Deterministic Models in Excel: Compliments to Large-Scale Simulation
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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 [email protected]
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
dRdtdBdt
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Common Lanchester Model ‘Flavors’
• For Aimed fire
• For Area fire
• For Ambush situations
= -
R
B
dB RdtdR Bdt
= -
R
B
dB RBdtdR RBdt
= -
R
B
dB RdtdR 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
dB P t RdtdR 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
dS SdtdI SI Idtd
I
R Idt
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Spreadsheet implementation
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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
dR t tdtdR 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:
212Pr{ } 1r
R r e
1.177ln 4CEP 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 k Ap BRhA
h
Lhit
T
A R pA
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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
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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 tB 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 dtdV
V t dt r tV 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
dR BdtdR 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
dR BdtdR BR t tdtdB R tdt
22 2 2
* 0 02 [ ] 2BN BN BNf f
B B R B
BNR B B RR
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You could also do this…
Iran
indonesia
India
Azerbaij
an
Pakista
n
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