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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 N00014-96-1-0315. - PowerPoint PPT Presentation
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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.
10
Optimization Model for Wireless Network Design: Signal-to-Interference Ratio (SIR)
Tower 3Tower 4
gPtar
13
PgPg tar
tar 213
14
gPtar
24gPtar
24
Pgg
P tartar
24
232
g
g
24
232
1SIR
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, 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
mintarget
TOT
target SIRPP
P
12
Optimization Model for Wireless Network Design: Quality of Service (QoS) Constraints
.0|}{\| if 0max where
max where
, )1(1
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becan constraint this1, ifbuilt is tower Since
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Mmm
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LySIR
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m
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
}\{
Cost Connectioncost MTSOtcosTower revenue
0
.max
Kj jKk
jkjkL Kk
kkKk
kkLMm C
m whsczbyaxrm
)3(,)1(
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Lysxg
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CMmydx
Mm Cjmj
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mmm
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m
14
Optimization Model for Wireless Network Design: Connection Constraints
)6(,
)5(,,
(4),
Kkαzs
KkLzs
Lys
kL
k
kk
Kkk
15
Optimization Model for Wireless Network Design: Flow Constraints for
Backbone Construction
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(10),)(
)9(,
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0
}\{0
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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
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 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
18
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.
19
Data Set 2: North Dallas Area|M| = 2,000, dm ~ u[1,10], |L| = 120, and |K| = 5
20
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
21
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%
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