jx)ints).
facilitates
section,
entering
traffic
must be processed
(i.e.,
network management and
at that
if
a
node contains
If
we assume
processed at most (or exactly) once, the
traffic
Most
from various nodes
that the traffic
Most
local access
to processors) decreases significantly, thus
(ii)
we
consider in Section 4.3
permits backfeed,
(iii)
assumes
that all
(i)
in this
traffic
originating at each node.
manner we can generate a diverse
modeling approaches
set of
By
-28-
assignment of
six
dimensions. For
that the physical
network
enhanced (high frequency) media are
and
(v)
considers at most one
selecting different combinations of assumptions
we outline some possible
network planning problem, and
using this framework.
can be
level of traffic processing.
assumes
models. In the next section,
for the single-p)eriod local
pjoint
traffic
reducing model complexity.
employ only one
point-to-point to the switching center, (iv) prohibits bifurcated routing,
processing step for
(i.e.,
network planning models can be differentiated along these
example, the tree location model that
has a tree structure,
from each distribution
number of f)Ossible homing patterns
existing second generation local access networks
a
node.
(0 Single versus multiple processing steps: Chir formulation of Section 3.3 permits multiple
processing steps in sequence.
use the
and the same nodal processors
maintenance. Another possible routing restriction has the following form:
traffic
linear
charges for cable and processor expansion, the
same transnussion medium on each
intermediate distribution
and are
by
difficult to solve.
telephone companies
Sf)ecifying that all the traffic
same feeder
separable costs are purely variable,
can apply a network flow model (possibly with gains)
network planning problem.
(e)
if all
costs that are shared
i.e.,
differentiate
them
Modeling Approaches for Local Access Network Planning
4.
some modeling approaches
This section describes
planning. For each approach,
differentiate
we
network
use the framework of Section 3 to identify the assumptior\s that
from other models, and
it
for single-period local access
briefly indicate the solution strategy.
and
existing models, while Sections 4.2
4.3 cover
two
Sections 4.1 reviews
alternative approaches. Section 4.1 considers the
general class of models called concentrator location models, and briefly discusses two other local access
network planning methods. In Section
4.2,
we discuss how
to view,
with appropriate assumptions, the
layered network model introduced in Section 3 as a fixed-charge network design problem and indicate
some
tree
]x>ssible solution approaches. Section 4.3 describes a
networks;
this
models
All the
single service type
Jack,
and Shulman
using
at
model can be solved
most two
to a single layer
and
we
levels in the layered
representation
network
p>oint-to-p>oint; thus,
(in
we can
represent these models
many, cases we can further reduce the representation
all
each concentrator
models ignore
by adding the
where
joint costs,
is
high
all
directly connected to the switching center
and assume
that traffic processors
and
have separable fixed and/or variable (volume dependent) costs (instead of more
each processor
processor cost at that
of determining
dynamic programming model of Helme,
because of the special cost structures). These models also assume that
facilities
(i.e.,
that applies only to
review, with the exception of the network design model, consider only a
general non-linear cost functions). Observe that,
for layer 2
model
when designing new networks.
a single level of traffic processing (the
network G^. Also,
transmission
restrictive
(1988) considers multiple processors in series). Thus,
frequency media are
in layer 2 of
that
efficiently
more
site.
ignore joint costs and assume a star topology
directly connected to the switching center),
cost of the point-to-point high frequency
Consequently, the
to locate
selected processor sites.
is
when we
new
local access
processors, and
Most models profX)sed
networks, and do not account for existing
in the literature
-29-
can simplify the
medium from
each
network planning task reduces
how to connect all
facilities.
we
site to
to the
the
problem
the distribution p>oints to the
apply only
to the
design of
new
4.1
Concentrator location and other design models
4.1.1
Centralized Teleprocessing Desig n
In the 1960's
networks
to
and
1970's centralized teleprocessing systems
were quite common, and configuring
connect users of the system to the central computer was an important design issue (see
Boorstyn and Frank (1977), Chandy and Lo (1973), Chandy and Russell (1972), Direlten and Donaldson
(1976),
Kershenbaum and Boorstyn
McGregor and Shen
(1977)).
(1975),
Kershenbaum and C3iou
These networks typically consist of
(1974), Mirzaian (1985),
many
and
(usually 100 or more)
geographically dispersed terminals that are coruiected to a central computer via communication
The
central
computer provides computational resources, and
acts as a switch to connect the terminal to a
wider distributed computation network.
When
located in clusters, using concentrators to
combine the communications
increase the utilization of communication lines)
planning models for these problems typically
potential concentrator sites,
and
the
number
becomes
start
lines.
of terminals
is
large,
traffic
and
the terminals are
from several terminals
The
a cost-effective interconnection strategy.
with a preselected
select concentrator locations
and
set of
(to
nodes of the network as
from the terminals
direct connections
to the concentrators.
The teleprocessing network design problem
number and
(terminal layout).
and
The
(3)
determining
how
to
connect every concentrator
similarity with the local access
treat terminals as distribution points,
teleprocessing network design
location
main components:
(1)
specifying the
location of concentrators (concentrator location), (2) assigning terminals to a concentrator
(terminal assignment),
when we
consists of three
Location Problem
central
computer as the switching
in the literature first
and terminal assignment decisions using a
assigned terminals
network planning problem becomes apparent
and the
methods proposed
to its
single model, called the Capacitated Concentrator
(separable) assignment costs. Subsequently, a terminal layout
first
-30-
concentrators by
terminal-to-
phase. (See, for example,
Rousset and Cameron (1986).)
relates to local access
to
method configures the
concentrator interconnections based on the assignments suggested by the
it
Most
determine the concentrator
(CCLP), that approximates the actual costs of connecting terminals
We first consider the CCLP as
center.
network planning.
Capacitated Concentrator Location Models
Given the
fixed installation costs
and
and the terminal-to-concentrator connection
locations,
and assigns each terminal
to
capacities of
new
concentrators at each jx)tential
(or assignment) cost, the
site,
CCLP selects concentrator
one of the selected concentrators
in
order to minimize the
total
concentrator and temninal assignment costs, subject to concentrator capacity constraints. The standard
CCLP formulation assumes a
single servde type,
and provides
terms of our local access network planning framework,
for only
CCLP models
one
level of concentration. In
typically
make
the following
additional assumptions:
(1)
CCLP models
Network topology:
design: each terminal
is
effectively
directly connected to
layers: the first layer is a
its
(2)
site
complete network v^th one direct connection from every terminal
is
a star
to
network connecting each potential
with the switching center.
Cost structure and capacities:
only to the design of
new
Essentially, all the
networks,
i.e.,
CCLP methods proposed
in the literature
apply
they do not account for existing transmission or cable
Most models incorporate only
capacities.
network
assigned concentrator, and each concentrator has a
each potential concentrator, and the second layer
concentrator
a double star topxjlogy for the final
The equivalent layered network representation contains
direct connection to the central computer.
two
assume
a fixed cost for each concentrator (that possibly includes
the cost of the concentrator-to-computer connection) and each terminal-to-concentrator connection,
and assume
The
total
capacity.
that the capacity of each
demand
Routing
Some models
restriction:
restrictions
by
concentrator
is
prespecified rather than expandable.
for all the terminals assigned to a particular concentrator
specialize the capacity constraint further
have equal demand; thus, they
(3)
new
limit only the
CCLP models do
number
must not exceed
by assuming
this
that all terminals
of terminals assigned to each concentrator.
not jjermit bifurcated routing. They can accomodate proxinuty
setting very high assignment costs for prohibited terminal-to-concentrator
assignments.
•31-
An
enhanced version of the
CCLP
model, called the multilevel concentrator
designs a hierarchical structure, where concentrators from one level
home on
location problem
concentrators at the next
higher level, and so on.
Even though most CCLP methods
implicitly
asssume a single concetrator type,
we can
incorporate multiple concentrator types (each with a different capacity, for instance) by replicating
the nodes corresponding to potential concentrator locations, and associating a different concentrator
type with each copy.
variables
Some models
(e.g.,
Pirkul (1986)) treat concentrator capacities as decision
by incorporating volume-dep)endent concentrator
assumes that the cost of connecting
costs.
a terminal to a concentrator
While the standard
CCLP model
does not depend on which other
terminals that concentrator serves (by assuming direct terminal-to-concentrator connections),
enhanced models
are connected
(e.g.,
Woo and Tang (1973)) indirectly account for the cost economies when terminals
by a spanning
Algorithms for the
heuristic local
tree.
CCLP
belong
to three
broad
is
(1989),
Efroymson and Ray
and Hamburger
Nemhauser
(1972), Erlenkotter (1978),
(1963), Sa (1969), Spielberg (1969)).
intractable, researchers
CCLP,
which has been studied
structurally related to the plant location problem,
extensively in the literature (Comuejols, Fisher, and
location
classes: optimization-based algorithms,
improvement methods, and clustering techniques.
The CCLP
Wolsey
some
(1977),
Comuejols, Nemhauser, and
Feldman, Lehrer, and Ray
Even though
this
problem
is
(1966),
Kuehn
theoretically
have successfully solved some relatively large-scale uncapacitated plant
problems optimally (Erlenkotter
(1978), Geoffrion
and Graves
For the
(1974), Korkel (1987)).
plants correspond to concentrators and customers to terminals, with the additional restriction
plant capacities.
Woo and Tang (1973)
propose a
CCLP algorithm based on
on
plant location solution
methods. Pirkul (1986) proposes an optimization-based solution method using Lagrangian relaxation
for
an enhanced
CCLP model
that incorporates concentrator sizing decisions (by including variable
concentrator costs). The method dualizes the terminal-to-concentrator assignment constraints of a
mixed integer programming problem formulation, and
subproblems corresponding
original problem.
to
iteratively solves single constraint 0-1
each concentrator location
To obtain good lower bounds,
to
knapsack
generate lower bounds on the cost of the
a subgradient optimization
method
iteratively
changes
the Lagrange multipliers; and, at each iteration, a heuristic procedure uses the Lagrangian subproblem
-32-
solution to construct a feasible solution, which provides an upper bound.
computational results for problems with up
to
The author
100 nodes and 20 concentrator
between the best upjjer and lower bounds vary from 0%
sites;
repxjrts
the percentage gaps
For a discussion of other optimization-
to 7.7%.
based and heuristic approaches to the capacitated plant location problem, see Comuejols, Sridharan,
and Thizy
(1987).
Polyhedral methods offer a potentially promising approach for solving capacitated plant and
concentrator location problems. These nnethods attempt to refine linear
programming
(i.e.,
polyhedral)
approximations (relaxations) of these problems by adding (strong) valid inequalities in a cutting plane
approach
(see,
Nemhauser and Wolsey
structure of plant
(1988)).
Surprisingly
and concentrator location problems
(for
Wolsey (1984a,1984b), Ward, Wong, Lemke, and Oudjit
Magnanti
some
(1988),
known about
little is
the polyhedral
partial results, see Barany,
LenJce and
Van Roy, and
Wong (1989) and Leung and
(1989)).
Local improvement procedures for
CCLP
start
with an
initial set
of concentrator locations
and
terminal assignments, and attempt to sequentially decrease the total cost by performing myopic
Add
changes. The
Add
heuristic
and the Drop
heuristic are
algorithm (Kuehn and Hamburger (1963))
is
two couinrxsn
a perturbation
local
improvement nttthods. The
method
that iteratively evaluates the
net savings (savings in terminal assignment costs less the cost of an additional concentrator) that accrue
by adding each unused
method
site to
the current set of concentrator locations.
terminates. CHherwise, the
reevaluates net savings for
iteratively
all
most
remaining
cost-effective site
sites.
is
added
If
no
site
produces net savings, the
to the current set,
and the method
Conversely, the Drop algorithm (Feldman
removes currently selected concentrators
until
no further cost reduction
Researchers have also proposed combined methods that alternate between
is
et al. (1966))
possible.
Add and Drop
phases.
The
ways.
One possible
initial
solution conasts of installing a concentrater at every potential location. Rousset
and (lameron
(1986)
have proposed a heuristic
starting solution for the local
improvement procedure can be generated
to
for each terminal, the difference in
concentrators.
possible
(i.e.,
It
in several
determine the terminal assignments; the procedure
first
calculates,
assignment costs between the nearest and second nearest
then sorts the terminals in decreasing order of this difference and assigns each one,
subject to the concentrator capacity constraints), to
-33
its
nearest concentrator.
if
Several researchers have proposed heuristic methods for the
(see, for
Dhas
CCLP based on clustering
concepts
example, McGregor and Shen (1977), Schneider and Zastrow (1982), and Konangj, Aidja, and
(1984)).
McGregor and Shen study the
single level concentrator location problem, while the other
papers address the multilevel problem. Konangi
et al.'s
method assumes
that the terminal-to-
concentrator assignment costs are proportional to the Euclidean distance between the two locations, and
do not vary
concentrator costs
clusters the terminals into
significantly
by
location. For each concentrator level, the
two groupjs based on geographical proximity, and
method
first
locates a concentrator at
the center (or the potential site closest to the center) of each cluster. Ousters are then successively split
if
savings result from this splitting, and concentrators are relocated at the centers of the
When cluster flitting does not give any further savings, a cluster merging
reduce cost by combining previously defined
for
one concentrator
current level as the
level, the
new
new
clusters.
procedure attempts to further
completing the clustering and merging steps
clusters. After
algorithm considers the next
level, treating the
concentrators in the
terminal locations. The authors report computational results for problems with
over 200 terminals.
Clustering methods can also exploit the underlying geometry. For example, under certain
circumstances, partitioning the plane into hexagonal cells with a service center located at (or near) the
center of each cell
is
approximately (or asymptotically) optimal. Several researchers have
established results of this nature (Fejes-Toth (1959), Fisher
Magnanti
(1988),
and Papadimitriou
and Hochbaum
(1980),
Haimovich and
(1981)).
Terminal Layout problem
Given the assignment of terminals
to concentrators, the terminal layout problem seeks the best
network topology connecting each concentrator
to its
assigned terminals. This model makes the
following assump>tions:
(1)
The
original network, containing
that is selected
must have a
all
available lines, has a general structure; the final topology
tree structure.
compression, the terminal layout problem
Effectively, since
is
it
does not consider
traffic
defined over a single layered network.
-34-
(2)
Each edge of the network carries a fixed charge; the model does not ctccount
and so
(3)
edge
for variable
costs,
effectively ignores cable sizing decisions.
The model incorporates only
multidrop
lines.
certain special types of capacity constraints that apply mostly to
(A multidrop
line is
analagous
to a route in the feeder
primary cable enunating from the concentrator
to
which auxiliary
network.
It
consists of a
lines attach individual
terminals at intermediate points.) These constraints include: (a) degree constraints, that specify an
upper
limit
on the number of incident
constraints, that restrict the
concentrator,
and
number
links at a braiKhing node, or at the concentrator, (b) order
of intermediate branching
that limit the total
(c) load constraints,
nodes between any terminal and the
number
of terminals that are connected
via a multidrop Une.
Severed authors have proposed heuristic
and Williams
and Sharma
(1966),
connecting each terminal directly
(1983)).
methods
for the terminal layout
problem
The Esau-Williams procedure begins with a
to the concentrator.
The method performs
a series of
(e.g.,
star
Esau
network
edge
interchanges to monotonically decrease costs while satisfying the multipoint line capacity restrictions,
and terminates when no further
cost reduction
is
possible.
A sp>ecial case of the termi lal
layout problem
that has received considerable attention in the optimization literature is the capacitated minimal
spanning
tree
problem. This
problem seeks the minimal sparuiing
tree connecting the concentrator to the
terminals, subject to degree constraints at the concentrator. Several optimal
heuristic algorithms
Gavish
(1983),
have been proposed
Gavish and Altinkemer
for this
(1986)).
problem
example, Chandy and Lo (1973),
Recently, researchers have
approaches (Hall (1989), Hall and Magnanti (1988))
4.1.2
(see, for
and optimization-based
for capacitated
begun
spanning
tree
to
study polyhedral
problems.
Other models
We now describe two other local access network design nxxiels, both of which apply only to the
design of
method
new
is
networks,
i.e.,
they
a heuristic proposed
do not accoimt
by Luna,
network planning problem which
we
Ziviani,
call
for existing cable or processor capacities.
and Cabral (1987)
first
to solve a variant of the local access
the service section connection problem.
-35-
The
The second method
is
a
dynamic programming algorithm developed by Hebne,
Jack,
and Shulman
network
(1988) for a tree
nrkodel that prohibits backfeed.
Service Section
Luna
We
Connection Model
et al. (1987) consider the following variant of the local access
network planning problenru
are given a partition of the set of distribution points into S subsets called service sections. Each
section contains a
service section,
number of potential
and
concentrator
sites.
We must select exactly one site from each
this site will serve all distribution jjoints within the section.
network has contains
all
The given physical
permissible interconnections between the potential concentrator sites and the
switching center. Each arc of the network carries a fixed cost (for using that arc) as well as a variable
cost that
depends on the volume of
traffic that is
routed on that arc; concentrators have fixed costs that
might vary by location. The planning problem consists
service section,
and
(ii)
of: (i)
selecting
designing a subnetwork that connects
all
one concentrator
the selected concentrator sites to the
switching center. The objective of this service section connection model
CCLP,
variable) arc costs plus the concentrator costs. Unlike the
from each
site
minimize the
is to
total (fixed
+
the service section connection problem
explicitly considers the topological design decisions for connecting concentrators to the switching center.
However, the model ignores the interconnections within each
service section,
i.e., it
does not consider
the topological design decisions for the subnetwork connecting the distribution fx)ints within a service
section to the selected concentrator site in that section.
concentrator
cost of this
site,
Effectively,
it
assumes
that, for
each potential
the designer has predetemnined the distribution jX)int-to-concentrator connections; the
subnetwork may be incorporated
in the concentrator cost for that site.
The model does not
consider multiple services, multiple concentrator tyjjes or different transnrdssion media, and
model economies of scale in concentrator and transmission
The layered network representation
layer containing an
suffices
augmented version
of the service section connection
A
problem consists
of a single
single layer representation
because the model considers only the topological design decisions for compressed
traffic
For the layered network representation,
traffic).
does not
costs.
of the given physical network.
ignores the flow pattern for the base rate
it
(and
we augment
the physical network as follows:
For each service section,
demands
we add
a super
node whose demand equals the sum of
for all distribution points in that service section.
-36-
We connect each
the
super node
to
every potential concentrator
away from
site in
The
the suf>er node.
the corresponding section using an edge directed
fixed cost of this
All the edges of the given physical
concentrator.
edge equals the cost of the
network (interconnecting potential
concentrator sites and the switching center) carry the fixed and variable costs specified
in the original problem;
all
original
nodes
(i.e.,
potential concentrator sites) serve as
transshipment points. The switching center node has demand equal
requirements in
all
service sections.
Since the model considers only
show
that
some optimal
new
without any capacity constraints,
facilities
solution to the layered network problem assigns each super
concentrator site in the corresponding service section; thus,
the
model from
we do not need
Because of the
spjecial (fixed
the fixed-charge network design
Observe
that this
potential concentrator site
Section
2.2),
node
possible to
to exactly
one
explicit constraints to prevent
model
model can
and so
will not bifurcate
plus variable) cost structure, this model
that
we
present in Section
do not belong
to the
do not meet
same
network as individual customer
is
any
a special case of
4.2.
implicitly incorporate proximity restrictions,
the service sections so that distribution f)oints that
the nodes of the
it is
selecting multiple concentrator sites in each service section. Also, the optimal topology
of the service section interconnection network will have a tree structure,
traffic flow.
to the total line
we
i.e.,
can choose
the proximity criterion with respect to a
service section as this
site.
Also,
if
we
interpret
locations, the service sections as allocation areas (see
the potential sites within each section as potential distribution points, and the switching
center as a concentrator, then
Luna
et al.'s
model applies
to the
design of the distribution network for
each concentrator, rather than the feeder network serving the switching center.
Luna
and projx)se
formulate the service section connection problem as a mixed integer program,
et al. first
a heuristic
method
to solve
it.
The
heuristic starts
each service section, select the distribution point that
is
by constructing the following design:
for
closest to the switching center; and, find the
shortest path tree (using only the variable arc costs) connecting the switching center to each selected
concentrator
solution.
it
site.
A
local
The method
improvement procedure then attempts
iteratively evaluates the cost savings as
it
to
improve
interchanges concentrator
performs profitable interchanges sequentially until no further savings
computational results for 3 problems ranging in size from 18 nodes, 54
-37-
this initial feasible
result.
arcs,
sites
or arcs;
The authors report
and 7 service
sections, to 263
nodes, 752 arcs, and 117 service sections. However, they do not evaluate the quality of these solutions
since the
method does not generate any lower bounds or alternative
Tree Network Model without
Helme et
al.
heuristic solutions.
backfeed
(1988) propose a
dynamic programming method
for a special class of local access
network planning problems. The method permits multiple processors
transmission
medium and
applies only to the design of
in series,
new networks.
It
but assumes a single
assumes
that the given
has a tree structure, does not permit backfeed, and does not account for economies of
processsor has a fixed cost (that
costs.
The solution method
node of
is
may
vary by location); transmission
facilities
scale.
network
Each
have only variable
based on a recursive procedure that exploits the tree structure. For each
the network, the recursive relationship determines the cost of connecting that
node
to the
switching center, for various possible combinations of downstream processor locations.
Helme et
al.
also propose a
Drop /Add
heuristic for a
more general
local access
network
planning model that considers general network topologies and multiple processor types in
fjermits bidirectional transmission
(i.e.,
backfeed) on links, and incorpyorates existing capacities as well
as fixed and volume-dependent processor costs.
assumes
Drop
The method ignores
that cable costs are directly proportional to the
heuristic starts with
an
initial
series,
design containing
number
all
fixed costs for cable expansion;
of (additional) cables required.
it
The
processor types at each node, and
successively eliminates processors to reduce the total cost. Because of the linear cable expansion cost
structure, the authors are effectively able to use a shortest path algorithm to
compute
the total
transmission cost for any given processor configuration.
4.2
Network Design Model
The general layered network framework introduced
in Section 3
can be specialized so that
it
takes the form of the fixed-charge network design problem which arises in a variety of distribution
planning, manufacturing and telecommunications contexts. Given the
destination
p>airs,
and
fixed
and variable
demand between
various origin-
costs for each arc of a network, the (fixed charge)
network
design problem involves selecting a subset of arcs, and routing the various commodities (subject to
-38-
conservation of flow, without gains or losses, at each node) over the selected arcs in order to n-iinimize
the total fixed plus variable arc costs.
The capacitated version
of this
problem accounts
for arc
capacity constraints as well. The network design problem generalizes several well-known optin-dzation
location, shortest path, Steiner tree, traveling salesman,
models including the plant
spanning
tree problems.
and solution methods
for the
In this section,
network framework
Magnanti and
we
and minimal
Wong (1984) and Minoux (1989) describe various applications
network design model.
demonstrate how, in certain circumstances,
for the local access
to cast the general layered
network planning problem as a fixed charge network design
model.
The
first
assumption concerns the transmission and processor cost structures The network design
.
model ignores joint costs between various transmission media, and assumes, as
II,
between transmission
a one-to-one correspondence
preferred transmission
medium
for
each transmission
rates
rate.
and media,
i.e.,
in the
example of Table
the planner preselects a
We also assume that all
processor and cable
installation/expansion costs are piecewise linear, consisting of fixed and variable components.
The network design model's second assumption concerns
processor types. Recall from our discussions in Section
the conversion ratios for different
3.1 that, in general,
certain input rate (or frequency), transmits output at a higher rate,
p (defined as the ratio of input to output lines).
assume
by employing
rate as type I's
The
and has a specified conversion
the network design
ratio
model formulation, we
that the conversion ratios for different processor types are compatible in the following sense.
Consider three processor
either
To develop
each processor requires a
tyf)es labeled 1, 2,
a type
output
1
rate),
and type
or
and
3,
and suppose we can compress
2 processor in series
by using
(i.e.,
rate l^ traffic to rate l^
the type 2 processor has the
a single type 3 processor (v^th input rate /^
same input
and output
conversion ratio compatibility assumption requires that the three conversion ratios
must
rate
l-^).
satisfy
the following equation:
P3 = Pl'P2'
where pj^ denotes the conversion
the rate l^
medium always
ratio for a type
m processor.
In other words, the
require exactly (x + P3) lines in the rate
the compression
was achieved using
Effectively, this
assumption permits us
a type
1
and type
/j,
messages on x
medium, regardless
of
lines in
whether
2 processor in tandem, or a single type 3 processor.
to associate a single conversion factor, call
-39-
it 5j,
with each
layer
/
We define
of the multi-layer network.
at the base rate (the
this factor as the
number
of channels (or circuits or lines)
lowest transmission rate, corresponding to the index
accommodate. In the example presented
in Table
/
the three processors
11,
=
1) that
do not
compatibility assumption since the product of the conversion ratios for processors
2)
does not equal the conversion
ratio (P3
each transmission rate enables us
to
each type
/
line
satisfy the
1
and 2(p^»p2 = 12x
= 48) of the type 3 processor. The single conversion factor
measure
equivalent base rate channels (rather than the
all
the traffic in every layer in terms of the
number
can
number
8j for
of
of lines of the corresponding medium). This
common traffic measurenvent unit preserves conservation of flow at each node, i.e., we do not require the
conversion factor p in the left-hand ade of constraint
compatibility assumption,
we can
(1) in
Section 33. Thus, with the conversion ratio
transform the network flow with gains flow equation
(1) to
a
standard network flow conservation constraint.
Ajsart
from these two assumptions, the network design model incorporates
the general layered
network framework described
network topologies, multiple service
tyjses (as
in Section 3.3.
In particular,
it
all
other features of
can handle general
long as these do not impose unique processing
requirements), sectional and pnaint-to-point cable types, econonnies of scale in processor and transmission
cost functions,
and existing transmission and processor
bifurcated routing.
If
capacities.
the cost functions are piecewise linear
It
and
also permits backfeed
and concave, and
if
the network does not
contain any existing capacities, the model reduces to an uncapacitated network design problem.
Existing resources
and non-concave
cost functions introduce arc capacities.
We first describe the cost
parameters for the uncapacitated fixed charge network design model with no existing processor and
transmission capacities, and with a fixed plus linear cost structure for each transmission and processing
facility.
Subsequently,
we discuss extensions to model
general piecewise linear cost functions and
existing capacities.
Uncapacitated Network Design Model
The transmission and processing costs
network. Let
layer
/"
Hj/'j--
signals,
are represented as edge cost functions in the layered
be the fixed cost of a processor located
and
let
Vj^.^..
be
its
and variable
node
i
that converts layer
/'
signals to
variable cost (p>er unit capacity, expressed in base rate channels).
Thus, for a processor with capacity of x (base
associate these fixed
at
rate) units, the total cost is Hj^.j..
costs with the processor
-40-
edge
(i,/',/")
+
We
Vj^.^"*x.
connecting layers
/'
and
/".
(Note that a per unit cost of
input
(medium
unit costs for a
vji.i..
medium
/
cost parameters define the fixed
medium
and variable
line
/
from
from node
to
i
j
(0,/')
and
multiple signal rates can enter the switching center. Otherwise
a
rate), the
processor edge
(0,/',/")
i
to
costs 5^*Cjj/)
costs for the transmission
edges connecting the switching center nodes
same transmission
implies that the processor cost per
i
represent, respectively, the fixed
Cj:;
(sectional or point-to-point) connection
cost of q:i implies that each additional
inter-layer
and
Similarly, let F^y
line is 5j*Vj^-j".)
I)
node
for the processor located at
(0,/")
(if all
edge
node
j.
(Again, a per unit
These two transmission
As
(i,j,/).
before, the
carry zero cost, assuming that
entering
traffic
must have the
and variable costs corresponding
carries the fixed
to
processor located at the switching center.
/'-to-/"
With
this set of
flow at each node, and
model parameters, the uncapadtated network design solution
satisfies all
demands
at
minimum
transmission edge
(i,j,/)
implies that the
Similarly, the flow
.
number of input
on processor edge
node
lines) of a processor at
Observe
signals.
number
that the transmission
i
of
medium
/
x^:^
units along
lines to install in feeder section
divided by
(i,/',/")
that conserves
plus flow costs corresponds to the
total fixed
optimal local access network plan. In the optimal network design solution, a flow of
Xj:j/5j
and per
is
gives the capacity (in terms of
5j.
that transforms layer
(i,j)
/'
input signals to layer
T'
output
edges might be either directed or undirected, and the model
permits multiple processors in series.
It is
possible to enrich this network design
economies of scale
in processor
and transmission
describes the cost of installing a
medium
R breakpoints.
r
contains
decreases from
Cj.
of
Fj.
and
R
r.
if
in various
in Figure
and
Fj.^^
parallel arcs
Bj.,
(i,j).
arc
(i,j),
and x
edge since
lies
this
by
in this figure
In general, this cost function
and the slope of the
cost function
denote the y-intercepts of the two line segments
between nodes
Cj..
i
and
j
in layer
/.
The
r"^ parallel arc carries a fixed charge
Because the overall transmission cost function
is
solution will automatically select the appropriate line segment even without
i.e., if
can model
We can incorporate this cost function in the network design model by
a vciriable cost of
constraints,
we
Suppose the function shown
6.
occurs at a capacity of
Fj.
ways. For instance,
these economies can be approximated
cable on feeder section
/
to c^^| at this point. Let
that define breakpoint
introducing
Breakpoint
costs
shown
piecewise linear, concave cost functions as
model
concave, the optimal
any
explicit capacity
the optimal local access network solution entails installing a capacity of x units
between
Bj.
and
edge minimizes
Bj.^-j,
the network design solution will route
total cost,
among
all
all
x units on the r*"
parallel edges, for the x units of flow.
-41-
on
We can
similarly nvxlel piecewise linear, concave processor cost functions
by introducing
parallel processor
edges. This type of cost function might represent the composition of alternative multiplexing or
concentration technologies with progressively increasing fixed costs but decreasing variable costs. As
mentioned
network formulation does not readily accommodate proximity
in Section 3, the layered
restrictions.
Certain properties of the optimal uncapacitated network design solution (with concave edge
cost functions
and no
existing capacities) have special significance for the local access
problem. For instance,
either processes all
both. Thus,
we
is
it
easy
incoming
to
show
traffic
or routes
cannot have both type
/
traffic
In particular, consider the route for, say, layer
on
an optimal solution
that
we
that
i
to
j
also
undergoes
the traffic to another
all
leaving node
1
this route containing, say, a 1-to-/ processor.
intermediate nodes from
some optimal network design
that, in
traffic
and a
/-to-/'
originating at
node
Then, the
1-to-/
i,
traffic
j.
solution, each
the
same
node
(i,/)
but not
level,
processor located at node
As
node be
the
node i and
all
Let
i.
originating at
processing at node
does not use bifurcated routing, and
define in Section
node on
network planning
j
a corollary, the
first
i.
node
problem has
satisfies the so-called contiguity
property
4.3.
Capacitated Netw^ork Design Model
The
uncap>acitated network design
existing capacities.
model applies
to local access
design problems with no
Modeling networks with existing processor and transmission
capacities, or with
piecewise linear but non<oncave cost functions requires explicit capacities on the edges of the layered
network. In particular, suppose the network already contains x type
we add
represent this capacity,
a parallel arc connecting
nodes
zero fixed and variable costs, but has a capacity of x*5j (basic
for
expanding the type
shown
/
transmission
in Figure 7a. Figure 7b
shows
facilities in
section
the equivalent
(i,j)
/
(i,/)
lines connecting
and
(j,/)
traffic) units.
is
nodes i and
(in layer
Similarly,
a piecewise linear
network representation with
/);
this arc
j.
To
has
suppose the cost
and convex
cost as
parallel arcs
and
appropriate arc fixed costs, variable costs, and capacities. Figure 8a shows a more general cost
structure,
and Figure 8b gives
As we have
its
representation.
seen, the specialization of the layered
network design formulation
for local access
network framework as a fixed charge
network planning
-42-
is
very versatile.
Its
assumptions are
less
restrictive
than previous local access network models, and indeed
we briefly outline solution methods
models. Next,
for the
it
generalizes
many
of the previous
network design problem.
Solution Methods for the Network Design Problem
The network design problem and
Lenstra and Rinnooy
methods
Gray
for solving the
(1978)).
variants are
its
known
to
Several authors have proposed heuristic
problem (Hoang
and Hinxman
(1973), Boffey
Wong
Kan
several of
(1979),
(1989a) propose a dual ascent
(1973), Boyce, Farhi,
Dionne and Florian
method
be NP-hard (Johnson,
and optinrdzation-based
and Weischedel
(1979)).
(1973), Billheimer
and
Balakrishnan, Magnanti, and
that generates provably near-optimal solutions to the
uncapacitated network design problem. The method essentially approximately solves the dual of the
linear
programming
relaxation of the network design integer
solution generates a starting design for a local
that can
be used
improvement
programming formulation. The dual
heuristic,
to verify the quality of the heuristic solutions.
and
also provides a lower
bound
This method generalizes several
previous algorithms for special cases of the network design problem, including the Steiner tree and
plant location problems.
containing
Mp
?o
The approach was successfully
C>ne possible solution strategy
by Lagrange
subproblem
is
good
much
harder to solve than the uncapacitated version.
dualize the arc capacity constraints
is to
multipliers,
and add these multiples
an uncapacitated network design problem
using, say, the dual ascent algorithm.
method such
on several randomly generated problems
45 nodes and 595 arcs.
Capacitated network design problems are
constraints
tested
By
iteratively
(i.e.,
to the objective function).
that can be solved at least
and lower bounds
The
resulting
approximately
modifying the Lagrange multipliers using a
as subgradient optimization (see, for example, Fisher (1981)),
heuristic solutions
multiply the capacity
for the original capacitated
we can
possibly generate
problem. However, previous
experience with this approach for the capacitated plant location, capacitated minimal spanning
and other
related
models suggests
that the addition of arc capacities significantly increases the
between the upper and lower bounds. Thus, we expect
capacities
and non-concave
One
of the
nodes and arcs
main
local access
cost functions to be computationally
limitations of the
in the layered
gaps
planning problems with existing
more
network design model
tree,
difficult.
is its size.
In particular, the
number of
network grows very rapidly with the number of different transmission
-43-
rates,
processor types, and distribution points. To model a problem involving 20 distribution points with
possible connections, 5 processor types,
all
and 3 transmission media, requires
a network with 63 nodes,
and over 600 (undirected) edges. These problem dimensions probably represent the
largest size that
current optimization-based network design algorithms can solve within a reasonable
computational time. In the next section
tractable since
it
assumes a
tree
we
amount
model
describe an altenutive specialized
network, and imposes certain restrictions on the
way
that
is
of
more
distribution points
are assigned to concentrators.
4.3
Tree Covering Model
In this section
we
a
a special case of the local access
call the tree covering model, that is
any existing
is
we describe
tree.
It
capacities.
also
solvable in polynomial time
The model assumes
makes some
applies to the design of
when
the network does not contain
network defining the permissible interconnections
that the
additional assumptions regarding the cost structure and routing pxDlicy.
The model does permit backfeed, and can incorporate economies of
it
network planning problem, which
new
scale.
We first describe the model as
networks, and subsequently describe an enhancement to account for
existing capacities.
We
at
node
j.
say that a node
Node homes on
i
i
homes on another node
j
if
the traffic from distribution p>oint
the switching center (node 0)
if its
traffic is
i
is
processed
not processed at any
intermediate node. The tree covering model makes the following assumptions:
(1)
The given physical network, containing
all
permissible cable sections, has a tree structure, rooted at
the svAtching center. Thus, a unique path connects each distribution p>oint to the switching center.
(2)
The model permits at most one
simplicity,
(3)
we assume
Contiguity assumption:
level of traffic processing
and assumes a single service
that traffic can arrive at different frequencies at the switching center.
The model assumes
that
if
a
node
i
homes on node
nodes lying on the (unique) path between nodes i and also home on node
j
home on
itself if
it
type. For
contains a processor.
We
j.
j,
then
all
intermediate
In particular,
node must
refer to this routing restriction as the contiguity
-44-
j
assumption since the
nodes homing on a partiailar processor induces a single contiguous or
set of all
connected subgraph of the original network.
(4)
The model does not permit bifurcated
must follow the same route
to the
routing,
the traffic originating at a particular
i.e., all
switching center
must use the same
(i.e.,
links,
node
and undergo
processing at the same node).
(5)
The model permits multiple processor types and transmission media. However,
costs
between media, and assumes
that all high frequency
media are
it
ignores any joint
point-to-p)oint to the
switching center. This assumption essentially permits us to include in the processor cost
transmission costs for
traffic
twisted wire pairs), which
(6)
Each processor type
is
emanating from each
we
will refer to as cables, is
assumed
to
and expansion
processor.
assumed
The base
rate
medium
(say,
be sectional.
to
have a fixed plus variable cost structure (including the
transmission cost for the processor's output) that
installation
traffic
all
entails a fixed
may
vary by location.
and variable cost
that varies
Sinnilarly, cable
by
section.
Like the
previous network design model, the tree covering model can also accommodate piecewise linear,
concave cost functions.
(7)
The model can
node
j.
By
also account for additional
selectively setting these
homing
homing
costs
when node homes on
costs to a high value,
i
we can
a processor located at
prohibit
homing
patterns
that violate proximity restrictions.
Without the additional homing
costs, the tree
fixed-charge network representation with
consists of the original physical tree
some
covering model has an equivalent single-layer
additional routing restrictions.
The
single layer
network with the following enhancements: each node of the
tree is
connected by a directed arc (directed away from the node) to the switching center. This arc models the
traffic
processor at the node as well as the jx)int-to-point cable connecting this processor to the
switching center. Every node of the network (except the switching center node) has supply equal to the
projected
is
number
the sink for
all
of channels required for the corresponding distribution p)oint; the switching center
flows.
The contiguity assumptions
node
translates into a routing restriction that specifies
-45-
that
each node (except the switching center) has outgoing flow (either
compressed
The
task into a
rate)
on exactly one
assumptions permits us
all
to
j
and
original tree
let T(j)
network
be the subgraph induced by
if
transform the local access network planning
the nodes of the original tree
network solution in which node contains a processor. Let
processor,
base rate or at the
arc.
tree covering model's
problem of covering
at the
nodes p and q both belong
this
N(j)
by
be the
node subset
to the
implies that T(j) must be a single connected component,
subtrees. Consider a local access
(i.e.,
node subset
i.e., it
set of
T(j)
N(j)).
must be
nodes
that
must be served by
a subtree of the original tree.
a single processor located at
subtree. For each potential processor location in the subtree,
the exact value of the cable expansion costs for
this processor
must serve the
total line
all
we can
subtree edges.
requirements for
all
each node
i
e T.
The node
i
that
minimizes
These properties enable us
total costs is the best
to solve the tree
the nodes of the
calculate the exact flows,
all
we
one of the nodes within the
cost
in the subtree.
easily calculate the total transmission plus processing cost of serving
all
Conversely, suppose
The processor
nodes
this
CXir contiguity assumption
network. (By convention, the switching center always contains a processor.)
T
home on
contains edge (p,q) of the
Thus, the union of the induced subtrees corresponding to each processor must sf>an
are given a subtree
that
is
also
and hence
known
ConsequenUy,
the nodes in subtree
since
we can
T from
processor location for this subtree.
covering model (without existing cable and
processor capacities) very efficiently using a dynamic programming algorithm based on a method
developed by Barany, Edmonds, and Wolsey (1986)
method
is
also closely related to the
for optimally covering a tree
p-median algorithm discussed by Kariv and Hakimi
algorithm starts from the leaves of the original
tree,
special case
Next we outline
when
A line
to
n so
(1989b) describe this
method
the given network
node
0.
Without
that the line
for
in
a different method, using a shortest path algorithm, to solve the
is
a line network.
network consists of a simple path connecting two end nodes, one of which
center node, say,
from
Wong
The
(1979).
and recursively builds the covering solution
successively larger subtrees. Balakrishnan, Magnanti, and
greater detail.
with subtrees. This
loss of generality,
assume
that the
network contains only (undirected) edges
Figure 9 shows this structure. Suppose
we want
to locate
is
the switching
nodes are indexed sequentially
of the
p processors on
form
this
(i-l,i),
for
i
=
l,2,...,n.
network, including the
processor at the switching center. Then, by our contiguity assumption, each processor induces a line
-46-
segment containing
the nodes
all
serves,
it
and the union of all p
line
segments cover
all
the nodes of the
network. Thus, the local access network planning problem for this special case reduces to the problem of
determining the number of processors
We
network into p segments.
way. Consider a
line
can formulate
segment from node
can easily determine the optimal
potential processor locations
Then
k >
i+1.
for
nodes
to-j line
Using
i
(i+1),
i.e.,
this information,
i
j,
(dj
the line
this
total cost of
serving
all
supply
determine the
segment
that
j
>
i.
in the following
As mentioned
the nodes in this segment
we can determine
located at
minimizes
sum of the
the
node
Let
k.
total cost) for
we
by enumerating
the flow
all
and suppose
k,
on each edge of the
supplies for
all
processor plus cable cost for serving
total
is
is
previously,
must carry the cumulative supply
dj, arc (i+l,i+2)
processor throughput
that the processor
with
(inclusive),
j
+ dj^p and so on. Thus,
total
fjartition of the original line
For instance, consider a potential location
j.
i's
and the optimal
problem as a shortest path problem
node
to
between i and
we can
assuming
node on
the
i
must carry node
segment. Also, the
between and
(i.e.,
arc {i,i+l)
and
(p) to locate,
nodes from
all
i
i-
to
j.
the nodes
be the best processor location
k^j
serving line segment
i-to-j,
and
let Cj:
be
the corresponding optimal cost.
To
find the best partition of the original network,
over the (n+1) nodes. For every
arc
is set
equal to
from node n
to
Cjj,
node
i
<
j,
this
we construct
a shortest path
network contains a airected arc from
the optimal cost of serving
all
nodes between
i
and
j
j
to (i-1).
(inclusive).
network defined
The
Then, every path
defines a partition of the line network. In particular, including arc
to-0 path corresponds to selecting the line
segment from node
i
to
j
cost of this
(i-l,j)
in the n-
as one element of the partition.
Consequently, the shortest n-to-0 path defines the optimal partition of the original network, and hence
identifies the optimal local access
network problem, the costs
existing cable
Cj:
and processor
simplified even further
and
network configuration. Because of the
can be computed
capacities.
is
efficiently.
special structure of the line
Also, the algorithm can easily accomodate
For situations without backfeed this approach can be
analagous to the well-known Wagner- Whi tin model of production
planning (Kubat (1985)).
For general tree networks, incorporating existing cable and processor capacities considerably
complicates the model and
approach
its
solution.
that formulates the local access
Balakrishnan et
al.
network planning problem as a mixed integer program, and
dualizes the capacity constraints. The resulting subproblem
planning problem that can be solved
(1989b) describe a Lagrangian-relaxation
efficiently
is
an uncapacitated
local access
using the dynamic programming procedure
-47-
network
we
mentioned previously. This solution gives a lower bound on the
authors
embed
the Lagrangian subproblem solution
Lagrange multipliers
in order to
also identifies a feasible
method
in
total cost of the original
an
iterative
problem. The
procedure that modifies the
improve the lower bound. The subproblem solution
at
each iteration
network expansion plan, which can be improved heuristically
to generate
good upper bounds. The authors describe various formulation and algorithmic enhancements
significantly
test
improve the method's performance, and report computational
results
that
based on some actual
networks.
In conclusion, this section has described various
models
for local access
network planning.
have seen several possible combinations of assumptions, each defining a separate model. Table
summarizes the main differences
in
local access
III
assumptions, feahjres, and solution methods for the models that
reviewed. The fixed<harge network design model
network design algorithms make
We
this
is
quite comprehensive, and recent advances in
modeling approach computationally
network planning problems, especially
we
for designing
feasible for
new networks. On
medium-sized
the other hand, the
general network design model does not exploit any special structure that specific application contexts
might
px)ssess.
For instance, in order to simplify the task of managing the network, some local
telephone companies might adopt routing policies that validate the contiguity assumption. Similarly,
if
each concentrator requires an umbilical (direct) connection
media
costs
especially
if
and assuming
a point-to-point
medium
for
to the
compressed
switching center, ignoring shared
traffic
these enhanced cables can be installed in existing ducts.
assumptions enables us
to
might be appropriate,
Making
these simplifying
use specialized algorithms, thus increasing the range of problem sizes that
can be solved.
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5.
Concluding Remarks
This paf)er has attempted to
(i)
set a
backdrop
for
economic modeling of
local access
telecommunication systems by tracing the evolution of relevant technology in public telecommunication
networks,
(ii)
provide a general framework for viewing planning and design models,
previous contributions from the literature, and
technological, economic,
the rising
demand
of digital
and
describe
for a variety of services
due
to the introduction of
economic models,
ISDN
for
differences in
modeling assumptions, and
we focused on
many
static (single
briefly outlined solution
modeling approaches has suggested, the general area of
Some
of the static
models
network planning continues
new developments
future periods.
pay
approaches.
to
decompose
we might
the multiperiod problem
be solved using one of the single-period methods.
to establish excess transmission
t
might represent the
and switching resources
at
time
t
'price'
for use in
Thus, the pricing mechanism accounts for the temporal coupling of plans by acting as an
incentive to exploit economies of scale.
final target
to
telecommunication
new modeling
In this scheme, the Lagrangian multipliers corresponding to a time period
veiling to
in
offer potential for use in multij^eriod settings. For instance,
decomposition method such as Lagrangian relaxation
we are
to increasing
challenging opportunities for modeling and algorithmic development, particularly for
into several single p>eriod problems, each that could
that
respond
period) models, emphasized the
standards and technologies should further stimulate the development of
a
to
methods. As our discussion of
local access
multiperiod and multiple service versions of the problem. The
employ
traditional local
telecommunication services.
For our review of planning models,
provide
particularly,
environment because the availability of electronic
compression devices for the feeder network has created new ways
demand
Several
standards, the installation
and mounting competitive pressures. The
tools are inadequate in the current
summarize
some new modeling approaches.
social forces are fueling interest in these
fiber optic technology,
network planning
traffic
and
(iv)
(iii)
network;
we might
An
alternative
approach
is to
use
static
models
to generate a
then apply a different model to plan the evolution, over the multiple
time periods of the planning horizon, from the current network to the target network. Shulman and
Vachani (1988) propose a related approach.
50-
The models
facilities
traffic
we discussed
that
did not include any special representation for fiber optic
partly because their current econoniic implications are comparable to those of other electronic
processing devices and high frequency media. Future developments and implementation of fiber
optics in the local loop might necessitate other distinctions in
The continuing evolution
formulate
new ways
research directions.
switches and other
of local access network technology
of utilizing this technology create a
One promising
'intelligent'
hardware
in the feeder
in
fiber optic technologies.
and ever increasing
of exciting
is
efforts to
and challenging future
the possibility of installing remote
and distribution networks. These devices can
many applications customers would need
communicate only with a nearby remote switch instead
and thus reduce the need
number
technological development
perform a number of switching center functions, and
center. This strategy of using
modeling
of connecting
the
all
remote switches would reduce the overall
way
to the switching
the feeder network,
traffic in
for additional cables or processors. Strategies for the
to
proper deployment of
these remote switches might be a fruitful topic for future studies.
The enormous bandwidth
new
of fiber optic networks creates intriguing opportunities for developing
services for households: for example, video
and home telemetry. With these new
services,
programming on demand,
customers might become
shopping
services,
much more dependent on
their
telecommunication system, and any disruption in service would be very undesireabie, pjerhaps,
local
comparable
to a
power
blackout. Thus, reliability issues should
assume
planning for future networks. Because of economic considerations
most
that
interactive
common current local network
any
design
is
a tree configuration.
single link failure will disconnect the network.
networks that can
Shallcross (1986)
offer
more
reliability
and Groetschel and
two paths between every
pjair
and resistance
Monma
of nodes) might
(1988)).
New
in
-51
much
greater importance in
minimizing the number of
The disadvantage
research
is
to failure (see, for
needed
links, the
of this design
is
to design local access
example,
Monma
and
Topologies such as ring networks (which provide
become more common
systems.
a
in future local
telecommunication
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fibers".
Figure
1
Hierarchical Structure of Telecommunication Networks
BACKBONE NETWORK
LOCAL ACCESS NETWO
Distribution point
customers
Figure 2
Typical Route of a Local Access Network
Customer
pre
Distribution
Network
Lateral Cable
Switching
Center
Distribution Points
Figure
3A
Local Access Network Planning Example
A
Switching Center
I
I
50
A40^^^
^=^00
50
90.#"
110
^00
Distribution
Point
100
demand
Numbers on edges denote
.vNXNxvv
current cable capacity
BOTTLENECK edges
Cum. demand > Capacity
Figure 3b
Cable Expansion Cost for Local Access Network Planning Example
Cable Expn
Cost
No. of additional cables
Figure 3c
Processor Cost for Local Access Network Planning Example
Processor
cost
Processor throughput
Figure 4
Sample Expansion Plan
Switching Center
^
^^ ii
40
350
50
Cable Expansion
.0-:'
^^==^00+100-^^''^
Distribution
Point
Concentrator
Dotted
lines
show concentrated
Numbers on edges denote number
traffic
of lines used
Figure 5
Layered Network Representation
Layer
1
Switching Center
demand
for
service type 2 H^
physical node
i
Layer 2
Layers
Figure 6
Cable Expansion Cost with Economies of Scale
Traffic
on edge
(i,j)
Figure 7A
Convex Cable Expansion Cost
No. of lines
Figure 7B
Equivalent Network Representation
variable
cost
F. ,c^ ,B.
Fixed
')
')
')
capacity
cost
2
0, C..,oo
2
1
Figure 8A
General Cable Expansion Cost Function
Cable
Expn. cost
Figure 9
Line Network
<^>-^D
Switching Center
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