Results

1… Fractal Results

                Runtime

                                2… Points in Set vs. Time (Separated)

                                4… Points in Set vs. Time (Combined)

                                6… Performance Analysis

                                7… Task Manager

                Performance Model

                                8… Overall Fractal Performance Model

                                9… Fractal Performance Model Time Constraints

                                9… Time Constraints for Fractal Performance Model

12… Solar Sim Results

                Runtime

                                12… Laptop Vs. Cluster Runtime

                                13… Nodes in Cluster

                                14… Start Time Analysis

                                14… Transmission Time Solar Sim

                Performance Model

                                16… Estimated Calculation Time for 3D Solar Sim

                                16… Performance Model (Solar Sim) Elements

22… Estimated Time for 3D Solar Sim With Multiple Spacecraft per Processor

23… Time Advantage of Adding a Node

                29… Cellular Automata Results

                                Runtime

                                                29…Time to Run 1200x1200 Cellular Automata

                                Performance Model

                                                30… Transmission and Calculation Time, Cellular Automata

                                                31… Transmission and Calculation Cross Effecting Total Time

                                Analysis

                                                31… Rate of Change: Derivatives of Calc, Trans, and Total

                                                32… Derivatives of Trans and Calc: Their Effect on Total Time

35… Fractal Pictures

46… Solar Sim Pictures

49… Cluster Pictures

 

 


Fractal Results

Here are two graphs recording the number of points that were in the Mandelbrot set vs. the time it took to calculate. The first accounts for the entire time vs. the entire number of points.  The second shows the three different processors and graphs their points independently. Both include trend graphs, the first is a polynomial and the second is linear.  

This graph shows the relative speeds of each node, a demonstration of the need for node weighting.

time p1

time p2

time p3

pixels p1

pixels p2

pixels p3

0.701

2.073

2.173

17976

69150

37358

0.741

1.262

2.383

18973

38962

64066

0.281

1.302

1.863

2035

40089

47039

0.24

0.601

2.974

66

11487

79449

0.25

0.841

2.774

87

19420

72048

1.292

1.933

3.996

36483

59913

113081

2.394

2.494

4.186

77237

74164

119939

2.453

2.553

4.176

79511

69133

119753

1.793

2.143

4.156

51657

60975

119996

0.711

2.073

2.183

17976

69150

37358

2.353

2.443

2.553

79920

43205

0

1.212

1.332

1.442

26980

27083

133

2.383

2.473

2.593

79518

68692

41919

3.315

3.415

3.545

115437

65992

42327

1.723

1.823

1.923

52522

37857

16776

1.612

1.722

1.832

47675

35736

4772

1.311

1.412

1.542

34798

23572

159

1.402

2.073

2.193

28783

60044

12196

2.474

2.584

2.684

74066

72324

3364

2.924

3.015

3.125

92150

89298

4142

3.225

3.325

3.425

109442

100402

11801

0.701

2.073

2.173

17976

69150

37358

0.981

1.883

1.993

28458

62661

36868

1.432

2.153

2.263

43551

71786

51017

0.701

2.073

2.164

17976

69150

37358

0.291

1.262

2.754

2309

38320

74529

0.861

1.843

2.659

22549

60582

70895

 


This Graph shows how not weighting the nodes during a fractal run, along with the variability of the program, affects the curve.

pixels

time

124484

2.174

96663

2.293

105763

2.453

85348

2.072

57893

1.763

159885

2.503

127775

2.073

336976

3.856

317507

3.576

225914

3.125

192411

2.684

48047

1.903

116447

3.175

94612

3.014

77301

2.714

67257

2.674

203569

3.735

95629

3.075

35249

2.834

8575

1.853

124484

2.194

68848

1.412

12456

0.641

3466

0.44

303

0.491

544

0.45

1112

0.451

25

0.49

5832

0.511

13036

0.842

241281

4.145

14822

0.981

38512

1.672

52784

2.002

73238

2.383

71133

2.373

86687

2.564

223312

3.676

316594

4.136

315132

3.886

311755

4.176

245226

4.146

223482

4.076

124484

2.173

157627

2.964

118489

2.704

194084

3.505

253344

3.896

278096

4.016

240083

3.856

243181

4.096

47359

2.224

154833

3.475

90133

3.005

71281

2.683

89370

2.664

113366

3.084

137027

3.255

144330

3.345

182494

3.735

124484

2.183

141322

3.475

123799

3.204

27058

0.991

108769

2.033

41838

1.412

66295

1.793

29475

1.252

24693

1.202

30277

1.352

 


This shows the difference the cluster makes in the calculation and the variability of the fractal calculation.

Comparison of different computers

 

Standardization:

 

 

Left:

-1.5

 

Right:

0.7

 

Top:

1

 

Bottom:

-1

 

Maxsteps:

254

 

Numcols:

900

 

Numrows:

900

 

 

 

 

One computer, 800 mz

Three nodes, not weighted

weighted, 3 nodes

14.591

4.366

2.964

14.521

4.376

2.974

14.801

4.416

3.024

14.551

4.376

2.964

14.32

4.386

2.984

14.331

4.386

2.974

15.132

4.386

2.964

14.771

4.376

2.964

14.701

4.357

2.975

14.711

4.376

2.975

14.751

4.366

2.974

14.631

4.386

2.965

14.771

4.367

2.975

14.641

4.386

2.964

 


This page freeze of task manager during a three-year simulation is descriptive of the tasks the
computer carries out during this time.

 



This graph shows the overall result for the fractal performance model.

 



This graph focuses on the effect nodes have on individual time calculations.


This graph shows that happens when the transmission time is greater than the calculation time, making the addition of a node take more time than not adding it.

Data for Fractal Performance Model:

FRACTALS

width

0.002777778

delay=

0.219272727

send time=

0.282181818

p=

640000

calc of one pixel

0.000401498

assumign equally shared between processors

 

n

trans time is (nodes-1)time delay plus send time

num sims= the negative of the log of desired width over log of scale factor (3)

celing function of the pervious

calc time= (pixels*calc of a pixel) over nodes

total= (trans+calc)*sims

 

 

 

 

 

 

 

2

0.501455

5.357763

6

128.4792

773.884

 

3

0.720727

5.357763

6

85.6528

518.2412

255.6428

4

0.94

5.357763

6

64.2396

391.0776

127.1636

5

1.159273

5.357763

6

51.39168

315.3057

75.77189

6

1.378545

5.357763

6

42.8264

265.2297

50.07605

7

1.597818

5.357763

6

36.70834

229.837

35.39271

8

1.817091

5.357763

6

32.1198

203.6214

26.21562

9

2.036364

5.357763

6

28.55093

183.5238

20.09756

10

2.255636

5.357763

6

25.69584

167.7089

15.81492

11

2.474909

5.357763

6

23.35986

155.0086

12.70028

12

2.694182

5.357763

6

21.4132

144.6443

10.36429

13

2.913455

5.357763

6

19.76603

136.0769

8.567379

14

3.132727

5.357763

6

18.35417

128.9214

7.15552

15

3.352

5.357763

6

17.13056

122.8954

6.026033

16

3.571273

5.357763

6

16.0599

117.787

5.108324

17

3.790545

5.357763

6

15.1152

113.4345

4.352564

18

4.009818

5.357763

6

14.27547

109.7117

3.722764

19

4.229091

5.357763

6

13.52413

106.5193

3.192406

20

4.448364

5.357763

6

12.84792

103.7777

2.741602

21

4.667636

5.357763

6

12.23611

101.4225

2.355198

22

4.886909

5.357763

6

11.67993

99.40102

2.021486

23

5.106182

5.357763

6

11.1721

97.66972

1.731301

24

5.325455

5.357763

6

10.7066

96.19233

1.47739

25

5.544727

5.357763

6

10.27834

94.93838

1.253948

26

5.764

5.357763

6

9.883016

93.8821

1.056287

27

5.983273

5.357763

6

9.516978

93.00151

0.880589

28

6.202545

5.357763

6

9.177086

92.27779

0.723716

29

6.421818

5.357763

6

8.860635

91.69472

0.583071

30

6.641091

5.357763

6

8.56528

91.23823

0.456491

31

6.860364

5.357763

6

8.288981

90.89607

0.34216

32

7.079636

5.357763

6

8.02995

90.65752

0.238548

33

7.298909

5.357763

6

7.786619

90.51317

0.144355

34

7.518182

5.357763

6

7.5576

90.45469

0.058473

35

7.737455

5.357763

6

7.341669

90.47474

-0.02005

36

7.956727

5.357763

6

7.137734

90.56677

-0.09202

37

8.176

5.357763

6

6.944822

90.72493

-0.15817

38

8.395273

5.357763

6

6.762063

90.94402

-0.21909

39

8.614545

5.357763

6

6.588677

91.21934

-0.27532

40

8.833818

5.357763

6

6.42396

91.54667

-0.32733

41

9.053091

5.357763

6

6.267278

91.92222

-0.37554

42

9.272364

5.357763

6

6.118057

92.34253

-0.42031

43

9.491636

5.357763

6

5.975777

92.80448

-0.46195

44

9.710909

5.357763

6

5.839964

93.30524

-0.50076

45

9.930182

5.357763

6

5.710187

93.84221

-0.53697

46

10.14945

5.357763

6

5.586052

94.41304

-0.57083

47

10.36873

5.357763

6

5.4672

95.01557

-0.60252

48

10.588

5.357763

6

5.3533

95.6478

-0.63224

49

10.80727

5.357763

6

5.244049

96.30793

-0.66013

50

11.02655

5.357763

6

5.139168

96.99428

-0.68635

51

11.24582

5.357763

6

5.0384

97.70531

-0.71103

52

11.46509

5.357763

6

4.941508

98.43959

-0.73428

53

11.68436

5.357763

6

4.848272

99.19581

-0.75622

54

11.90364

5.357763

6

4.758489

99.97275

-0.77694

55

12.12291

5.357763

6

4.671971

100.7693

-0.79653

56

12.34218

5.357763

6

4.588543

101.5843

-0.81507

57

12.56145

5.357763

6

4.508042

102.417

-0.83263

58

12.78073

5.357763

6

4.430317

103.2663

-0.84929

59

13

5.357763

6

4.355227

104.1314

-0.8651

60

13.21927

5.357763

6

4.28264

105.0115

-0.88011

61

13.43855

5.357763

6

4.212433

105.9059

-0.89439

62

13.65782

5.357763

6

4.144491

106.8139

-0.90798

63

13.87709

5.357763

6

4.078705

107.7348

-0.92092

64

14.09636

5.357763

6

4.014975

108.668

-0.93326

65

14.31564

5.357763

6

3.953206

109.6131

-0.94502

66

14.53491

5.357763

6

3.893309

110.5693

-0.95625

67

14.75418

5.357763

6

3.8352

111.5363

-0.96698

68

14.97345

5.357763

6

3.7788

112.5135

-0.97724

69

15.19273

5.357763

6

3.724035

113.5006

-0.98705

70

15.412

5.357763

6

3.670834

114.497

-0.99643

71

15.63127

5.357763

6

3.619133

115.5024

-1.00543

72

15.85055

5.357763

6

3.568867

116.5165

-1.01404

73

16.06982

5.357763

6

3.519978

117.5388

-1.0223

74

16.28909

5.357763

6

3.472411

118.569

-1.03023

75

16.50836

5.357763

6

3.426112

119.6069

-1.03784

76

16.72764

5.357763

6

3.381032

120.652

-1.04515

77

16.94691

5.357763

6

3.337122

121.7042

-1.05218

 


Solar Sim Results

This graph shows the relative speeds of a single laptop against the cluster, both running the same Solar Simulation.


The estimated time you save when using the cluster over the laptop is 243.293x+11.283.

Data:

Laptop

Cluster

392

133

711

208

1003

282

1351

357

 

429

 

502

 

575

 


This graph shows the relative speeds of each node, and how they diverge with the number of calculations.

n1

n2

n3

total in sec

78.26

73.51

82.11

133

153.32

143.687

160.872

208

227.147

214.048

239.815

282

302.194

284.079

317.687

357

374.158

352.437

394.137

429

447.203

420.795

470.546

502

519.727

489.244

547.117

575

 



This graph focuses on the startup time of transmission for the smallest possible structure.

 


Solar Sim tests of startup time using the minimum possible structure to send.

The second graph extends the transmission analysis to include sending differing amounts of information. Since Solar Sim sends back an integer containing RGB and the rocket ID, the amount of information transmitted is measured in ‘ints’. This allows the estimate of the time to transmit any given amount of data.

Raw data for both graphs:

K1000

timenum for nodes+transfer nide 205

208

211

1

0.13

0.13

0.14

1

0.12

0.14

0.15

1

0.14

0.151

0.161

1

0.15

0.16

0.16

1

0.13

0.14

0.15

2

0.14

0.15

0.16

20

0.131

0.141

0.151

100

0.14

0.14

0.15

200

0.12

0.13

0.14

200

0.14

0.15

0.15

200

0.17

0.17

0.18

300

0.13

0.13

0.14

300

0.15

0.15

0.16

300

0.131

0.141

0.151

400

0.15

0.16

0.16

400

0.14

0.14

0.15

1000

0.12

0.13

0.14

1500

0.13

0.14

0.14

2343

0.131

0.141

0.151

3000

0.14

0.14

0.15

4687

0.141

0.151

0.151

5000

0.19

0.2

0.22

6000

0.15

0.17

0.18

7000

0.19

0.21

0.23

9375

0.17

0.2

0.24

18750

0.24

0.32

0.41

37500

0.31

0.51

0.701

75000

0.541

0.952

1.352

150000

1.062

2.023

3.025

300000

1.873

3.475

5.108

600000

3.475

6.76

10.385

1200000

6.93

13.379

20.099

 



                To determine how scalable this system is to large numbers of computers, this graph shows the estimated runtime for Solar Sim using tens of nodes and therefore allowing for 3D simulation.

The graph is not smooth because the number of times needed to repeat the calculation for a desired result cannot be fractional, and so the ceiling function is needed. However, sometimes when one node is added, such as the break between 28 and 29 nodes, the extra node will throw the number of repeats just under a integer, and give an added calculation time decrease to that number of nodes.


                This shows how the variables in the 3D estimate change according to the number of nodes.

Raw data for both Solar Sim 3D performance model with one spacecraft per node:

n

trans time is (nodes-1)time delay plus send time

num sims= the negative of the log of desired width over log of rockets

celing function of the pervious

calc time= number of rockets, aka nodes, times one calc

total= (trans+calc)*sims

 

 

 

 

 

 

2

2.37075

#NUM!

#NUM!

21.2

#NUM!

3

3.7072

#NUM!

#NUM!

31.8

#NUM!

4

5.04365

#NUM!

#NUM!

42.4

#NUM!

5

6.3801

-17.5449

#NUM!

53

#NUM!

6

7.71655

-31.6747

#NUM!

63.6

#NUM!

7

9.053

-57.9146

#NUM!

74.2

#NUM!

8

10.38945

-134.564

#NUM!

84.8

#NUM!

9

11.7259

#DIV/0!

#DIV/0!

95.4

#DIV/0!

10

13.06235

168.4278

169

106

20121.54

11

14.3988

92.07947

93

116.6

12182.89

12

15.73525

66.43667

67

127.2

9576.662

13

17.0717

53.49565

54

137.8

8363.072

14

18.40815

45.65021

46

148.4

7673.175

15

19.7446

40.36195

41

159

7328.529

16

21.08105

36.54121

37

169.6

7055.199

17

22.4175

33.64207

34

180.2

6888.995

18

23.75395

31.3605

32

190.8

6865.726

19

25.0904

29.51362

30

201.4

6794.712

20

26.42685

27.98467

28

212

6675.952

22

29.09975

25.59214

26

233.2

6819.794

24

31.77265

23.7969

24

254.4

6868.144

26

34.44555

22.39303

23

275.6

7131.048

28

37.11845

21.26038

22

296.8

7346.206

30

39.79135

20.32397

21

318

7513.618

32

42.46425

19.53445

20

339.2

7633.285

34

45.13715

18.858

19

360.4

7705.206

36

47.81005

18.2706

19

381.6

8158.791

38

50.48295

17.75471

18

402.8

8159.093

40

53.15585

17.29719

18

424

8588.805

42

55.82875

16.88801

17

445.2

8517.489

44

58.50165

16.51935

17

466.4

8923.328

46

61.17455

16.18505

17

487.6

9329.167

48

63.84745

15.88015

16

508.8

9162.359

50

66.52035

15.60064

16

530

9544.326

52

69.19325

15.34321

16

551.2

9926.292

54

71.86615

15.10513

16

572.4

10308.26

56

74.53905

14.88411

15

593.6

10022.09

58

77.21195

14.67822

15

614.8

10380.18

60

79.88485

14.48582

15

636

10738.27

62

82.55775

14.30551

15

657.2

11096.37

64

85.23065

14.13607

15

678.4

11454.46

66

87.90355

13.97646

14

699.6

11025.05

68

90.57645

13.82575

14

720.8

11359.27

70

93.24935

13.68316

14

742

11693.49

72

95.92225

13.54799

14

763.2

12027.71

74

98.59515

13.41959

14

784.4

12361.93

76

101.2681

13.29744

14

805.6

12696.15

78

103.941

13.18103

14

826.8

13030.37

80

106.6139

13.06993

14

848

13364.59

82

109.2868

12.96374

13

869.2

12720.33

84

111.9597

12.86211

13

890.4

13030.68

86

114.6326

12.76472

13

911.6

13341.02

88

117.3055

12.67127

13

932.8

13651.37

90

119.9784

12.58152

13

954

13961.72

92

122.6513

12.49522

13

975.2

14272.07

94

125.3242

12.41214

13

996.4

14582.41

96

127.9971

12.3321

13

1017.6

14892.76

98

130.67

12.25491

13

1038.8

15203.11

100

133.3429

12.1804

13

1060

15513.46

102

136.0158

12.10842

13

1081.2

15823.8

104

138.6887

12.03883

13

1102.4

16134.15

106

141.3616

11.97149

12

1123.6

15179.54

108

144.0345

11.90629

12

1144.8

15466.01

110

146.7074

11.84311

12

1166

15752.49

112

149.3803

11.78185

12

1187.2

16038.96

114

152.0532

11.72241

12

1208.4

16325.44

116

154.7261

11.6647

12

1229.6

16611.91

118

157.399

11.60864

12

1250.8

16898.39

120

160.0719

11.55415

12

1272

17184.86

122

162.7448

11.50115

12

1293.2

17471.34

124

165.4177

11.44958

12

1314.4

17757.81

126

168.0906

11.39938

12

1335.6

18044.29

128

170.7635

11.35047

12

1356.8

18330.76

130

173.4364

11.30281

12

1378

18617.24

132

176.1093

11.25635

12

1399.2

18903.71

134

178.7822

11.21103

12

1420.4

19190.19

136

181.4551

11.1668

12

1441.6

19476.66

138

184.128

11.12363

12

1462.8

19763.14

140

186.8009

11.08146

12

1484

20049.61

142

189.4738

11.04026

12

1505.2

20336.09

144

192.1467

11

11

1526.4

18904.01

146

194.8196

10.96063

11

1547.6

19166.62

148

197.4925

10.92213

11

1568.8

19429.22

150

200.1654

10.88445

11

1590

19691.82

152

202.8383

10.84758

11

1611.2

19954.42

154

205.5112

10.81148

11

1632.4

20217.02

156

208.1841

10.77612

11

1653.6

20479.62

158

210.857

10.74148

11

1674.8

20742.23

160

213.5299

10.70754

11

1696

21004.83

162

216.2028

10.67427

11

1717.2

21267.43

164

218.8757

10.64164

11

1738.4

21530.03

166

221.5486

10.60964

11

1759.6

21792.63

168

224.2215

10.57825

11

1780.8

22055.24

170

226.8944

10.54745

11

1802

22317.84

172

229.5673

10.51721

11

1823.2

22580.44

174

232.2402

10.48752

11

1844.4

22843.04

176

234.9131

10.45837

11

1865.6

23105.64

178

237.586

10.42974

11

1886.8

23368.25

180

240.2589

10.4016

11

1908

23630.85

182

242.9318

10.37395

11

1929.2

23893.45

184

245.6047

10.34678

11

1950.4

24156.05

186

248.2776

10.32006

11

1971.6

24418.65

188

250.9505

10.29379

11

1992.8

24681.25

190

253.6234

10.26795

11

2014

24943.86

192

256.2963

10.24253

11

2035.2

25206.46

194

258.9692

10.21752

11

2056.4

25469.06

196

261.6421

10.19291

11

2077.6

25731.66

198

264.315

10.16869

11

2098.8

25994.26

200

266.9879

10.14484

11

2120

26256.87

202

269.6608

10.12136

11

2141.2

26519.47

204

272.3337

10.09823

11

2162.4

26782.07

206

275.0066

10.07546

11

2183.6

27044.67

208

277.6795

10.05302

11

2204.8

27307.27

210

280.3524

10.03091

11

2226

27569.88

212

283.0253

10.00912

11

2247.2

27832.48

214

285.6982

9.987652

10

2268.4

25540.98

216

288.3711

9.966485

10

2289.6

25779.71

218

291.044

9.945617

10

2310.8

26018.44

220

293.7169

9.925041

10

2332

26257.17

222

296.3898

9.904749

10

2353.2

26495.9

224

299.0627

9.884734

10

2374.4

26734.63

226

301.7356

9.864991

10

2395.6

26973.36

228

304.4085

9.845512

10

2416.8

27212.08

230

307.0814

9.826292

10

2438

27450.81

232

309.7543

9.807325

10

2459.2

27689.54

234

312.4272

9.788605

10

2480.4

27928.27

236

315.1001

9.770127

10

2501.6

28167

238

317.773

9.751885

10

2522.8

28405.73

240

320.4459

9.733874

10

2544

28644.46

242

323.1188

9.716089

10

2565.2

28883.19

244

325.7917

9.698525

10

2586.4

29121.92

246

328.4646

9.681178

10

2607.6

29360.65

248

331.1375

9.664043

10

2628.8

29599.37

250

333.8104

9.647114

10

2650

29838.1

252

336.4833

9.630389

10

2671.2

30076.83

254

339.1562

9.613863

10

2692.4

30315.56

256

341.8291

9.597532

10

2713.6

30554.29

258

344.502

9.581391

10

2734.8

30793.02

260

347.1749

9.565437

10

2756

31031.75

262

349.8478

9.549667

10

2777.2

31270.48

264

352.5207

9.534076

10

2798.4

31509.21

266

355.1936

9.518662

10

2819.6

31747.94

268

357.8665

9.50342

10

2840.8

31986.66

270

360.5394

9.488347

10

2862

32225.39

272

363.2123

9.47344

10

2883.2

32464.12

274

365.8852

9.458697

10

2904.4

32702.85

276

368.5581

9.444113

10

2925.6

32941.58

278

371.231

9.429686

10

2946.8

33180.31

280

373.9039

9.415413

10

2968

33419.04

282

376.5768

9.401291

10

2989.2

33657.77

284

379.2497

9.387318

10

3010.4

33896.5

286

381.9226

9.37349

10

3031.6

34135.23

288

384.5955

9.359805

10

3052.8

34373.95

290

387.2684

9.346261

10

3074

34612.68

292

389.9413

9.332855

10

3095.2

34851.41

294

392.6142

9.319584

10

3116.4

35090.14

296

395.2871

9.306446

10

3137.6

35328.87

298

397.96

9.293439

10

3158.8

35567.6

300

400.6329

9.280561

10

3180

35806.33

302

403.3058

9.267809

10

3201.2

36045.06

304

405.9787

9.255182

10

3222.4

36283.79

306

408.6516

9.242676

10

3243.6

36522.52

308

411.3245

9.230291

10

3264.8

36761.24

310

413.9974

9.218024

10

3286

36999.97

312

416.6703

9.205873

10

3307.2

37238.7

314

419.3432

9.193836

10

3328.4

37477.43

316

422.0161

9.181912

10

3349.6

37716.16

318

424.689

9.170098

10

3370.8

37954.89

320

427.3619

9.158393

10

3392

38193.62

322

430.0348

9.146795

10

3413.2

38432.35

324

432.7077

9.135302

10

3434.4

38671.08

326

435.3806

9.123913

10

3455.6

38909.81

328

438.0535

9.112626

10

3476.8

39148.53

330

440.7264

9.10144

10

3498

39387.26

332

443.3993

9.090353

10

3519.2

39625.99

334

446.0722

9.079363

10

3540.4

39864.72

336

448.7451

9.068468

10

3561.6

40103.45

338

451.418

9.057669

10

3582.8

40342.18

340

454.0909

9.046962

10

3604

40580.91

342

456.7638

9.036347

10

3625.2

40819.64

344

459.4367

9.025823

10

3646.4

41058.37

346

462.1096

9.015387

10

3667.6

41297.1

348

464.7825

9.005039

10

3688.8

41535.82

350

467.4554

8.994777

9

3710

37597.1

352

470.1283

8.9846

9

3731.2

37811.95

354

472.8012

8.974508

9

3752.4

38026.81

356

475.4741

8.964498

9

3773.6

38241.67

358

478.147

8.954569

9

3794.8

38456.52

360

480.8199

8.944721

9

3816

38671.38

362

483.4928

8.934952

9

3837.2

38886.23

364

486.1657

8.925261

9

3858.4

39101.09

366

488.8386

8.915647

9

3879.6

39315.95

368

491.5115

8.906109

9

3900.8

39530.8

370

494.1844

8.896646

9

3922

39745.66

372

496.8573

8.887257

9

3943.2

39960.52

374

499.5302

8.877941

9

3964.4

40175.37

376

502.2031

8.868697

9

3985.6

40390.23

378

504.876

8.859523

9

4006.8

40605.08

380

507.5489

8.85042

9

4028

40819.94

382

510.2218

8.841385

9

4049.2

41034.8

384

512.8947

8.832419

9

4070.4

41249.65

386

515.5676

8.82352

9

4091.6

41464.51

388

518.2405

8.814687

9

4112.8

41679.36

390

520.9134

8.80592

9

4134

41894.22

392

523.5863

8.797218

9

4155.2

42109.08

394

526.2592

8.788579

9

4176.4

42323.93

396

528.9321

8.780003

9

4197.6

42538.79

398

531.605

8.77149

9

4218.8

42753.64

400

534.2779

8.763037

9

4240

42968.5

402

536.9508

8.754646

9

4261.2

43183.36

404

539.6237

8.746314

9

4282.4

43398.21

406

542.2966

8.738042

9

4303.6

43613.07

408

544.9695

8.729828

9

4324.8

43827.93

410

547.6424

8.721671

9

4346

44042.78

412

550.3153

8.713572

9

4367.2

44257.64

414

552.9882

8.705529

9

4388.4

44472.49

416

555.6611

8.697541

9

4409.6

44687.35

418

558.334

8.689609

9

4430.8

44902.21

420

561.0069

8.68173

9

4452

45117.06

422

563.6798

8.673905

9

4473.2

45331.92

424

566.3527

8.666134

9

4494.4

45546.77

426

569.0256

8.658414

9

4515.6

45761.63

428

571.6985

8.650746

9

4536.8

45976.49

430

574.3714

8.64313

9

4558

46191.34

432

577.0443

8.635564

9

4579.2

46406.2

434

579.7172

8.628048

9

4600.4

46621.05

436

582.3901

8.620581

9

4621.6

46835.91

438

585.063

8.613163

9

4642.8

47050.77

440

587.7359

8.605793

9

4664

47265.62

442

590.4088

8.598471

9

4685.2

47480.48

 

               

 


One way to massively parallelize Solar Sim is to allow for many computers and have each run the
minimum process- the calculation of one spacecraft. However, since in some cases the transmission time will dwarf the calculation (as when there are huge numbers of computers increasing the delay time or a shorter simulation time) it is also important to look at how to balance the two factors of many computers with each computer calculating more spacecraft. This shows the calculation time when the work is shared evenly between all computers, as is used in this project.


The purpose of this analysis is to determine when the time benefits of adding a node are negligible. Each node costs money, time, and maintenance, plus boot time. When there are huge numbers of nodes but only a finite number of rockets the fixed delay transmission time will make the additional node only add to the
transmission wait without contributing to solving the problem faster. This graph shows the time improvement of adding another node (the derivative of the previous graph).

Raw data for 2D Solar Sim with multiple spacecraft per node:

n

trans time is (nodes-1)time delay plus send time

num sims= the negative of the log of desired width over log of rockets

celing function of the pervious

calc time= number of rockets times one calc over the nodes

total= (trans+calc)*sims

 

 

 

 

 

 

 

2

225

36.54121

37

1192.5

52447.5

 

3

225

23.05494

24

795

24480

27967.5

4

225

18.2706

19

596.25

15603.75

8876.25

5

225

15.73744

16

477

11232

4371.75

6

225

14.13607

15

397.5

9337.5

1894.5

7

225

13.01624

14

340.7143

7920

1417.5

8

225

12.1804

13

298.125

6800.625

1119.375

9

225

11.52747

12

265

5880

920.625

10

225

11

11

238.5

5098.5

781.5

11

225

10.56278

11

216.8182

4860

238.5

12

225

10.19291

11

198.75

4661.25

198.75

13

225

9.874829

10

183.4615

4084.615

576.6346

14

225

9.597532

10

170.3571

3953.571

131.044

15

225

9.353016

10

159

3840

113.5714

16

225

9.135302

10

149.0625

3740.625

99.375

17

225

8.939827

9

140.2941

3287.647

452.9779

18

225

8.763037

9

132.5

3217.5

70.14706

19

225

8.602126

9

125.5263

3154.737

62.76316

20

225

8.45484

9

119.25

3098.25

56.48684

21

225

8.319346

9

113.5714

3047.143

51.10714

22

225

8.19414

9

108.4091

3000.682

46.46104

23

225

8.077972

9

103.6957

2958.261

42.42095

24

225

7.969795

8

99.375

2595

363.2609

25

225

7.868721

8

95.4

2563.2

31.8

26

225

7.773998

8

91.73077

2533.846

29.35385

27

225

7.684979

8

88.33333

2506.667

27.17949

28

225

7.601105

8

85.17857

2481.429

25.2381

29

225

7.521892

8

82.24138

2457.931

23.49754

30

225

7.446917

8

79.5

2436

21.93103

31

225

7.37581

8

76.93548

2415.484

20.51613

32

225

7.308242

8

74.53125

2396.25

19.23387

33

225

7.243924

8

72.27273

2378.182

18.06818

34

225

7.1826

8

70.14706

2361.176

17.00535

35

225

7.124038

8

68.14286

2345.143

16.03361

36

225

7.068035

8

66.25

2330

15.14286

37

225

7.014404

8

64.45946

2315.676

14.32432

38

225

6.962979

7

62.76316

2014.342

301.3336

39

225

6.91361

7

61.15385

2003.077

11.26518

40

225

6.86616

7

59.625

1992.375

10.70192

41

225

6.820505

7

58.17073

1982.195

10.17988

42

225

6.776532

7

56.78571

1972.5

9.695122

43

225

6.734137

7

55.46512

1963.256

9.244186

44

225

6.693226

7

54.20455

1954.432

8.823996

45

225

6.653712

7

53

1946

8.431818

46

225

6.615515

7

51.84783

1937.935

8.065217

47

225

6.578562

7

50.74468

1930.213

7.722017

48

225

6.542785

7

49.6875

1922.813

7.400266

49

225

6.508121

7

48.67347

1915.714

7.098214

50

225

6.474511

7

47.7

1908.9

6.814286

51

225

6.441902

7

46.76471

1902.353

6.547059

52

225

6.410244

7

45.86538

1896.058

6.295249

53

225

6.37949

7

45

1890

6.057692

54

225

6.349596

7

44.16667

1884.167

5.833333

55

225

6.320522

7

43.36364

1878.545

5.621212

56

225

6.292229

7

42.58929

1873.125

5.420455

57

225

6.264683

7

41.84211

1867.895

5.230263

58

225

6.23785

7

41.12069

1862.845

5.049909

59

225

6.211699

7

40.42373

1857.966

4.878726

60

225

6.1862

7

39.75

1853.25

4.716102

61

225

6.161326

7

39.09836

1848.689

4.561475

62

225

6.137051

7

38.46774

1844.274

4.414331

63

225

6.113351

7

37.85714

1840

4.274194

64

225

6.090202

7

37.26563

1835.859

4.140625

65

225

6.067582

7

36.69231

1831.846

4.013221

66

225

6.045471

7

36.13636

1827.955

3.891608

67

225

6.02385

7

35.59701

1824.179

3.775441

68

225

6.002699

7

35.07353

1820.515

3.664399

69

225

5.982003

6

34.56522

1557.391

263.1234

70

225

5.961743

6

34.07143

1554.429

2.962733

71

225

5.941904

6

33.59155

1551.549

2.879276

72

225

5.922472

6

33.125

1548.75

2.799296

73

225

5.903432

6

32.67123

1546.027

2.722603

74

225

5.884771

6

32.22973

1543.378

2.649019

75

225

5.866475

6

31.8

1540.8

2.578378

76

225

5.848533

6

31.38158

1538.289

2.510526

77

225

5.830932

6

30.97403

1535.844

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225

5.813663

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30.57692

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225

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30.18987

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29.8125

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225

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6

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225

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225

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225

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225

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