Title Page
ABSTRACT
Contents
EXECUTIVE SUMMARY 14
Ⅰ. INTRODUCTION 17
A. BACKGROUND 17
B. GENERAL SCENARIO 18
C. SCOPE AND OBJECTIVES 19
D. LITERATURE REVIEW 20
Ⅱ. ONE-ON-ONE MODEL 23
A. THE SCENARIO 23
B. DISCRETE-TIME (ABSORBING) MARKOV CHAIN 24
1. Parameters 25
2. General Assumptions 26
3. Transition Matrix for the Model 27
Ⅲ. ANALYSES FOR THE ONE-ON-ONE MODEL 29
A. SENSITIVITY ANALYSIS TO BLUE'S SCOOTING POLICY 29
1. Fixed Parameters 29
2. Sensitivity to the Probability of Blue's offensive scooting (b) 30
3. Sensitivity to the Probability of Blue's defensive scooting (c) 33
B. SENSITIVITY OF BLUE'S WIN PROBABILITY TO THE PERFORMANCE OF COUNTER-BATTERY RADAR 37
Ⅳ. TWO-ON-ONE MODEL 39
A. THE SCENARIO 39
B. DISCRETE-TIME MARKOV CHAIN FOR THE TWO-ON-ONE MODEL 39
1. General Assumptions 41
2. Transition Matrix for the Two-on-One Model 42
Ⅴ. ANALYSES FOR THE TWO-ON-ONE MODEL 43
A. SENSITIVITY ANALYSIS TO BLUE'S SCOOTING POLICY 43
1. Fixed Parameters 43
2. Sensitivity to the Probability of Blue's offensive scooting (b) 43
3. Sensitivity to the Probability of Blue's defensive scooting (c) 47
B. SENSITIVITY OF BLUE'S WIN PROBABILITY TO THE PERFORMANCE OF COUNTER-BATTERY RADAR 49
Ⅵ. MANY-ON-MANY MODEL 51
A. THE SCENARIO 51
B. CONTINUOUS-TIME (ABSORBING) MARKOV CHAIN 52
1. Parameters 52
2. General Assumptions 53
3. Transitions Diagram for the Model 53
4. Sample Transitions 54
Ⅶ. ANALYSES FOR THE MANY-ON-MANY MODEL 57
A. FIXED PARAMETERS FOR SENSITIVITY ANALYSIS 57
B. SENSITIVITY TO THE RATE OF BLUE'S OFFENSIVE SCOOTING 58
C. SENSITIVITY TO THE RATE OF BLUE'S DEFENSIVE SCOOTING 58
Ⅷ. CONCLUSIONS 61
A. SUMMARY 61
B. FUTURE WORK 62
APPENDICES 63
APPENDIX A. THE SITUATIONS AND ASSUMPTIONS FOR TRANSITIONS OF THE ONE-ON-ONE MODEL 63
APPENDIX B. THE SITUATIONS AND ASSUMPTIONS FOR TRANSITIONS AND TRANSITION PROBABILITIES OF THE TWO-ON-ONE MODEL 67
LIST OF REFERENCES 109
INITIAL DISTRIBUTION LIST 111
Table 1. Possible states for Blue and/or Red 24
Table 2. Descriptions of each state of DTMC for the one-on-one model 25
Table 3. Definitions and notations of probability parameters 25
Table 4. Descriptions of each state of DTMC for two-on-one model 40
Table 5. Descriptions of each state of Continuous-Time Markov Chain (CTMC) for many-on-many model 52
Table 6. Sample transitions for the many-on-many model 54
Figure 1. Transition matrix for the one-on-one model 27
Figure 2. The parameter values for the one-on-one analysis 29
Figure 3. Blue's win probability for two scenarios of scooting policies of Red (a) and two scenarios of detection... 31
Figure 4. Blue's win probability for four scenarios of Red's scooting policies 33
Figure 5. Blue's win probability for four scenarios of Red's scooting policies 35
Figure 6. Blue's win probability for four scenarios of Red's scooting policies 36
Figure 7. Graph of sensitivity analysis to β₂ and δ₂ 37
Figure 8. Transition matrix for the two-on-one model 42
Figure 9. Blue's win probability for four scenarios of Red's scooting policies 45
Figure 10. Blue's win probability as a function of scooting policies of Red (a) and detection probability (b) 47
Figure 11. Blue's win probability for four scenarios of Red's scooting policies 49
Figure 12. Graph of sensitivity analysis to β₂ and δ₂ 50
Figure 13. Flow diagram for the many-on-many battle 51
Figure 14. Transitions diagram for the many-on-many model 54
Figure 15. The parameter values of sensitivity analysis for the many-on-many model 57
Figure 16. Blue's win probability for four scenarios of Red's scooting policies in the many-on-many model 59