# Maximal covering wit..

```Gunhak Lee , Alan T. Murray
Present by Limin Zheng
Agenda
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Introduction
Backgroud
Problem description
Mathematical formulation
Application details
Results & Discussion
Conclusions
Intorduction
Introduction
Many US cities and Countries are attempting to build wireless broadband
networks for communication and service in their communities as basic
infrastructure to facilitate local economic development and enable much
wider service provision to more people.
 wireless networks in municipalities have been widely utilized for
a range of public applications, such as public hotspots, public
safety and general communication
 Wireless broadband networks would play an important role in
improving the quality of our life, giving people the freedom and
capability to communicate with the world anytime, anywhere
cellular systems, relying on mesh or multi-hop networking.
 Wireless broadband is attractive to municipalities willing to construct
their own communication network given limited budgets.
Introduction
What is the primary concern in providing
 When local governments attempt to provide wireless broadband
services to their communities, the primary concern is where to
place relevant facilities and how to connect them.
What is the purpose of this paper ?
 In this paper, we address location modeling approaches for
planning citywide wireless broadband services. More specifically,
we propose a mathematical formulation of the maximal covering
problem with survivability constraints based on wireless mesh
network topology.
Backgroud
Background
• Survivable network design
 What is survivable network ?
 What is disjoint path?
Background
• Wi-Fi based mesh networks
 What is mesh networks
Problem Description
Problem Description
For the purpose of this paper, we specifically address
two issues:
(1) how to locate Wi-Fi equipment to maximally
cover demand given a specified number of units.
(2) how to connect Wi-Fi equipment to ensure
survivable networking.
Solution:
For(1)Maximal Covering Location Problem (MCLP)
For(2) Number of node disjoint paths for any pair of nodes
Problem Description
 Regarding the architecture of a mesh wireless network, some of
nodes (gateways) must be connected to hard, land based
infrastructure and thus reliable performance of network is
dependent on the existence of duplicated paths between a general
node and gateway node.
Mathematical Formulation
Mathematical Formulation
Model:
Maximal Covering problem with Survivability Constraints (MCSC)
Assumed based on equipment capabilities :
1. Maximum distance of wireless access form the Wi-Fi router.
2. Maximum distance for wired access from the exiting backbone infrastructure.
3. Maximum distance for interconnection between the Wi-Fi routers.
Mathematical Formulation
Something need to predefined or given when using this model :
1. Potentially eligible sites to provide wireless broadband services
constitute a discrete set of locations.
2. Set of points is predefined to represent aggregate population to be
covered by the Wi-Fi router.
3. A number of facilities, p, is given exogenously.
4. a number of Wi-Fi routers, q, providing wired connection to the
existing backbone infrastructure, is also specified in advance.
Mathematical Formulation
Based on the hierarchy of a wireless broadband network,
parameters and sets are defined as follows:
I
set of demand nodes
J
set of potential sites for Wi-Fi router
M
set of existing DSL central offices
ai
population at demand node I
p
required number of Wi-Fi routers to be deployed
q
required number of Wi-Fi routers for wired connections to
existing central offices
K
required number of disjoint paths
dij
shortest distance from demand node i to Wi-Fi router at j
djc
shortest distance from Wi-Fi router at j to DSL central
office at c
djl
shortest distance between Wi-Fi routers at j and l
Mathematical Formulation
Ni
{j єJ|dij ≤ R}
Ψ
{j єJ|dcj ≤ L , c є M}
Ωj
{l єJ|djl ≤ W , j ≠ l}
R
coverage standard for Wi-Fi;
L
coverage standard for DSL central office
W
maximum distance for Wi-Fi router point to point
interconnection
Mathematical Formulation
Decision variables are defined as follows:
Mathematical Formulation
Mathematical Formulation
Mathematical Formulation
Application Details
Application Details
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It is assumed that wireless routers must be within 12,000 feet (L)
from a central office.
The coverage standard of a Wi-Fi router (R) is specified as 3465
feet, so 6930 feet is used for the maximum distance for Wi-Fi point
to point interconnection (W)
The required number of Wi-Fi routers (p) is specified to be in the
range of 8–29.
It is assumed that 20% of Wi-Fi routers satisfy the required
number of wired connections to existing central offices (q).
Two cases of disjoint paths, K = 1 and K = 2, are examined for
survivable network design.
Application Details
The MCSC was solved exactly using a commercial optimization solver,
named CPLEX 10.0 (ILOG) on an Intel Xeon 3 GHz CPU with 3 GB
memory. ArcGIS 9.1 was used to manage needed input information (Ni,
W, xj) through spatial analysis functionality. Also, Visual Basic
Application (VBA) with ArcObjects was used to create the necessary
text file of the MCSC that is read into CPLEX. Further, GIS provides
capabilities for visualizing and evaluating solutions.
Results & Discussion
Discussion
In this paper, we focus on system reliability for comparison between
two different network configurations.
For each network configuration, there are specific source and
destination nodes. Accordingly, sets are defined as follows:
Evaluating system reliability
Reliability of a node can be defined as the probability that it functions
during a specified time period. Given node reliability, the probability
of a disjoint path for a pair of source and destination nodes can be
derived by the joint probability of nodes along the disjoint path, based
upon the assumption of independence.
Since there could be a number of disjoint paths between source and
destination nodes, system reliability is the sum of the probabilities of
all possible disjoint paths between source and destination nodes.
Evaluating system reliability
The standard mathematical formulation of system reliability can
be found in Shier (1991), and stated as follows:
Evaluating system reliability
Average system reliability for entire network, Raverage, is computed
by averaging the reliabilities for all pairs of source and destination
nodes as follows:
where Q is the number of all pairs of source and destination nodes.
Evaluating system reliability
Node reliability probabilities are assumed (0.8 in our case)
S-t pairs
K p(s) q(t)
(Q)
1
11 6
60
2 11 4
40
Disjoint
path
71
80
Raverage
0.69
0.41
For examining these two configurations in cases of a specific
node failure, we calculate reliability after simulating any single
node failure.
K=1 , average reliability = 0.5
K=2, average reliability = 0.61
Conclusions
Conclusions
Adequately positioning wireless access points is crucial in order to
extend service coverage with a given budget limit. Another significant
consideration for building wireless broadband networks is the
provision of reliable broadband service. However, it is difficult to cover
a large area reliably because a more reliable broadband network
ecessarily requires a more interconnected network topology to ensure
redundancy in routing.
Conclusions
To deal with these considerations simultaneously, we introduced the
maximal covering problem with survivability requirements (MCSC).
This approach extends classical facility location and network design
problems by explicitly integrating covering and network survivability.
For more practical use of this approach, several related technical issues,
such as radio coverage planning, traffic and routing controls and
channel assignment, must be taken into account. This paper, however,
focuses on general methodological issues concerning maximal
covering and survivable network design. Thus, this paper is expected to
help decision makers and network planners understand coverage and
design issues through the use of a method for obtaining solutions and
presenting expected network configurations.
Conclusions
The application found that a wireless network can be designed to
provide citywide wireless broadband services to an urban area,
ensuring network survivability to a high degree. Comparatively, we also