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

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

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

Wireless Network Design Tool

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

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

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

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

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

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.

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.

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.

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

Integer Programming Model

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:

Quality of Service (QoS) Constraints

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

Global Valid Inequalities

Global Valid Inequalities

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

Optimality Cuts: Derivation

Optimality Cuts: Derivation

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

Phase II: Eliminating Infeasibilities

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 9.10.27

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 0.03125 Minimum signal-to-interference ratio required.    Based on data generation used by Amaldi et al. [2001a,b]

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.

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.

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.

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