Deterministic Models in Excel:
Compliments to Large-Scale Simulation
CDR Harrison Schramm
hcschram@nps.edu
831.656.2358
Operations Research Department
Naval Postgraduate School, Monterey, CA
N81 Brown Bag
24 July 2012
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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,
dR
dt
dB
dt
  B
  R ,
where the gammas represent arbitrary functions.
We explore specific choices, and their
consequences subsequently
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Common Lanchester Model ‘Flavors’
• For Aimed fire
dB
dt
dR
dB
• For Area fire
dt
dR
dt
   R RB
dt
 RR
= - B B
= - B R B
dB
• For Ambush situations
dt
dR
dt
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 RR
= - B B R
A note about scaling
• Understanding Scaling is important in
differential equation models.
E [ B ( t  h )]  B ( t )  E [  R ( t )]h
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Lanchester Model Vs. Simulation
R 0  100, B 0  100,   .1,   .3 N  50
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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 Jima
Engel’s Analysis
P (2)  6, 000
P (5)  13, 000
 R  .0544
dB
dt
P (0)  54, 000
dR
 B  .01066
dt
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 P (t )   R R
 BB
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
equations
3. 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
  SI
dt
dI
  SI   I
dt
dR
 I
dt
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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)
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*
dR
dt
dR
dt
dB
dt
  BN t  t
  B t  t
  R
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t
*
*
The effect of the Network on Targeting
• If ordnance errors are equal and uncorrelated,
we may say that they are circularly distributed,
and
Pr{ R  r }  1  e
1 r 
  
2  
2
Where the common unit of error is Circular Error
Probable (The radius that encloses ½ of the
rounds fired), which may be converted by:
 
CEP
ln 4

CEP
1.177
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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:
E [ k ( h )]   p kill |hit p hit B h
• For Area fire:
E [ k ( h )]   p kill |hit
AL
BRh
AT
• We should prefer area fire iff:
AL
AT
R  p hit
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Case Study II: Networked Battle of Iwo
Jima
P (2)  5, 000
 R  .0544
P (5)  11, 000
 R  .0544
 BN  .0544
 B  .0106
t 3
*
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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
Stuxnet Propagation by Country
40000
35000
30000
Iran
Machines Infected
indonesia
25000
India
Azerbaijan
20000
Pakistan
Malaysia
15000
USA
Uszbekistan
10000
Russia
Great Britain
5000
0
0.00
50.00
100.00
150.00
200.00
250.00
Days since zero
300.00
350.00
400.00
450.00
500.00
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 ~ B inom  B (t ), R (t )  h 

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B (t ) 
Stochastic Diffusions and Lanchester
• We may consider this as a stochastic diffusion,
with the Stochastic Differential Equations:
dX ( t )    Y ( t ) dt 
 r ( t ) dW R ( t )
dY ( t )    X ( t ) dt 
 b (t ) d W B (t )
• Which lead to the Ordinary Differential
Equations: d V   2  V ( t ) d t   r ( t )d t
XX
XY
d V YY   2  V X Y ( t ) d t   b ( t ) d t
d V X Y    V X X   V YY
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Variance
Dashed 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:
dR
dt
dR
dt
dB
dt
= -  BN B t < t
*
 BN B0   B B f   R ( R0  R f )
2
 BB t  t
*
B* 
= -  R R t
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2
 BN   B
2
2
Aimed Fire → Area Fire:
Model and Result
• Conversely, in this situation, network
reduction causes us to go from aimed fire to
area fire
dR
dt
dR
dt
dB
dt
= -  BN B t < t
*
   B BR t  t
= -  R R t
*
R* 
 BN
B
 BN
2


2
B
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R 
2
0
 BN
R
[B  B ]  2
2
0
2
f
 BN
B
Rf
You could also do this…
Stuxnet infectivity parameters (Least Squares Fit)
0.018
0.016
0.014
0.012
0.016-0.018
0.014-0.016
0.01
0.008
Iran
India
Pakistan
USA
Russia
0.006
0.012-0.014
0.01-0.012
0.008-0.01
0.006-0.008
0.004
0.002
0
0.004-0.006
0.002-0.004
0-0.002