General Rate Theory and its Application to Attrition Modeling

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General Rate Theoryand its Application to
Attrition Modeling

• Bruce W. Fowler, Ph. D.
• U. S. Army Aviation and Missile Command
• Aviation Missile Research, Development, and
  Engineering Center
• Advanced Systems Directorate
• Redstone Arsenal, AL 35898
• Bruce.fowler_at_rdec.redstone.army.mil

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The views expressed in this presentation are
those of the author and do not represent any
official position of the United States Army

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Outline of Presentation

• General Rate Theory
• Basic Components
• Combining the Components
• Attrition Rate Theory
• Assumptions and Overview of Attrition Combat
• Combat Process Derivations
• Attrition (Lanchester) Differential Equations

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General Rate Theory
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General Rate Theory 1

• Many Physical processes manifest as punctuated
  events.
• Want to represent these as continuous
• Time is usually the independent variable
• But not necessarily!
• Causative processes may be stochastic or
  deterministic

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General Rate Theory 2

• Causative processes may be repetitive
• Events caused by actions of a collection of
  agents (elements)
• Collection population may be time dependent
• Collection population may be one agent
• Collection population may be regenerative
• May be multiple collections
• This is the basic idea of General Rate Theory

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Examples

• Physics
• Nuclear Decay
• Light (photon) scattering
• Lasers
• Chemistry
• Chemical Reactions

• Ecology
• Predator-Prey
• Operations Research
• Queues
• Attrition
• Communications

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Basic Components
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GRT Components

• Probability Theory
• Processes/events may be stochastic
• Renewal Theory
• Processes/events may be repetitive
• Order Statistics
• Have collections of agents
• Functional Theory
• Events are punctuated

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Probability Theory
We identify the Cumulative Probability Function
or Cumulative Distribution Function (cdf), and
the Probability Density Function (pdf).
Negative Exponential Distributions (NED), A
variation on the NED is the Gamma
Distribution, which occurs naturally in the
Renewal Theory of NED.
is RATE, properly defined by postulate
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Renewal Theory
The mathematical formalism used to describe this
repetition of a process is Renewal Theory. ref.
Renewal Theory is a probabilistic counting
methodology. That is, it counts the occurrence of
repeated events whose evolution is described in
terms of probability functions, usually functions
of time. Each event is referred to as a renewal,
and has a cdf. Events are sequential, that is,
the first renewal occurs before the second, and
so forth, and independent.
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The expected number of renewals by time is
expressed by the Renewal Function, which is
defined as
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where the parenthetical superscripted R
indicates renewal. Because the cdf are assumed to
be continuously differentiable, we may
immediately define a renewal density function
(rdf) as
General Form
Generator/ Initiator
Propagator
Shorthand
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Simple Generator
Pdf follows directly
Uniform propagator pdf
Evolution Equation
NED special case
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Order Statistics

• Where we used Renewal Theory to describe the
  dynamics of a single element, we use Order
  Statistics to describe the behavior of the
  several as a group..
• We want to consider two pictures of probability,
  which we may call the implicit and explicit
  temporal representation pictures. The explicit
  picture is the more familiar. In this picture,
  the probability functions are seen as dependent
  variables of (nominally) time. In the explicit
  time picture, we operate directly on the time
  variable. In the implicit picture, we operate
  directly on the probability functions as
  variables.This requires that the probability
  functions (at least the cdfs) are monotonically
  increasing over time and that there is a
  one-to-one correspondence between probability and
  time. This is not particularly difficult, but
  there is one idea that we must accept. In this
  picture, we may speak of the idea that a
  probability may have moments (e.g., expected
  value) and because of the one-to-one
  relationship, this implies a time.

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If we write the unit function
Binomial Expansion
Probability that event r of N has occurred
More on next slide
pdf of event r of N
Note Implicit Picture!
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Order Statistics Event pdf

• Divide view into three groups
• N-r events have not occurred
• r-1 events have occurred
• 1 event occurs, but cant tell which of r it is
• Derivative results in form given
• More details in Herbert Davids book Order
  Statistics

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Examples of Explicit and Implicit Expectation
Values
Inherent Normalization
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Extension to Non-Integer number of Elements
for values of kgtR, where indicates the
integer value of the argument, the exponent of y
becomes negative, and the sign of the product
begins to oscillate. The series, equation thus
becomes oscillatory beyond this value of k.
Further, for this part of the series, the product
will grow in magnitude slower than k!, so that
each succeeding term not only oscillates, but
also gets smaller.
Uniformly Convergent
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Setting Up for Later
cdf of kth event
pdf of kth event
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Functional Theory
How do we interpret
When we have been defining
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The basic idea of functional (or distribution)
theory is that instead of describing a function
with a value at every point , we use the
real number for every belonging to a
class of accessory functions. In this
representation, is a functional on the class of
accessory functions.
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For developing rate theory, the class of
accessory functions is comprised of the
individual probability functions.
The functional, designated by is defined by the
integral relationship which may be
multi-dimensional. This integral spans the space
of and the accessory functions must span this
space as well. The derivative of the functional
is given by This obviously means that the
accessory functions be zero or constant at the
boundary of the spanned space.
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that if the function has discrete jumps that
occur at points , then its functional
derivative has the form where is the
piecewise continuous derivative, and is the
signed magnitude of the ith jump. We shall refer
to this as the jump factor or function, depending
on form..
Example of Weak Delta Function Gaussian
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Let us consider the differential equation . If
is a continuous function, then we may find
a classical (or strict) solution
that obeys the differential equation continuously
at every point . If, however, is a
functional, then is a solution of the
differential equation if and only if
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Combining the Components
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Putting It Together

• Event times described by Probability Theory
• Deterministic events have strong delta function
  pdfs
• Repeated Events described by Renewal Theory
• Collective actions described by Order Statistics
• Discrete to Continuous described by Functional
  Theory
• But we still have to fit it all together!

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Basic Rate Theory

• Write derivative using functional theory,
• Note that we are going to ignore continuous
  terms,
• Separate into gain, loss components for
  simplicity of thought.

Unit jumps for now!
Have NOT specified what x is!
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Nature of OS Delta Functions

• Number of interacting (attriting) elements
  changing we dont care since we established
  validity of Binomial for real number of elements
  (and were gaping to sum it away!)
• Realness of attriting elements ditto
• But still have to show that we have at least weak
  delta functions here.

Graphical Evidence in P picture not good enough!
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Differential that r of N events has occurred
Change of variable nice integration range!
Simple rewrite, then a McWheeny or Saddle Point
Expansion
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Set first derivative to zero to get extremum
behavior
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Now write the expansion to second order
Definition of convenience
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Write out exponents
Manipulate a bit
Use Stirlings Approximation
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Non-Exponent Piece
Yields
Which behaves like weak delta function
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Bottom Line
This behaves like weak delta function
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Comparison of Curves
Shape off comes from only using second order
expansion
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Back to the DE
Still unit jump!
Differential equation in OS probabilities and time
Change index of summation
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General Form (unit Jump!)
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Continuous Population Behavior
Same Behavior!
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Jump Functions
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Attrition Rate Theory
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ART Outline

• Background of Previous Attrition Theory
• Underview of Attrition Theory
• Attrition Rate Theory Specifics
• Combat Processes
• Attrition Rate Differential Equations

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Previous Attrition Theory
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Ad hoc ADE Theory
Basic Homogeneous Lanchester Attrition Theory ADE
postulated
Quadratic ADE
Linear ADE
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Lanchester 1

• The ad hoc nature of these LADE (and of most ADE
  in general) follow from their postulation.
• The basic idea is that the rate at which
  attrition occurs (the rate of change of a force
  strength,) depends on the force strengths times
  an attrition rate coefficient that is constant.
• Lanchester originally associated the Quadratic
  LADE with Modern combat, characterized by
  long-range, direct fire where targets were always
  plentiful.

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Lanchester 2

• Lanchester associated the Linear LADE with
  Ancient combat, characterized by short-range
  physical contact between shooter and target
  so that the availability of targets entered
  into the rate.
• Because the rate at which a shooter fired
  (Modern combat) was most easily represented as a
  constant (shots per time,) then attritions per
  time per shooter would be constant.
• Similarly, for Ancient combat, when blows were
  being dealt, which depended on the availability
  of targets, they were also presumed constant in
  rate.

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Lanchester 3

• Models of contemporary land combat associate
  indirect (artillery) fire with area lethality,
  and direct fire with point lethality.
• The literature tends to treat these synonymously,
  although there are structural differences between
  indirect/direct fire and area/point lethality.
• Since indirect fire is normally conducted out of
  Line-of-Sight (LOS), indirect fire shooters do
  not generally perform target assessment.

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Lanchester 4

• Target assessment is generally performed by
  direct fire shooters.
• Area lethality attrition depends on the number of
  targets in the lethal area and thus attrition
  depends on the density of force strength.
• Point lethality has a lethal area smaller than
  an individual target, so the density effect is
  one or zero.

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Lanchester 5

• Examination of these LADEs within this
  contemporary view and the simple two sub process
  model gives two sets of interpretations of the
  Quadratic and Linear LADE.
• The Quadratic LADE is applicable to
• point lethality, direct fire combat when the
  target shooting sub process is limitingly slow
  compared to the target acquisition sub process,
  and to
• area lethality, indirect fire when the target
  density remains constant over time (occupied area
  shrinks.)

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Lanchester 6

• The Linear LADE is applicable to
• point lethality, direct fire combat when the
  target acquisition sub process is limitingly slow
  compared to the target shooting sub process, and
  to
• area lethality, indirect fire when the target
  density varies over time (occupied area remains
  constant.)
• Thus, the two LADE forms apply to both types of
  combat although the common association is of the
  Quadratic LADE with point lethality, direct fire
  combat and the Linear LADE with area lethality,
  indirect fire combat

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Stochastic Lanchester

• Integer valued force strength
• Probability that i,j enemy, friendly alive at
  time t
• Normalization
• Initial Condition
• Expected Value may be used,
• But commonly use evolution equation

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Evolution Equation
Other Moments Too!

• Leads to interesting things, like Conclusion
  Conditions where probability accumulates in
  conclusion states.
• The Evolution Equations are still ad hoc!

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Heterogeneous Lanchester

• Move from one collection of identical elements
  per side to several collections of identical
  elements per side
• Refer to each collection as a Force Strength
  Component
• Component may be system or organizational
  aggregation
• Extend Homogeneous Quadratic Equations

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• From conclusion of TA, shooter is paired or tied
  to its target, thus only part of each force
  strength component is engaged with a particular
  force strength component
• To recover the general form of the ADE, and
  simplify the problem, introduce Fire Assignment
  Factors
• fraction of force strength component engaging
  opposing force strength component at time t.

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• This recovers the form of the ADE
• Can then write in vector/tensor form
• Note naming convention some call components
  ARCs, some distinguish two factors

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Conjugate Rate Theories

• Basic Lanchester Attrition Theory lacks means to
  calculate Attrition Rate Coefficients.
• Primarily concerned with heterogeneous problem
• Two major pieces
• ARC calculation
• Fire Assignment Factor calculation
• Minor (?) piece
• LOS loss

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Bonder-Farrell Theory of ARCs

• Based in Renewal Theory
• Loose connection appears to assume average
  renewal cycle representative of ARC
• Uses Blackwells Theorem to calculate ARC,
• Form of ADE

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• For simple, alternating (effectively uniform) two
  process serial attrition process, gives
  homogeneous ADE
• Presumes linearity of shooters ad hoc nature of
  ADE
• Infinite time limit?

Simplification based on Open Literature
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Fire Assignment

• Natural Fire Assignment w/wo rejection
• No prioritization
• No Look Ahead/Around
• No Target Queue

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LOS Gain/Loss

• Model of targets coming in, going out of LOS
• Measure average length of LOS segment, Average
  distance between LOS segments, divide into
  average speed
• Form Evolution Equations

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• Convert these evolution equations (EE) in numbers
  to densities, interpret densities as
  likelihood's,
• Define renewals of coming in, going out of LOS
• Identify equivalent renewal for combat process
  mechanics

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Differentiate definitions
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Simplify
Identify pdfs
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Effect of LOS on Combat Processes

• Target Acquisition First
• TA can end either by acquisition or LOS loss
• OR process
• Generate pdf

• Fraction of LOS interrupted cycles or renewal
  is, from Taylor,

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Important Result
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• Since will not quit TA when lose target to LOS
  loss, resume this is a renewal
• Renewal will continue till have target can
  calculate effective pdf of target acquisition
  from terminating renewal

LOS Loss has no Effect on TA!
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• Target Shooting not the same.
• If shooting interrupted by LOS loss, must go back
  for new target.
• Combine to get OR pdf, and introduce
  probability of LOS interruption as likelihood
  renewal ends in attrition.
• Now counting attritions and LOS loss in Attrition
  Renewal

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(Attrition) Rate Theory
This is the problem How do we smoothly map from
this to a continuous ADE? (or any type of
differential equation.)
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Background Information

• Proper Aggregation
• Combat Processes
• Probability Theory
• Functional Theory
• Ad hoc ADE Theory

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Proper Aggregation

• Aggregation is the combining of disparate
  elements into a mass..
• Proper Aggregation is a somewhat ad hoc process
  by which the modeler decides that the members of
  some collection of observable things may be
  considered identical.,
• Proper Aggregation is an aggregation of
  definition while Formal Aggregation is an
  aggregation of transformation.
• Proper Aggregation generally has associated with
  it a Resolution of some sort

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• The test of a Proper Aggregation is largely
  statistical. For example, there are two
  components to Proper Aggregation for an attrition
  thing. How the thing attrits other things and
  how it is attrited.
• Proper Aggregation should be based on application
  of the precepts of Systems Engineering
• In general, we shall refer to the things that
  have been Properly Aggregated together as
  elements. This terminology is intended to
  reinforce the essentially fundamental nature of
  these class-aggregated things within the modeling
  context. In Classical Lanchester Theory,
  Lanchester uses the term units, but we have
  chosen not to use this terminology to avoid
  confusion with the military organizational use of
  this term.

Identicality of Elements!
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Combat Processes
Combat Processes are the activities performed by
the elements within the context of the modeling
problem

• The primary sub processes of attrition are
  usually considered to be
• Target Acquisition the detection, recognition,
  classification, identification, etc. of an
  observed entity to be an enemy element to be
  engaged in combat.
• Target Communication the communication of the
  position, type, etc. information of a target from
  the Target Acquisition sub process
  instrumentality to the Target Shooting sub
  process instrumentality. This sub process may be
  simple and direct, and thus often not explicitly
  considered, or complex and indirect as we shall
  consider later when we address Phase Aggregation.

Usually Slide over TC
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Combat Processes 2

• Target Shooting the engagement of the target
  element until it is attrited, or the sub process
  is prematurely terminated.
• Target Assessment the assessment that the
  target has been attrited. This sub process is
  usually lumped with Target Shooting for Close
  Combat elements, but is separate for other forms
  of engagement. This sub process is closely
  associated with Battle Damage Assessment at the
  Operational Level of War

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Why ART?

• Why do we want to do this?
• Subject the ad hoc nature of ADE to closer
  scrutiny
• Look at the constant ARC interpretations
• Put a firmer body of theory under Attrition
  Theory shore up its foundation
• Demonstrate there is another way of getting to
  ADE than Stochastic Lanchester and Duel Theory
  which do not agree with LAT!

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Assumptions 1

• Combat elements of a given type are identical
  (Proper Aggregation)
• Combat is described by processes
• Processes have temporal duration and are
  described by probability distributions
• Processes are often (generally) repetitive in
  combination.

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Assumptions 2

• Standard Lanchester Assumptions Kerr
• Renewal Theory, in some general form, is a
  reasonable approximation of what a combat
  elements does in combat
• Order Statistics is a reasonable means of
  collecting the individual actions into a
  collective picture
• Functional Theory is the Hammer, Renewal Theory
  Order Statistics are the anvil

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Combat Processes(attrition)

• Target Acquisition
• Target Communication
• Target Shooting
• Target Assessment

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Target Acquisition
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Target Acquisition(more detail)
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Target Shooting
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Target Shooting(more detail)
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Target Communication
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Glimpse Theory
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Glimpse Theory 2
Only reflects constant glimpse time
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ART of Target Acquisition 1
1 Target
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Art of Target Acquisition 2
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ART of Target Acquisition 3
The pdf is NED, but with non-linear argument
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ART of Target Acquisition 4
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ART of Target Shooting 1
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ART of Target Shooting 2
pdf is NED
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Serial Architecture Attrition Process
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Local Parallel Architecture Attrition Process
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Example Serial TATS
Simple Alternating Renewal on NED
Combined pdf
Laplace Transform
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Renewal Evolution Is Linear
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Set or Summation
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Where we are

• Up to now, fairly general (except unit jump!)
• Include renewal/repeated nature,
• Specify for ADE form

Recognize rdf

• Confirms linearity
• Accommodates other LAT type assumptions
• Inherently time dependent rdf ARC?

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Recovering BASIC Lanchester ADE
Do serial geometry since were going to take
limits Context Put ARC in expected time form
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Look at limit TA becomes glacially slow
Now get serious with ART Recall NED can act like
delta function
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Serial Geometry renewal
Apply delta function limit
Thus
Giving rdf
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Point Lethality
Slow Target Acquisition Already worked Gives ADE
Slow Target Shooting Similar to above Will not
repeat Gives ADE
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Area Lethality

• Change from attrition renewal to fall of shot
  renewal rate of fire in rich target environment
  (adjusted rate of fire otherwise)
• Pdf still NED
• Change in jump function
• Was magnitude one for point attrition
• Now function of target density does not depend
  on delta function argument

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Further generalization of ART ADE
Must calculate jump functions
ADE (general) is
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Constant Target Area
Lanchester Linear ADE
Constant Target Density
Lanchester Quadratic ADE
ART replicates Basic Lanchester
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ART Results

• Constant Coefficient Lanchester ADE result when
• One combat process Attrition process
• Combat process NED
• Otherwise Variable Coefficient?

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Two Combat Process ADEs
Basic ADE
Point Lethality unit jump function
Serial Geometry Problem Renewal function follows
from before
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ART ADE
Lanchester Bonder Farrell (LBF) ADE
Look at differences in ARCs and Force Strengths
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Parallel Geometry Problem More complicated
renewal TA runs by itself
But TS has a complicated generator
Do NOT know general solution, but can take slow
sub process limits If (e.g.,) TA very slow
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Reduces renewal to
Which is now linear and solvable Similar
situation for other limit (slow TS) Get compact
form for rdf for both cases
ART and LBF ADE follow as before (You figure it
out!) But well still look at rdf/ARCs and Force
Strengths
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Note that they seem to converge slower!
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A Simple Approximation
rdf looks like
Look at Lanchester Quadratic Explicit FS solutions
Postulate
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Varies slowly with time! Write
And recall that FS losses are small for large
forces
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Serial Geometry
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Parallel Geometry
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Reinforcements
Basic ad hoc LADE form
Now have to introduce FS densities because we
care when elements come into battle
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Primitive of density rate equation
Must address preference for engagement assume
cant tell how long on battlefield
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By definition of total FS
If the renewals simple NED,
And the total FS is
Which recovers LADE
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Approach to EquilibriumConnection to Lanchester
Bonder Farrell (?)

• LADE results only if renewal is simple, uniform
  NED
• Sizable differences between ART and LBF ADE
• Is this important?
• Wed really like to have constant ARCs
• Is outer time variation short lived?
• What is approach to equilibrium

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Renewal Function for two sub process is
If constant ARC, equivalent is
Ratio first to second
And remove time as variable
This shows the approach to equilibrium or
equivalence
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Back to explicit serial, parallel geometry
problems
These are symmetric, so we can look at either
limit, but we have to satisfy limits!
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Looks good for Serial!
Not so good for Parallel!
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Heterogeneous Two Process Renewal
Back to Basic ART DE
And generalize to components
More explicit ART form
Have to pull out jump functions because of LOS No
fratricide!
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Markov Renewal Theory
Force Strength Components Also number of
component elements engaging opposing component
elements
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Serial Geometry Problem Pdf has form
Target rejection by zero parameter
value Otherwise natural fire assignment with
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Jump functions must account for loss of LOS,
otherwise would be one for point lethality
Evolution equations of the renewals are
Reflects contribution for renewals in other
engagement channels Look at Picture
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Each of the (e.g.,) second Red arrows has a
component from all of the first arrows
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Now collapse the ADE and put in the renewals
Which we again collapse to rdf
And put in the fire assignments, and jump
functions
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If we make the fire assignments explicit for
natural selection with (?) rejection
Still working on the rdfs, but we can make a
quick approximation Write out the first three
renewals explicitly
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Note the common summations all were really
interested in here is the overall gross flow of
probability. Introduce the approximation for the
shooting
And make the expected times match
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Invert this
And rewrite the renewals as
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This doesnt look like a very good approximation
But it gives an interesting result!
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The LT rdf is
Since all we have is an alternating renewal that
is non-uniform at start and stop. This has a
final form for the rdf of
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That gives us an LBF ARC of
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And an LBF ADE of
That can be compared to other work
Research Continues
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Back-Up Slides
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The Map of Attrition Fowlers View with a slant
to this presentation