Deterministic Models in Excel:Compliments to Large-Scale Simulation

Operations Research DepartmentNaval Postgraduate School, Monterey, CA

N81 Brown Bag24 July 2012

THIS PRESENTATION IS UNCLASSIFIED

CDR Harrison Schrammhcschram@nps.edu

831.656.2358

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My Intro

• N-81 Alumnus, currently on Faculty at NPS• Current work with Deterministic Modeling:

– Application to cyber– Applications to Infectious Disease

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Format

• Three Blocks, increasing in technicality• Block I : Fundamentals• Block II: Next Steps• Block III: The Frontiers

• After Block I, semi-open ended.

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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

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BLOCK I

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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?

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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

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Lanchester Model Story

Tabletop / Whiteboard

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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

,

B

R

dR

dtdB

dt

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Common Lanchester Model ‘Flavors’

• For Aimed fire

• For Area fire

• For Ambush situations

= -

R

B

dBR

dtdR

Bdt

= -

R

B

dBRB

dtdR

RBdt

= -

R

B

dBR

dtdR

BRdt

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A note about scaling

• Understanding Scaling is important in differential equation models.

[ ( )] ( ) [ ( )]RE B t h B t E t h

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Lanchester Model Vs. Simulation

0 0100, 100, .1, .3 50R B N

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Application: Spreadsheet Implementation

• We’ll do this in real time.

• How it can go wrong– Negative force levels

• Extensions and applications:– Reinforcements– Network Application

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Case Study: The battle of Iwo JimaEngel’s Analysis

( ) R

B

dBP t R

dtdR

Bdt

.0544R

.01066B

(0) 54,000P

(2) 6,000P (5) 13,000P

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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

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Block II: Next Steps

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Review:Telling a Story with Math

• These are the steps:1. Tell the story (stick figures, Legos, etc)2. Write discrete time, discrete space

equations3. See what happens.

• In this section, we will tell a new story and look at Lanchester Applications.

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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).

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The Story

• Whiteboard

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The Math

dSS

dtdI

SI Idtd

I

RI

dt

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Spreadsheet implementation

Sapphire Growth as an S-I Process

Courtesy: Stefan Savage.

DShield is the Distributed Intrusion Detection System Project (www.dshield.org)

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Lanchester with Shocks: An application to Networked Forces

These slides are shamelessly stolen from my MORS presentation

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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*

*

*

BN

B

R

dRt t

dtdR

t tdtdB

tdt

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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:

21

2Pr{ } 1r

R r e

1.177ln 4

CEP CEP

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Reduction in as a function of CEP

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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:

|[ ( )] kill hit hitp pE h Bk h

|[ ( )] Lkill hit

T

E kA

p BRhA

h

Lhit

T

AR p

A

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Case Study II: Networked Battle of Iwo Jima

.0544R

.0544BN

.0106B

* 3t

.0544R

(2) 5,000P (5) 11,000P

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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?

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The Impact of Network Failure

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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.

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Block III: The Frontiers

This section contains current research.

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S-I and Stuxnet

Applying S-I model to Stuxnet…Unclassified data from W.32 Stuxnet Dossier,

Symantec Corporation White Paper

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 500.000

5000

10000

15000

20000

25000

30000

35000

40000

Stuxnet Propagation by Country

IranindonesiaIndiaAzerbaijanPakistanMalaysiaUSAUszbekistanRussiaGreat Britain

Days since zero

Mac

hine

s Inf

ecte

d

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Best-fit Cross-Infectivity Rates

This is a notional sketch to show what you could do with this data if you had it.

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Stochastic Lanchester

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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 ( )

~ ( ),( )

R t hBinom B t

B t

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Stochastic Diffusions and Lanchester

• We may consider this as a stochastic diffusion, with the Stochastic Differential Equations:

• Which lead to the Ordinary Differential Equations:

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) R

B

Y t dt r t dW t

dY t X t dt

dX

b d

t

t W t

2 ( ) ( )

2 ( ) ( )XX XY

YY XY

XY XX YY

dV dt

dV

V t dt r t

V t dt b t dt

dV V V

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VarianceDashed lines are simulation, Solid lines SDEs

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Covariance

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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.

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Fin.

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Backups

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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:

*

*

= - t < t

= -

BN

B

R

dRB

dtdR

B t tdtdB

R tdt

2 2 2 20 0

*

( )BN B f R f

BN B

B B RB

R

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Aimed Fire → Area Fire: Model and Result

• Conversely, in this situation, network reduction causes us to go from aimed fire to area fire

*

*

= - t < t

= -

BN

B

R

dRB

dtdR

BR t tdtdB

R tdt

22 2 2

* 0 02[ ] 2BN BN BN

f fB B R B

BNR B B RR

You could also do this…

Iran

indonesia

India

Azerbaij

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Pakist

an

Malaysi

aUSA

Uszbek

istan

Russia

Great B

ritain

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Iranindonesia

IndiaAzerbaijan

PakistanMalaysia

USAUszbekistan

RussiaGreat Britain

Stuxnet infectivity parameters (Least Squares Fit)

IranindonesiaIndiaAzerbaijanPakistanMalaysiaUSAUszbekistanRussiaGreat Britain