Deterministic Models in Excel: Compliments to Large-Scale Simulation

1. CDR Harrison Schramm hcschram@nps.edu 831.656.2358 Deterministic Models in Excel:Compliments to Large-Scale Simulation Operations Research Department Naval Postgraduate School, Monterey, CA N81 Brown Bag 24 July 2012 THIS PRESENTATION IS UNCLASSIFIED

2. My Intro • N-81 Alumnus, currently on Faculty at NPS • Current work with Deterministic Modeling: • Application to cyber • Applications to Infectious Disease UNCLASSIFIED

3. Format • Three Blocks, increasing in technicality • Block I : Fundamentals • Block II: Next Steps • Block III: The Frontiers • After Block I, semi-open ended. UNCLASSIFIED

4. References: • Aircraft in War: Dawn of the Fourth Arm. F. L. Lanchester • The Pleasures of Counting. T.W. Korner • Epidemic Modeling: Daley and Gani • Lanchester Models of Warfare (vol. 1 and 2), James G. Taylor UNCLASSIFIED

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6. BLOCK I UNCLASSIFIED

7. Why are we doing this? • The usefulness of doing “Paper-and-Pencil” analysis • As a supplement to simulations • To guide the right questions! • Fast, Transparent • Where does this not apply? UNCLASSIFIED

8. Three Steps for mathematical modeling • Tell a story • Draw a picture or use Legos • Write discrete time, discrete space model • How would you play this game with two people and some dice? • Take limits* • Analytic Results* *These steps are not always necessary UNCLASSIFIED

9. Lanchester Model Story Tabletop / Whiteboard UNCLASSIFIED

10. Lanchester Models Pit two sides, Blue and Red, against each other, and analyze the resulting combat as a deterministic model. In their most general form, where the gammas represent arbitrary functions. We explore specific choices, and their consequences subsequently UNCLASSIFIED

11. Common Lanchester Model ‘Flavors’ • For Aimed fire • For Area fire • For Ambush situations UNCLASSIFIED

12. A note about scaling • Understanding Scaling is important in differential equation models. UNCLASSIFIED

13. Lanchester Model Vs. Simulation UNCLASSIFIED

14. Application: Spreadsheet Implementation • We’ll do this in real time. • How it can go wrong • Negative force levels • Extensions and applications: • Reinforcements • Network Application UNCLASSIFIED

15. Case Study: The battle of Iwo JimaEngel’s Analysis UNCLASSIFIED

16. Part I Wrap-up • In Block I we discussed: • How to tell a story with mathematics • How to implement this in Microsoft Excel • With added emphasis on: • What can go wrong • Where these methods do not apply UNCLASSIFIED

17. Block II: Next Steps UNCLASSIFIED

18. Review:Telling a Story with Math • These are the steps: 1. Tell the story (stick figures, Legos, etc) 2. Write discrete time, discrete space equations 3. See what happens. • In this section, we will tell a new story and look at Lanchester Applications. UNCLASSIFIED

19. New Model: Infectious Diseases:The S-I and S-I-R Models • The Story: A fixed population of N individuals who interact with each other at some intensity has a pathogen introduced • May be ‘simple’ (S-I) epidemic, or epidemic with removals (S-I-R). UNCLASSIFIED

20. The Story • Whiteboard UNCLASSIFIED

21. The Math UNCLASSIFIED

22. Spreadsheet implementation UNCLASSIFIED

23. Sapphire Growth as an S-I Process Courtesy: Stefan Savage. DShield is the Distributed Intrusion Detection System Project (www.dshield.org)

24. Lanchester with Shocks: An application to Networked Forces These slides are shamelessly stolen from my MORS presentation UNCLASSIFIED

25. Shock Action - modification • Consider a model in which the dynamics of combat change suddenly and irrevocably at a deterministic time, t*. • Our solutions to follow are implicit in the corresponding variables, which we call B* or R* UNCLASSIFIED

26. The effect of the Network on Targeting • If ordnance errors are equal and uncorrelated, we may say that they are circularly distributed, and Where the common unit of error is Circular Error Probable (The radius that encloses ½ of the rounds fired), which may be converted by: UNCLASSIFIED

27. Reduction in as a function of CEP UNCLASSIFIED

28. When should we just switch from Aimed to Area fires? • Let be the firing rate. For Aimed fire: • For Area fire: • We should prefer area fire iff: UNCLASSIFIED

29. Case Study II: Networked Battle of Iwo Jima UNCLASSIFIED

30. We may ask… • Suppose that Blue has a vulnerable network, but plans like his network was invulnerable, uses Lanchester for his planning and plans for a 10% casualty rate. • Suppose further that the quality of his network gives him parity with the advantage for being ‘dug in’ • We may ask: What’s the impact of having a his network fail? UNCLASSIFIED

31. The Impact of Network Failure UNCLASSIFIED

32. Part II Wrap-up • In this section, we: • Derived the model for infectious diseases from first principles • Applied in a spreadsheet • Showed how Lanchester models may be adapted for Cyber Effects. UNCLASSIFIED

33. Block III: The Frontiers This section contains current research. UNCLASSIFIED

34. S-I and Stuxnet UNCLASSIFIED

35. Applying S-I model to Stuxnet…Unclassified data from W.32 Stuxnet Dossier, Symantec Corporation White Paper

36. Best-fit Cross-Infectivity Rates This is a notional sketch to show what you could do with this data if you had it. UNCLASSIFIED

37. Stochastic Lanchester UNCLASSIFIED

38. Lanchester Equations: A probabilistic Approach • We said earlier that we’re using the Expected value (or mean field) approximation to the process. • Expectation of what? • Following the assumptions of the Lanchester Model, the Distribution for the blue losses in a ‘small’ interval is UNCLASSIFIED

39. Stochastic Diffusions and Lanchester • We may consider this as a stochastic diffusion, with the Stochastic Differential Equations: • Which lead to the Ordinary Differential Equations: UNCLASSIFIED

40. VarianceDashed lines are simulation, Solid lines SDEs UNCLASSIFIED

41. Covariance UNCLASSIFIED

42. Block III Wrap up • In this section, we moved ‘into the frontiers’: • Cyber Applications of S-I • Stochastic Lanchester • Thank you for your time and interest. UNCLASSIFIED

43. Fin. UNCLASSIFIED

44. Backups UNCLASSIFIED

45. Aimed Fire → Aimed Fire Model and results • In this situation network loss causes us to go from highly effective aimed fire to less accurate aimed fire. The model is specified as: UNCLASSIFIED

46. Aimed Fire → Area Fire: Model and Result • Conversely, in this situation, network reduction causes us to go from aimed fire to area fire UNCLASSIFIED

47. You could also do this…