1. Base Station Location and
Service Assignment in
W-CDMA Networks
Joakim Kalvenes1
Jeffery Kennington2
Eli Olinick2
Southern Methodist University
1Edwin L. Cox School of Business
2School of Engineering

2. Wireless Network Design: Inputs
Potential locations for radio towers (cells)
“Hot spots”: concentration points of users/subscribers (demand)
Potential locations for mobile telephone switching offices (MTSO)
Locations of access point(s) to Public Switched Telephone Network (PSTN)
Costs for linking towers to MTSOs, and MTSOs to PSTN

3. Wireless Network Design: Problem
Determine which radio towers to build (base station location)
Determine how to assign subscribers to towers (service assignment)
Determine which MTSOs to use
Maximize profit: revenue per subscriber served minus infrastructure costs

4. Wireless Network Design Tool

5. Code Division Multiple Access (CDMA) Technology
The basis for 3G cellular systems
Channel (frequency) allocation is not an explicit issue since the full spectrum is available in each cell
New calls cause incremental noise (interference)
New calls admitted as long as the signal-to-noise ratio stays with in system limit
Power transmitted by handset depends on distance to assigned radio tower
Tower location and assignment of customer locations to towers must be determined simultaneously

6. Power Control Example Signal is attenuated by a factor of g13 Tower 3
Received signal strength must be at least the target value Ptar
Tower 3
Signal is attenuated by a factor of g13
Subscriber at Location 1 Assigned to Tower 3

7. Signal-to-Interference Ratio (SIR)
Tower 3
Tower 4
Subscriber at Location 1 assigned to Tower 3
Two subscribers at Location 2 assigned to Tower 4

8. Some Related CDMA Literature: Base Station Location & Service Assignment
Galota, Glasser, Reith, and Vollmer (2001)
Profit maximization
Polynomial-time approximation scheme
Amaldi, Capone, and Malucelli (2001a, 2001b)
Minimize cost to serve all users
Randomized add-drop heuristic
Tabu search to improve solutions
Mathar and Schmeink (2001)
Maximize system capacity for a fixed budget
Simplified interference model

9. Our New Model for CDMA Base Station Location and Service Assignment
Integer linear program (ILP)
Maximizes profit
Enforces hard constraints on signal-to-interference ratio
Incorporates FCC licensing rules for US providers

10. Constants and Sets Used in the Model
L is the set of candidate tower locations.
M is the set of subscriber locations.
gmℓ is the attenuation factor from location m to tower ℓ.
is the set of tower locations that can service customers in location
is the set of customer locations that can be serviced by tower ℓ.

11. More Constants and Sets Used in the Model
dm is the demand (channel equivalents) in location
r is the annual revenue generated per channel.
is the FCC mandated minimum service requirement.
is the cost of building and operating a tower at location
SIRmin is the minimum allowable signal-to-interference ratio.
s = 1 + 1/SIRmin.

12. Decision Variables Used in the Model
yℓ =1 if a tower is constructed at location ℓ; and zero, otherwise.
The integer variable xmℓ denotes the number of customers (channel equivalents) at that are served by the tower at location
The indicator variable qm =1 if and only if location m can be served by at least one of the selected towers.

13. Integer Programming Model
The objective of the model is to maximize profit:
subject to the following constraints:

14. Integer Programming Model

15. Quality of Service (QoS) Constraints
For known attenuation factors, gml, the total received power at tower location ℓ, PℓTOT , is given by
For a session assigned to tower ℓ
the signal strength is Ptarget
the interference is given by PℓTOT – Ptarget
QoS constraint on minimum signal-to-interference ratio for each session (channel) assigned to tower ℓ:

16. Quality of Service (QoS) Constraints

17. Enhancing the Basic ILP Model
Global Valid Inequalities
Optimality Cuts
Post-Processing Procedure
Branching Rule

18. Global Valid Inequalities

19. Global Valid Inequalities

20. Optimality Cuts
If customers at site m are served, then profit is maximized by assigning them to the available tower that has the largest attenuation factor (i.e., the nearest available tower).
We can add the following cuts to formulation

21. Optimality Cuts: Derivation

22. Optimality Cuts: Derivation

23. Post Processing
Values of the attenuation factors (gij) have a large range
Coefficients in QoS constraints (6) may differ in magnitude by as much as 109.
Causes scaling problems so that solutions returned by CPLEX are not always feasible
Post-processing procedure ensures feasibility within a reasonable tolerance

24. Phase II: Eliminating Infeasibilities

25. Computational Experiments
Branching on tower-location decisions (y's) before customer assignment (x's) reduces branch-and-bound time.
Number of (8) cuts depends on |Cm|; may actually increase solution time if too large.
Computing resources used
Compaq AlphaServer DS20E with dual EV6.7 (21264A) 667 MHz processors and 4,096 MB of RAM
CPLEX version 6.6.0
AMPL release

26. Parameters for Dense Data Set
Grid size
400 m by 400 m
Coverage area for test cases.
|L| 22
Number of potential base stations randomly placed in the coverage area.
|M| 95
Number of potential subscribers randomly placed in the coverage area. Cm L
The potential base stations to which subscriber m can be assigned. dm 1
The number of potential subscribers in location m. r
$42,820
Annual revenue for each subscriber serviced. 
0.25
Mandated minimum service requirement. al
$145,945
Annualized cost for installing a base station in location l.
SIRmin
Minimum signal-to-interference ratio required. 
Based on data generation used by Amaldi et al. [2001a,b]

27. Results for Dense Test Data Set
Average for Phase I Over 20 Problem Instances
Optimality Gap
Run
(7)
(8)
Demand
Satisfied
Profit
CPU Time
mm:ss PP
Average
Max 1 no
61.21%
$2,154,309
17:47 15
39.25%
65.09% 2
yes
63.05%
$2,221,947
27:55 14
37.34% 3
93.79%
$3,238,779
04:30 9
8.76%
33.85% 4
97.47%
$3,381,352
22:05
4.76%
15.82%
PP gives number of problems (out of 20) requiring post processing.

28. Parameters for Sparse Data Set
Grid size
Market in US Plains State
Coverage area for test cases.
|L| 40
Number of potential base stations randomly placed in the coverage area.
|M|
250
Number of potential subscribers randomly placed in the coverage area.
|Cm|
Between 1.7 and 2.4 on average
The potential base stations to which subscriber m can be assigned. dm
Uniform [1,{8,16,32}]
The number of potential subscribers in location m.
300 problem instances: 100 sets of subscriber locations each with 3 distributions for demand at each location.

29. Results for Sparse Data Set
* Only 10 instances were attempted.
** Only 69 of 100 instances were solved within the limits of 8 hours of CPU time and 1.8 Gb RAM.

30. Conclusions Sparse network structure
Branching rule and cuts enable CPLEX to solve realistically sized problem instances of our model
Dense network structure
Previous work uses randomized local search
Our solutions provably within 5% of optimal on average
Sparse network structure
Solved a set of 300 test problems with |L| = 40 and |M| = 250
Average optimality gap = 1.2% and average CPU time = 13 minutes
Extensions (suggested by a well-known cellular provider)
Maximize capacity of existing 2G system to provide CDMA
Capacity expansion of an existing 3G system subject to budget constraint