W-CDMA Network DesignQibin Cai1

Joakim Kalvenes2

Jeffery Kennington1

Eli Olinick1

Dinesh Rajan1

Southern Methodist University

1 School of Engineering 2Edwin L. Cox School of Business

Supported in part by Office of Naval Research Award N00014-96-1-0315

2

Wireless Network Design: Inputs• “Hot spots”: concentration points of users/subscribers (demand)• Potential locations for radio towers (cells)• 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, – MTSOs to each other or to PSTN

3

Wireless Network Design: Problem

• Determine – Which radio towers to build (base station location)

– How to assign subscribers to towers (service assignment)

– Which MTSOs to use

– Topology of MTSO/PSTN backbone network

• Maximize profit: revenue per subscriber served minus infrastructure costs

4

Wireless Network Design Tool

5

Optimization Model for Wireless Network Design: Sets

• L is the set of candidate tower locations.• M is the set of subscriber locations.

• Cm is the set of tower locations that can service subscribers in location m.

• Pl is the set of subscriber locations that can be serviced by tower ℓ.

• K is the set of candidate MTSO locations– Location 0 is the PSTN gateway

– K0 = K {0}.

6

Optimization Model for Wireless Network Design: Constants

• dm is the demand (channel equivalents) in subscriber location m.

• r is the annual revenue generated per channel.

• al is the cost of building and operating a tower at location .

• bk is the cost of building an MTSO at location k.

• clk the cost of providing a link from tower ℓ to MTSO k.

• hjk the cost of providing a link from MTSO j to MTSO/PSTN k.

is the maximum number of towers that an MTSO can support.

7

Optimization Model for Wireless Network Design: Constants

• SIRmin is the minimum allowable signal-to-interference ratio. – s = 1 + 1/SIRmin.

• gmℓ is the attenuation factor from location m to tower ℓ.

– Ptarget is the desired strength for signals received at the towers.

– To reach tower l with sufficient strength, a handset at location m transmits with power level Ptarget / gmℓ.

8

Optimization Model for Wireless Network Design: Power Control Example

Subscriber at Location 1 Assigned to Tower 3 Tower 3

gPtar

13

PgPg tar

tar 13

13

Received signal strength must be at least the target value Ptar

Signal is attenuated by a factor of g13

9

Optimization Model for Wireless Network Design: Decision Variables Used in the Model

• Binary variable yℓ=1 iff a tower is constructed at location ℓ.

• The integer variable xmℓ denotes the number of customers (channel equivalents) at subscriber location m served by the tower at location l.

• Binary variable zk=1 iff an MTSO or PSTN is established at location k.

• Binary variable slk=1 iff tower l is connected to MTSO k.

• Binary variable wjk= 1 iff a link is established between MTSOs j and k.

• ujk= units of flow on the link between MTSOs j and k.

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Optimization Model for Wireless Network Design: Signal-to-Interference Ratio (SIR)

Tower 3Tower 4

gPtar

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PgPg tar

tar 213

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gPtar

24gPtar

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P tartar

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232

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1SIR

Subscriber at Location 1 assigned to Tower 3

Two subscribers at Location 2 assigned to Tower 4

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Optimization Model for Wireless Network Design: Quality of Service (QoS) Constraints

• For known attenuation factors, gml, the total received power at tower location ℓ, Pℓ

TOT , 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:

.targetTOT

mjMm Cj mj

m xg

gPP

m

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target SIRPP

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12

Optimization Model for Wireless Network Design: Quality of Service (QoS) Constraints

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13

Optimization Model for Wireless Network Design: Integer Programming Model

The objective of the model is to maximize profit:

subject to the following constraints:

Cost Backbone

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Cost Connectioncost MTSOtcosTower revenue

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jkjkL Kk

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14

Optimization Model for Wireless Network Design: Connection Constraints

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15

Optimization Model for Wireless Network Design: Flow Constraints for

Backbone Construction

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16

Computational Experiments

• Computing resources used– Compaq AlphaServer DS20E with dual EV6.7 (21264A) 667 MHz

processors and 4,096 MB of RAM– Latest releases of CPLEX and AMPL

• Computational time – Increases substantially as |L| increases from 40 to 160– Very sensitive to value of

• Lower Bound Procedure– Solve IP with l = 0 for all l – Stop branch-and-bound process when the optimality gap (w.r.t LP) is

5%

• Estimated Upper Bound Procedure– Relax integrality constraints on x, y, and s variables.– Solve MIP to optimality with l = 0 for all l

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Data for Computational Experiments

• Restrict • Two Series of Test Problems:

1. Candidate towers placed randomly in 13.5 km by 8.5 km service area

– 1,000 to 2,000 subscriber locations dm ~ u[1,10]

– |L| drawn from {40, 80, 120, 160}

– |K| = 5, placed randomly in central 1.5 km by 1.0 km rectangle

2. Simulated data for North Dallas area– |M| = 2,000 with dm ~ u[1,10]

– |L| = 120

– |K| = 5

}.10:{ 15 mm gLC

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Sample Results for Data Set 1Upper Bound Procedure Best Feasible Solution from Lower Bound

Procedure

Problem |L| |M| Towers Demand

Profit CPU Towers Demand

Profit CPU Gap

R110 40 1,000 35.6 92.60% 18.330:00:0

2 37 92.80% 18.220:00:2

0 0.60%

R160 80 1,000 42.0 92.20% 17.550:08:4

3 39 87.50% 16.740:01:4

0 4.62%

R210 120 1,000 50.0 94.20% 17.660:43:1

8 51 91.50% 16.970:08:4

8 3.91%

R410 160 1,000 53.1 93.10% 16.810:57:0

2 53 90.30% 16.210:15:0

7 3.57%

R260 40 2,000 37.0 65.30% 26.720:00:1

4 38 65.30% 26.60:01:1

7 0.45%

R310 80 2,000 62.4 87.60% 34.930:10:0

4 65 86.80% 34.330:03:5

1 1.72%

R360 120 2,000 N/A N/A N/A2:00:0

0 75 93.40% 36.420:14:5

2 5.00%

R460 160 2,000 N/A N/A N/A2:00:0

0 88 93.70% 35.240:56:4

0 5.00%

• Solution times for Lower Bound Procedure varied from 30 seconds to 1 hour of CPU time.

•Average value of 2.0 ≤ |Cm| ≤ 8.4.

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Data Set 2: North Dallas Area|M| = 2,000, dm ~ u[1,10], |L| = 120, and |K| = 5

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Results for North Dallas

ProblemName MTSOs Towers Demand Profit CPU MTSOs Towers Demand Profit CPU Gap

ND100 2 75.4 82.6% 31.55 0:39:05 2 77 82.0% 31.13 0:02:12 1.3%ND200 2 75.4 85.5% 32.33 0:55:18 3 78 84.8% 31.71 0:03:35 1.9%ND300 3 84.2 87.3% 33.00 1:13:05 2 82 85.2% 32.17 0:03:15 2.5%ND400 2 79.5 86.3% 32.92 0:45:29 2 80 85.6% 32.45 0:03:04 1.4%ND500 2 81.5 87.0% 32.93 0:40:46 2 82 85.0% 32.00 0:03:10 2.8%ND600 2 80.7 86.1% 32.86 0:46:06 2 82 85.9% 32.63 0:06:00 0.7%ND700 2 81.4 85.6% 32.36 0:53:35 2 82 84.7% 31.93 0:03:31 1.3%

Upper Bound Lower Bound Procedure

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Sample Results with Heuristics

Heuristic 1: |Cm| ≤ 1   Heuristic 2: |Cm| ≤ 2

Problem |L| |M| Towers Demand

Profit CPU Gap Towers Demand

Profit CPU Gap

R110 40 1,000 40 93.50% 18.090:00:0

1 1.31% 37 92.80% 18.220:00:1

4 0.60%

R160 80 1,000 67 92.40% 15.050:00:0

1 14.25% 47 90.40% 16.530:00:3

3 5.81%

R210 120 1,000 94 93.00% 13.030:00:0

1 26.22% 67 93.90% 15.880:01:1

4 10.08%

R410 160 1,000 94 83.90% 10.530:00:0

1 37.36% 76 92.10% 14.260:00:5

4 15.17%

R260 40 2,000 40 65.30% 26.380:00:0

3 1.27% 38 65.30% 26.60:00:4

5 0.45%

R310 80 2,000 79 89.80% 33.900:00:0

2 2.95% 65 86.50% 34.210:01:5

2 2.06%

R360 120 2,000 113 96.50% 34.390:00:0

2 10.30% 85 94.30% 35.980:03:3

6 6.15%

R460 160 2,000 141 96.40% 31.510:00:0

1 15.06% 100 93.30% 33.990:06:2

3 8.37%

Geo. Mean 3.55%

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The Power-Revenue Trade-Off

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Downlink Modeling

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Conclusions and Directions for Future Work

• IP model for W-CDMA problem– Too many variables to be solved to optimality with commercial solvers– Developed cuts and a two-step procedure to find high-quality solutions with

guaranteed optimality gap.– Largest problems took up to 1 hour of CPU time– Heuristic 2 reduces computation times by an order of magnitude and still finds

fairly good solutions • Results for North Dallas problems on par with randomly generated data

sets.• Model can be integrated into a planning tool; quick resolves with new

tower locations added to original data• Extensions

– Construct a two-connected backbone with at least two gateways– Consider sectoring– Tighten the l parameters