W-CDMA Network Design Qibin Cai 1 Joakim Kalvenes 2 Jeffery Kennington 1 Eli Olinick 1 Dinesh Rajan 1 Southern Methodist University 1 School of Engineering 2 Edwin L. Cox School of Business Supported in part by Office of Naval Research Award N

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. C m is the set of tower locations that can service subscribers in location m. P l 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 –K 0 = K  {0}.

6 Optimization Model for Wireless Network Design: Constants d m is the demand (channel equivalents) in subscriber location m. r is the annual revenue generated per channel. a l is the cost of building and operating a tower at location. b k is the cost of building an MTSO at location k. c l k the cost of providing a link from tower ℓ to MTSO k. h jk 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 SIR min is the minimum allowable signal-to-interference ratio. –s = 1 + 1/SIR min. g mℓ is the attenuation factor from location m to tower ℓ. –P target 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 P target / g mℓ.

8 Optimization Model for Wireless Network Design: Power Control Example Subscriber at Location 1 Assigned to Tower 3 Tower 3 Received signal strength must be at least the target value P tar Signal is attenuated by a factor of g 13

9 Optimization Model for Wireless Network Design: Decision Variables Used in the Model Binary variable y ℓ =1 iff a tower is constructed at location ℓ. The integer variable x mℓ denotes the number of customers (channel equivalents) at subscriber location m served by the tower at location l. Binary variable z k =1 iff an MTSO or PSTN is established at location k. Binary variable s l k =1 iff tower l is connected to MTSO k. Binary variable w jk = 1 iff a link is established between MTSOs j and k. u jk = units of flow on the link between MTSOs j and k.

10 Optimization Model for Wireless Network Design: 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

11 Optimization Model for Wireless Network Design: Quality of Service (QoS) Constraints For known attenuation factors, g ml, the total received power at tower location ℓ, P ℓ TOT, is given by For a session assigned to tower ℓ –the signal strength is P target –the interference is given by P ℓ TOT – P target QoS constraint on minimum signal-to-interference ratio for each session (channel) assigned to tower ℓ:

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

13 Optimization Model for Wireless Network Design: Integer Programming Model The objective of the model is to maximize profit: subject to the following constraints:

14 Optimization Model for Wireless Network Design: Connection Constraints

15 Optimization Model for Wireless Network Design: Flow Constraints for Backbone Construction

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

17 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 d m ~ 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 d m ~ u[1,10] –|L| = 120 –|K| = 5

18 Sample Results for Data Set 1 Upper Bound Procedure Best Feasible Solution from Lower Bound Procedure Problem|L||M|TowersDemandProfitCPUTowersDemandProfitCPUGap R110401, %18.330:00: %18.220:00:200.60% R160801, %17.550:08: %16.740:01:404.62% R , %17.660:43: %16.970:08:483.91% R , %16.810:57: %16.210:15:073.57% R260402, %26.720:00: %26.60:01:170.45% R310802, %34.930:10: %34.330:03:511.72% R ,000N/A 2:00: %36.420:14:525.00% R ,000N/A 2:00: %35.240:56:405.00% Solution times for Lower Bound Procedure varied from 30 seconds to 1 hour of CPU time. Average value of 2.0 ≤ |C m | ≤ 8.4.

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

20 Results for North Dallas

21 Sample Results with Heuristics Heuristic 1: |C m | ≤ 1 Heuristic 2: |C m | ≤ 2 Problem|L||M|TowersDemandProfitCPUGapTowersDemandProfitCPUGap R110401, %18.090:00:011.31% %18.220:00:140.60% R160801, %15.050:00: % %16.530:00:335.81% R , %13.030:00: % %15.880:01: % R , %10.530:00: % %14.260:00: % R260402, %26.380:00:031.27% %26.60:00:450.45% R310802, %33.900:00:022.95% %34.210:01:522.06% R , %34.390:00: % %35.980:03:366.15% R , %31.510:00: % %33.990:06:238.37% Geo. Mean3.55%

22 The Power-Revenue Trade-Off

23 Downlink Modeling

24 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