Battlespace/Information War (BAT/IW): A System-of-Systems Model of a Strike Operation

Abstract

This paper presents a low-resolution, high-level modeling methodology for the analysis of the effectiveness of a Blue system of systems operating in a battlespace. The methodology enables quick turn around and efficient exploration of sensitivities of measures of Blue combat success to realistically imperfect Blue intelligence, surveillance and reconnaissance capabilities: limited and imperfect sensor surveillance and reconnaissance, particularly battle damage assessment (BDA), and finite, hence saturable, communications and weapons delivery capability. The model explicitly represents aircraft sorties, fires, sensor/shooter latencies, target losses, imperfect target type classification, imperfect weapon assignment, and BDA; various levels of the above imperfections can be applied, facilitating tradeoff studies, The model is deterministic/expected value in nature, although it analytically represents time-dependent stochastic features such as system saturability. Model experimentation suggests the following results. Decreasing shooter latency can result in greater attrition than correspondingly increasing the probabilities of correct BDA or weapon assignment, although at the expense of a greater number of weapons fired per target killed. Erroneous BDA returns dead targets to the shooter targeting list. These dead targets not only result in wasted weapon expenditure but also take sensor/shooter resources away from legitimate live targets. Increasing the probability of correct BDA can result in a greater number of targets killed during a time period than increasing the probability of correct weapon assignment.

Full text

NPS-OR-02-005
NAVAL
POSTGRADUATE
SCHOOL
Monterey,
California
BATTLESPACE
/
INFORMATION
WAR
(BAT/IW):
A
System-of-
Systems
Model
of
a
Strike
Operation
by
Donald
P.
Gaver
Patricia
A. Jacobs
August
2002
Approved
for
public
release;
distribution
is
unlimited.
Prepared
for.
The MOVES
Institute
Naval Postgraduate
School
Monterey,
California
93943
20020912
042Z

NAVAL
POSTGRADUATE
SCHOOL
MONTEREY,
CA
93943-5000
RADM
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R.
Ellison
Richard
Elster
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Provost
This
report
was
prepared
for the
MOVES
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Postgraduate
School,
Monterey,
CA 93943,
and
funded
by
the
Wayne
E.
Meyer
Institute
of
Systems
Engineering,
Naval
Postgraduate
School,
Monterey,
CA
93943;
Space-C2-Information
Warfare,
N6, 2000
Navy
Pentagon,
Washington,
DC
20350-2000;
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CA
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BATTLESPACE
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INFORMATION
WAR
(BAT/IW):
A
System-of-Systems
Model
of
a
Strike
Operation
N0001402WR20147
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Donald
P.
Gaver
and
Patricia
A.
Jacobs
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DISTRIBUTION/AVAILABILITY
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.
ABSTRACT
(Maximum
200
words.)
This paper
presents
a
low-resolution,
high-level
modeling
methodology
for
the
analysis
of
the
effectiveness
of
a
Blue
system
of
systems
operating
in
a
battlespace.
The
methodology
enables
quick
turn
around
and
efficient
exploration
of
sensitivities
of
measures
of
Blue
combat
success
to
realistically
imperfect Blue
intelligence,
surveillance
and
reconnaissance
capabilities: limited and
imperfect sensor surveillance
and
reconnaissance,
particularly battle
damage assessment
(BDA), and
finite,
hence
saturable,
communications
and weapons
delivery
capability.
The
model
explicitly represents
aircraft
sorties,
fires,
sensor/shooter
latencies,
target
losses,
imperfect
target
type classification,
imperfect
weapon
assignment,
and
BDA;
various
levels
of
the
above
imperfections
can be
applied,
facilitating
tradeoff
studies.
The
model
is
deterministic/expected
value
in
nature,
although
it
analytically represents
time-dependent
stochastic
features
such
as
system
saturability.
Model
experimentation
suggests the
following
results.
Decreasing
shooter
latency
can
result
in
greater attrition
than
correspondingly
increasing
the
probabilities
of
correct BDA
or weapon
assignment,
although
at
the
expense
of
a
greater
number
of
weapons
fired
per
target
killed.
Erroneous
BDA
returns dead
targets
to
the
shooter
targeting list.
These
dead
targets
not only
result
in
wasted
weapon
expenditure
but
also
take
sensor/shooter
resources
away
from
legitimate
live
targets.
Increasing
the
probability
of
correct
BDA
can
result
in
a
greater
number
of
targets
killed
during
a
time period
than
increasing
the
probability
of
correct weapon
assignment.
1.
SUBJECT
TERMS
15.
NUMBER
OF
deterministic network
of
queues
with
saturation; intelligence, surveillance,
and
reconnaissance; imperfect
sensors;
PAGES
target losses
64
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BATTLESPACE
/
INFORMATION
WAR
(BAT/IW):
A
System-of-Systems
Model
of
a
Strike Operation
Donald
P.
Gaver
Patricia
A.
Jacobs
Abstract
This
paper
presents
a
low-resolution,
high-level
modeling
methodology
for the
analysis
of
the
effectiveness
of
a
Blue
system
of
systems
operating
in
a
battlespace.
The
methodology
enables
quick
turn
around
and
efficient
exploration
of
sensitivities
of
measures
of
Blue combat
success
to
realistically
imperfect
Blue
intelligence,
surveillance
and
reconnaissance
capabilities:
limited
and
imperfect
sensor
surveillance
and
reconnaissance,
particularly
battle
damage
assessment
(BDA),
and
finite,
hence
saturable,
communications
and
weapons delivery
capability.
The
model
explicitly
represents
aircraft
sorties,
fires,
sensor/shooter
latencies,
target
losses,
imperfect target
type
classification,
imperfect
weapon
assignment,
and
BDA;
various
levels
of
the
above
imperfections
can
be
applied,
facilitating
tradeoff
studies.
The model
is
deterministic/expected
value
in
nature,
although
it
analytically
represents
time-dependent
stochastic
features such
as
system
saturability.
Model
experimentation
suggests
the
following
results.
Decreasing
shooter
latency
can
result
in
greater attrition than
correspondingly
increasing
the
probabilities
of
correct
BDA
or
weapon
assignment,
although
at
the
expense
of
a
greater
number
of
weapons
fired
per
target
killed.
Erroneous
BDA
returns
dead
targets
to
the
shooter
targeting
list.
These
dead
targets
not
only result
in
wasted
weapon
expenditure
but
also
take
sensor/shooter
resources
away
from
legitimate
live
targets.
Increasing
the
probability
of
correct
BDA can
result
in a greater
number
of
targets
killed
during
a
time
period
than
increasing the
probability
of
correct
weapon
assignment.

1.
Introduction
This
paper
provides
a
low-resolution,
high-level
methodology
"scoping
model"
for
analysis
of
a
Blue
strike
operation:
missile-shooting
and
manned
aircraft
sortie
response
to
a
Red
ground
force
that
enters
a
region
for
hostile
purposes
(regional
occupancy
with
territorial
objective,
staging
for
missile,
e.g.,
SCUD,
shots,
etc.).
For
example,
the
beginning
of
a
Major
Regional
Contingency
(MRC).
The
model
proposed
facilitates
efficient
initial
exploration
of
sensitivities
of
measures
of
Blue
combat
success
to
realistically
imperfect
Blue
intelligence,
surveillance
and
reconnaissance
(ISR),
and
information
warfare
(IW)
capabilities:
limited
and
imperfect
sensor
surveillance
and
reconnaissance,
particularly
BDA,
and
finite,
hence
saturable,
communications
and
weapons
delivery
capability.
The effect
of
both
Red
and
Blue
(aircraft)
decoys
and
false
targets
are
implicitly
present
and
readily
analyzed.
It
is
proposed
that
somewhat
more
detailed
representation
of
Red
air defense
(AD)
and
Blue
suppression
of
enemy
air
defense
((J)SEAD)
be
included
in
the
present
formulation.
The
ultimate
aim
of
the
model
is
to
suggest
the
value
of
alternative
operational
architectures
and
investment
programs.
The
model
realistically,
but
economically,
assesses
the
capability
of
limited
Blue
system-of-system
capacities
to
provide
needed
services:
sensor
regional
coverage
and
potential
target
classification
capability;
communication,
with
delays
(bandwidth
limits);
and
shooter
firing
rate
and
lethality.
Highly
nonlinear
unfavorable
responses
to
deficiencies
in
such
capabilities,
and
in
their
relative
balance
and
mutual compatibility,
are
easily
and
quickly
revealed
by
exploration
of
the
model.
These
deficiencies
can
be
potentially
rectified
by
suitably
improving,
modifying,
and
balancing
Sensor-Shooter
capabilities.
In
particular
the
effect
of
reducing
bandwidth
requirements,
e.g.,
by
storing
and
occasionally
updating
slowly
changing
information
appropriately,
is
implicitly
reflected
in
the
model:
the
above
effect
simply
increases
service
rates
(decreases
latency).
Sensitivity
to
such
an
architectural
design
possibility
is
traceable
using
the
model;
the

effect
may
be
considerable
whenever
the
original
system
operates
near
a
saturation
level,
wherein
message
traffic
nearly
reaches
or
exceeds
available
processing
capacity
(bandwidth).
The
model
does
not, however,
address
any
downside
issues
associated
with
increased
local
information
storage.
1.1
Model
Features
In
the
models
of
the
present
paper
realism
and
needed
detail
are
introduced,
but
parsimoniously.
In
particular
(see
Figure
1),
Spatial
considerations
are
accounted
for:
the
region
9Th
is
viewed
as
a
collection
of
non-overlapping
and
inclusive
subregions
{1
9
ý,
i
=
1,
2,
...
,
I};
for
instance,
choice
of
these
allow
for
explicit
range
dependencies
(introduction
of
range
bands
is
a
convenient
simple
device).
The
exact
specification
of
the
subregions
is
left
to
analyst
discretion;
it
may
be
reasonable
to
let
these
be
internally
homogeneous
in
the
sense
of
terrain,
hence
visibility.
The
number
of
subregions
is
arbitrary
in
principle,
but
limited
by
computational
considerations.
Some
averaging
within
and
between
subregions
is
inevitable.
If
desired,
and
meaningful,
a
subregion
can
be
an
established
route,
e.g.,
road,
from
one
point
to
another.
*
Red
force
type
variability
is
recognized:
types
are
broad
but flexible,
being
at
present
and
only
for
example,
Heavily
Armored
Vehicles
(generically
Tanks
or
Hard
Targets),
Light
Armored
Vehicles
(Armored
Personnel
Carriers),
Unarmored
Vehicles
(Trucks,
etc.),
Infantry,
...
generically
classified
as
Soft
Targets.
The
state
of
Red forces
is
taken
to
be
characterized
(at
minimum)
by
the
numbers
of
each
of
its
force
elements
(see
above)
in
each
of
the
designated
subregions
at
a
particular
time.
If
desired,
the
above
state
description
can
be
extended
or
expanded
(for
instance,
the
numbers
of
Red targets
in
a
particular
formation,
moving
or
still
at
a
given
time,
can
be
recorded
as
state
variables
that
evolve
together
as
time
elapses);
the
above
is
essential
to a
dynamic
description
of
2

system
evolution.
The
state
of
Red
is
a
(vector-valued)
description
that
changes
dynamically
in
time,
responding
to
Red
territorial
objectives,
and
in
reaction
to
actions
by
the
Blue
force.
"*
Red
force
mobility/maneuver
across
subregions
(e.g.,
advance
or
retreat)
is
modeled.
This
motion
can
well
be
a
response
to
perceived
Blue
actions.
"*
Blue
Sensor
and
ISR
("Network-Centric')
force
capabilities
and
actions
are
modeled
(for
operational
effect,
without
engineering
systems-level
detail):
the
Sensor
effort
("servicing
capacity")
allocated
to
subregion
91ý-
at any
time
determines
the
rate
at
which
data
on
Red
occupancy
of
that subregion
can be
obtained
and
transferred
to
a
Blue
candidate-for-targeting
list;
note
that
the
model
recognizes
that
it
takes
time
for
Blue
Sensor/ISR
assets
to
discover/process
data
on
Red
presence
in
a
subregion,
so
if
insufficient
Blue
assets
are
available
there,
an
effective
queue/waiting
line
develops
made
up
of
undetected
Reds.
The
variation
and
adaptation
of
Blue
Sensor
assets
across
space
(subregions)
in
response
to
Red
movements
and
actions
is
a
feature
of
modem
Network-Centric
warfare.
It
is
also
realistic
to
model
the
realistic
errors,
delays,
and
susceptibility
to
deception
and
jamming
inherent
in
any
information-gathering
system;
this
is
done
in
such
a
way
as
to allow
assessment
of
the value
of
increased
Sensor
asset
capability,
and
tactics
for
employment
thereof.
The
fact
that
Red
forces
(types
and
locations)
are
known
by
Blue
only after
delay
and
with
error
is
represented
by
the
Sensor/ISR
submodel.
In
reality,
Blue
must
act
on
the
basis
of
an
imperfect
quantitative
perception
of
the
true
Red
state.
The
consequences
of
that
imperfection
are
that his
subsequent
actions
("shooting,"
maneuvering)
are
affected.
Our
model
indicates
the
value
of
altering
the
system
components
of
our
system-of-systems.
*
Blue
Shooter's
varied
assets
and
capabilities
are
represented
in
adjustable
detail:
surface-surface
and
air-surface
missiles
or
bombs
are
viewed
as
tailored
to,
or
3

optimized
for,
particular
target
types.
For
instance,
a
weapon
suited
to
destruction
of
Heavy
Armor
can
certainly
kill
Light
Armor,
but
at
greater
than
necessary
expense
per
round,
but
a
weapon
optimized
for
Light
Armor
or Infantry
will
likely
have
small
chance
of
killing
Heavy
Armor.
The
effect
of
misallocating
weapons
to
targets,
largely
caused
by
ISR
mistakes
and
delays,
but
also
influenced
by
logistics
(weapon
supply),
can
be
substantial,
as
measured
by
cost
to
Blue
in
own
casualties,
in
dollars,
and
in
campaign
length.
Of
course
the
number,
hence
shooting
("service")
rate
of
the
Blue
Shooter
(missile
firing
tubes,
plus
aircraft
sorties)
influences
the
length
of
the
target
queue:
the greater
the
gross
Shooter
rate,
the
shorter
the
delay
in
targeting
a
Red
that
has
been
detected
and
classified
(although
possibly
incorrectly)
by
the
Blue
Sensor.
As the
rate
of
target
addition
to
the
Shooter
Target
List/Queue
increases,
the
greater
the
(missile)
Shooter's
effective
delay
(latency)
and
the
smaller
the
chance
of
a
successful
shot:
this
effect
is
highly
nonlinear
as
potential
target
input
approaches
Shooter
capacity,
and
thus
is
a source
of
important
detrimental
sensitivity.
Certainly
additions
to
the
target
list/queue
that
are
the
result
of
Sensor
mistakes
(initial
misclassifications,
or
incorrect
BDA
leading
to
re-targeting
of
killed
Red
assets)
can
be
seen
to
have
cascading
detrimental
effects
on
system
effectiveness,
particularly
when
the
overall
system
becomes
heavily
loaded,
i.e.,
as
increasingly
many
targets
are
placed
on
the
Shooter's
list.
The
model
helps
identify
requirements
for
controlling
such
bottleneck
situations.
The
present
model
is
deterministic/expected-value,
although
it
reflects
time-dependent
stochastic
features
such
as
system
saturation
analytically,
using
a
mathematical
device;
cf.
Filipiak
(1988).
The
basic
model
formulation
logic
can
be
made
to
govern
a
discrete-event
simulation
if
desired,
or
as
the
basis
for
probabilistic
analysis;
see
Gaver
and
Jacobs
(1999)
for
a
simplified,
but
4

analytically
treatable,
version.
The
present
modeling
technique
facilitates
quick
approximate
investigations
and
model
browsing
for
interesting effects.
2.
Example
Sensor-Shooter
Architectures
Consider
two
specific
examples
of
Sensor-Shooter
architectures:
(A)
Architecture
1:
General
Regional
Coverage
(GRC),
Delayed
Battle
Damage
Assessment
(BDA),
in
which
prosecuted (shot-at) targets
are
released
back
into their
present
(sub)region
for
eventual
discovery
and
re-classification
by
a
Regional
Sensor
System,
or
(B)
Architecture
2:
Local,
Quick
Follow-up
(LQF)
and
General
Regional
Coverage
(GRC),
Battle Damage
Assessment
(BDA),
in
which
a
Local
Quick
Follow-up
sensor
capability
exists
to
immediately
assess
the
results
of
a
shot
(but
with
potential error)
and the
just-targeted
Red
then
re-targeted
immediately
if
judged
alive;
if
alive
and
judged
dead
it is
released
for
eventual
later
rediscovery
by
the
GRC
(until
such
discovery
it is
free
to
engage
in
hostile
acts).
Tomahawk,
Block
4,
has
some
LQF
capabilities.
Both
Architectures
are
represented
as
generic.
It
will
be
the
objective
of
subsequent
work
to
characterize
the
operational
capabilities
of
potential
new
systems
and
architectures
by
adjusting
the
parameters
of
our
model.
Then
model
experimentation
can
be
used
to
suggest
specific
system investments
to
best
add
operational
value.
3.
Illustrative
Invasion-Strike
Scenario
Figure
1
depicts
a
region,
9R,
into
which
Red
units
are
envisioned
to
pour
from the
North.
Their
objective
is
to
move
Southwards,
but
perhaps
to
punctuate
the
journey
with
occasional
hostile
actions
such
as
missile
(e.g.,
SCUD)
shots.
The
Figure
1
legend
describes
illustrative prototypical
Red
units
and
their
general
states
of
perception
by
the
Blue
ISR.
The
Red
units
may
be
in
any
of
the
range
bands
shown:
R
4
to
R
1
.
The types shown
here
are
(1)
Heavy
Armor
Units
(Hard
Targets)
both
5

detected
and
undetected,
and
hence
on
a
Blue
virtual
target
list;
(2)
Light Armor
and/or
Infantry,
undetected,
but
also
detected
and on
the Blue
virtual
target
list;
(3)
Decoys
(deliberate)
and
False
Targets
(natural/environmental,
e.g.,
civilian
non-combatants),
undetected
and
also
detected
and on
the
Blue
virtual target
list.
These
can
be
both
undetected
and
detected,
and
placed on
a
target
list.

Invasion-Strike
Scenario
Red Input
x
1
Ax
A
\
A
V\
(missile
shot)
Blue
CV
Amphibious
Troop/
Blue
Missile
(Fighters, UAVs)
Weapons
Carriers
Ship
Legend:
n
=
armor
unit (undetected
&
unclassified)
El
=
armor
unit
(detected/classified, targetable)
MzA
X
=
light
armor
and/or
infantry
(undetected
& unclassified)
X
= light
armor
and/or
infantry
(detected/classified,
targetable)
A
=
decoys
or
false
targets
RJj.
=
subregions
(defined geographically);
range
bands
Figure
1
7

4.
Flow
Charts
of
Operational
Architectures
and
Blue
Perception
In
Figures
2
and
3
we
introduce
flow
charts
to
describe
the
general
motion
of
Red
units
between
states
of
concern
to
the
Blue
system
of
systems.
Figure
2
depicts
the state
transitions
for
units
under Architecture
1,
for
which
the
sensor
coverage
is
of
the various
regions.
Note
that this
coverage
may be
made
variable
and
adaptive,
and
that
detections occur
at
effective
(sweep)
rates
determined
by Blue
decision makers.
Presumably
the
Blue
allocation
of
detection
effort
should
be made
to
conform
to
anticipated
and
predicted
Red
motion
in
the
region;
likewise
that
Red motion
can
react
to
perceived
Blue
sensor presence,
which
is
of
analyst-adjustable
acuity.
The
model
equations
(see
Appendix
I)
can
be
made
to
represent
the
various
feedbacks
that
adjust
Red and
Blue
behavior.
SENSOR
SYSTEM
(REGIONAL
COVERAGE)
"SHOOTER
SYSTEM
UNDETECTED
"'.
REDS
.-
-
NEW
RED
ILoss
of
Track
I
DETECTED
SHOOTERS
ARRIVALS
TARGET
REDS
l
Missiles
CANDIDATES
-
Target
Gunnery
I
~
1
ls/q
eu
Manned
a/c
I
"
"SHOT-AT
-"" --
t
- -. .-- -
_.1-
"
"(ALIVE,
CLASS.
DEAD)
SHTATIV
(HIT,
ED
ALVE
(MISSED)
(ALIVE.
CLASS.
ALIV
(MISSED
ALIVE)
" • "
~~(DEAD,
CLASS.
ALIVE),r
DAD
%
DEAD)
S
....
(DEAD,
CLASS.
DEAD)
0
All
target
classification
and
BDA
decisions
are
error-prone
or
fallible
to
adjustable
degree.
Hence
the
DETECTED
REDS
target
queue
contains
alive
targets,
and
also
dead
(presumed
alive)
and dead
(unclassified
or
BDAed)
Figure
2.
Architecture
1:
General
Regional
Coverage
Sensors
(GRC)
(Delayed
Shooter
BDA)
Manned
a/c
have
immediate
BDA
as
in
Figure
3
Figure
3
represents
state
transitions
of
units
governed
by
Architecture
2.
Here
we
augment
the
GRC
capability
by
an
Immediate
Shooter
BDA
capability
associated
with
8

the
missile
shooters
(only):
when
a
Blue
missile
is
fired
at
a
Red
target
in
9ki.
the
act
is
presumed
to
be
immediately
followed
by
a
BDA
sensor
(possibly
on
the missile);
if
(1) a
Blue miss
or
failure
to
kill
is
registered,
an
immediate
return
to
the target
list
is
scheduled,
while
(2)
if
a
Blue
kill
is
registered,
the
target
is
immediately
declared
dead
and new
targets
are
engaged. We
model
the
effects
of
the
several
possible
errors:
in
case
(1)
the
mistake
of
registering
a
miss
when
a
kill
actually
occurred results
in
at
least
one
wasted
shot,
while
in
case
(2)
the error
of
mistakenly
registering
a
kill
means
that the
targeted
Red
is
free
to
engage
in
further
operations
hostile
to
Blue.
SENSOR
SYSTEM
(REGIONAL
COVERAGE)
.""............................."'.
S
O
T R
S
S
E
UNDETCTEDSHOOTER
SYSTEM
"
UNDETECTED
'
: REDS
- ---
- --
- --
- " --
REDS-
NEW
RED
Loss
ofTrack
'
DETECTED
SHOOTERS
I
ARRIVALS
TARGET
REDS
Missiles
I
CANDIDATES
Gunney
I
~
lis/queu
*.~Manned
a/c
(ALIVE.
CLASS.
DEAD)
--
-
SESRSSE
1
(LOCAL;CLOSE
- • -•.• I -
<ISSD.
LIVE
-- /
FOLLOW-UP)
"(MISSED,
ALIVE
(DEAD(DEAD
CLASS.
ALIVE)
CLASS.
DEAD)
adjustable
...........
(DEAD,
CLASS.
DEAD)
(MISSED.
ALIVE
CLASS.
DEAD)
All
target
classification
and
BDA
decisions
are
error-prone
or
fallible
to
adjustable
degree.
Figure
3.
Architecture
2:
GRC
and
Local
Quick
Follow-Up
(LQF)
(Immediate
Shooter
BDA)
5.
Strike
Dynamics
We
present
the details
of
the
state variables,
rate
parameters,
and
dynamical
equations
in
Appendix
I.
The
equations
are
annotated
term
by
term
so
that
the
contribution
of
each
term
is
explained.
In
broad
outline,
Reds
enter
the
region,
possibly
migrating
Southwards,
are
detected
by
Blue
sensors
in
each
region
(with
delay),
then
placed
on
a
target
list
(from
which
track
losses
can occur), are
eventually
shot
at
by
either
missiles
(or
gunnery)
or
9

manned
aircraft.
Those
missed
are
subject
to
re-detection
(BDA) and
reclassification
(without
reference
to
individual
past,
except
by
regional
location).
The
aim is
to
understand
the
effectiveness
of
Blue
sensor-shooter
assets
of
given
force
size,
composition,
and
effectiveness
against
a
composite
Red
force.
Our
current
formulation
is
the deterministic
equivalent
of
a
continuous-time
multivariate
stochastic Markov
process.
Such
a
stochastic process
is
one attractive
modeling
option,
but
has
not
been
adopted
for
reasons
of
computational
economy.
Monte
Carlo
simulation has not been
adopted
here for
the
same
reason.
Ample
precedent
for
the
present
style
of
modeling
occurs
in
ecology,
epidemiology,
and
population
biology
(see
Murray
(1989)),
not
to
mention
early
military
operations
research; see
Dockery
and
Woodcock
(1993),
Taylor
(1980),
and
Anderson
(1995).
One
difference
in
our model
is
the
explicit
representation
of
congestion,
particularly possible
saturation,
of
Blue's
sensor,
communication,
and
shooter
resources.
These
latter
resources
are
bound
to
be
limited,
so
backlogs
("queues")
effectively
develop
of
Red
units
that
await
detection
("service"),
and
subsequently
are
placed
virtually
in
line
for shooter
"service."
However,
that
service can
be
much
reduced
in
effective
speed
because
of
the
presence
of
dead,
misclassified
targets,
and
decoys.
Some
of
these
are
lost from
track,
and
hence
the
target
list.
Classical
queuing
theory,
see
Kleinrock
(1975),
leads
us
to
expect
that
some backlog
will
tend
to
develop
at
both
the sensor
and
communication/shooter
stages,
even
when
demand
for service
does
not
exceed
service
rate
at
a
stage;
this
is
the
result
of
short-term
random fluctuations
in
the communications
and
shooter system.
We
choose
to
represent
the
latter
effect
approximately
by
modifying
a
simple
mathematical device
put
forward
by Agnew
(1976)
and
Rider
('1976),
more
recently discussed
by
Filipiak
(1988).
These
papers
introduce the
non-linear
H-function
defined,
along
with
parameters.
The
modification
required
for sensors
has
recognized
that
(a)
the sensor "server"
must itself
be
multiple
(several
sensor
platforms potentially
cover
each
subregion
whose
capacities
are
finite
and
saturable,
and
(b)
must
accommodate
multiple
Red
target
types
and
states;
10

the latter
is
accomplished
by
proportional
processor
sharing, and
the
former
by
increasing
effective
service
(here
sweep)
rate.
The
modification
for the
communication/shooter
system
is
the
same,
but
modifies
the
processor
sharing
service
intensity
across the
target
list
by
priority
weighting:
wik
represents
the fraction
of
time
made
available
to
process
(shoot)
targets
in
region
i
judged
to
be
of
type
k;
Cik
represents
the
sensor
skill,
it
being
the
probability
that
a
target
of
typej
in
region
i
is
believed
to
be,
and
treated
as
of
type
k,
where
coj
=
1
represents perfect skill.
Each
of
the
above
parameters
is
treated
in
the
present
report
as
a
static
analyst-decision maker
choice:
a
constant.
However,
the
possibility,
and
attraction,
is
great
to
let
these
"parameters"
be
replaced
by
functions,
and
so
be
automatically
adjusted,
i.e.,
governed
by
feedback
from Blue
perception
of
Red
states and
inferred course
of
action.
6.
Case
Study
1:
Constant
Red
Arrival
Rate
We
first
consider
an
extremely
simple
scenario,
wherein
a
constant
number
of
Reds
per
day
(500
Hard,
500
Soft)
pour
into
the
northernmost
region,
IR
4
.
The
basic
sensor
sweep
rate
is
ý=
5000. We
present
graphs
of
the
model
output, beginning
with
cumulative input
and
number
undetected
Figure 4(a)
without
Blue attrition. The
subsequent
figures
appear
in
Appendix
II.
Note
that
in
the
non-attrited
deterministic/fluid
approximation
the
cumulative
entry
level
of
Red
targets
is
strictly
linear,
and
the
cumulative
number
undetected
quickly
approaches linearity
in
time.
11

Total
Number
of
Red
targets
Constant
Input
Rate
: 500 H
tgts,
500
S
tgts
per
day
No
Attrition;
No
loss
from
track
Number
of
Hard
targets=Number
of
Soft
targets
60000
.. . ....
...... .. .
.... . ...
.. ... .
50000
-o
40000
E 1 -
Number
undetected
30000
----
Total
number
of
Reds
in
region
z
20000
10000
0
0
10
20
30
40
50
Time
Figure
4(a)
Case
Study
2:
Accelerated Red
Arrival
Rate
to
a
Maximum
Number
of
Reds
In
this
case
study
Reds
enter
the
region
at
an
increasing
rate
until
the number
of
Hard
Reds
that
enter reaches 30,000
and
the
number
of
Soft Reds
that
enter
reaches
10,000.
The
number
of
Hard
Reds
that
enter
the
region
by
time
t
is
(100)t
2
/2
for
t
<
24.5
and
30,000
for
t
>
24.5;
the
number
of
Soft Reds
that
enter
the region
by
time
t
is
(100)t
2
/2
for
t
<
14.1
and
10,000
for
t
>
14.1.
Figure
5(a)
displays the
assumed
arrival
of
Red
targets
into
region
9tR
and
the
number
undetected
for the
basic
sensor
sweep
rate
•=
9,000.
12

Number
Live
Targets
No
Attrition
;
No
loss
from
track
Accel: 100
H
tgts,
100
S
tgts
per
day
to
a
Maximum
of
30000
Hard
Tgts.
and
10000
Soft
Targets
35000
30000
i
-----------
---------
25000
21
2
No.
Hard
Live
Targets
_
20000
A
No.
Soft
Live
Targets
"0
-a-
No.
Undetected Targets
_15000
E
z
10000
5000
0
0
10
20
30
40
50
Time
Figure
5(a)
13

Tabled
Results
The
tables
below
and
on
the
following
page
summarize
various
model
combat
outcomes
of
interest
at
t
=
25
and
t
=
50
(days).
One
can
see
at
a
glance
the
effect
of
the
presumed
system
architectures
and
parameters.
More
detailed
graphical
presentations
are
also
possible.
They
are
presented
and
discussed
in
Appendix
II.
TABLE
6.1
Summary
of
Combat
Outcomes
(Measures
of
Effectiveness)
Combat
Duration
=
25
days
Linear
Input
C=0.5;
BDAO0.5
C=1.0;
BDA7O.5
C=0.5;
BDA--.0
C=1.0;
BDAI-.0
Arch.
I
Arch
11
Arch.
I
Arch
1I
Arch.
I
I
Arch
11
Arch.
I
Arch
11
Number
Undetected
Latency=
1
hr
1.06E+04
8.872+03
8.33E+03
5.17E+03
3.55E+03
3.512+03
2.87E+02
2.73E+02
Latency=
0.5
hr
1.51E+03
3.66E+02
3.68E+02
2.31E+02
1.50E+02j
1.29E+02
1.07E+01
6.40E+00
Number
weapons
expended
Latency=
I
hr
5.23E+04
4.89E+04
5.43E+04
4.95E+04
3.92E+04
3.92E+04
3.47E+04
3.47E+04
Latency=
0.5
hr
9.05E+04
7.08E+041
8.38E+04.
5.89E+04
4.70E+04
4.70E+04
3.54E+04
3.54E+041
Number
Reds
killed
(out
of
2.50E+04)
Latency=
1
hr
1
.42E+04
1.60E+04
1.65E+04
1.97E+04
2.12E+04
2.12E+04
2.46E+04
2.46E+04
Latency=
0.5
hr
2.342+04
2.46E+04
2.46E+04
2.472+04
2.482+04
2.48E+04
2.50E+04
2.50E+04
Number
weapons
expended
per
target
killed
Latency=
1
hr
3.68E+00
3.04E+00
3.29E+00
2.51E+00
1.85E+00
1.85E+00
1.41E+00
1.41E+00
Latency=
0.5
hr
3.86E+001
2'88E+00
3.41E+00
2.38E+0Q
1.90E+00
1.90E+00
1.42E+001
1.42E+00
Number
erroneous
weapons
expended
Latency=
1
hr
3.27E+04
2.93E+04
1.55E+04
1.092+04
1.932+04
1.93E+04
0.002+00
0.002+00
Latency=
0.5
hr
5.66E+04
4.112+04
2.44E+04
1.19E+041
2.32E+041
2.31E+04
0.002+00
0,00E+00
Number
Blue
Aircraft
Killed
(out
of
100)
Latency=
1
hr
7.74E+00
6.33E+00
6.11E+00
3.60E+00
2.90E+001
2.89E+00
5.54E-01
5.54E-01
Latency=
0.5
hr
1.40E+001
6.01E2-01
5.68E-01
3.93E-01
3.08E-011
3.07E-01
3.75E-021
3.25E-02
Number
Red
Weapons
Expended
Latency=
1
hr
2.69E+03
2.20E+03
2.12E+03
1.26E+03
1.00E+03
1.002+03
1.89E+02
1.88E+02
,Latency=
0.5
hr
4.81E+02
2.002+02
1.93E+02,
1.31E+021
1.052+02I
1.04E+02
1.28E+01
1.11E+01
Parametric
Values
Constant
Input
Rate:
500
H
tgts,
500
S
tgts
per
day
pHIH=pHIS=pS
IS=0.7;
pSIH=O.
1;
100
shooters
Time
until
tracked
target
loss=1
hr.
100
aircraft;
1
hr. on
station;
11
hrs.
off
station
14

TABLE
6.2
Summary
of
Combat
Outcomes
(Measures
of
Effectiveness)
Combat
Duration
=50
days
Linear
Input
C=0.5;
BDAO-.5
C=1.0;
BDA=O.5
C=0.5;
BDA--1.O
C=1.0;
BDAI1.O
________
Arch.
I
jArch
11
-Arch.
I
JArchI11
Arch.I
I
Archil1
Arch.
I
jArchII1
Number
Undetected
Latency=
1
hr
2.19E+04j
1.88E+04
1.72E+04!
1.17E+04
7.47E+031
7.42E+03
2.88E+021
2.74E+02
Latency=
0.5
hr
2.79E+031
3.67E+02
3.68E+021
2.32E+02
1.51E+021
1.29E+02
1.07E+01
j6.41
E+00
Number
weapons
expendedI
Latency=
1
hr
1.03E+053-
9.75E+04
1.07E+0511
1.00E+05
7.82E+041
7.83E+04
7.OOE+04i
7.00E+04
Latency=
0.5
hr
1.83E+05
1.43Et+05
1.69E+051
1.19E+05
9.45E+041
9.415E+04
7.08E+04!
7'.08E+04
Number
Reds
killed
(out
of
5.00E+04)I
Latency=
1
hr
2.79E+04f
3.10E+04
31.26E+04
1
3.81
E+04
4.22E+04i
4.23E+04
4.96E+04
4.96E+04
Latency=
0.5
hr
4.71E+04l
4.96E+04
4.96E+041
4.97E+04
4.98E+041
4.98,E+04
5.00E+04
1
5.00E+041
Number
weapons
expended
per
target
killedI
Latency=
1
hr
3.69E+001
3.14E+00
3.28E+001
2.62E+00
1.85E+001
1.85E+00
1.41
E+00I
1.41
E+00
Latency=
0.5
hr
3.89E+001
2.88E+00
3.40E+001
2.39E+00
1.90E+003
1.90E+00
1.42E+001
1.42E+00
Number
erroneousj
weapons
expended
Latency=
1
hr
6.43E+04
5.88E+04
3.07E+041
2.33E+04
3.85E+04i
3.85E+04
0.00E+00I
0.00E+00
Latency=
0.5
hr
1.14E+051
8.30E+04
4.94E+041
2.40E+04
4.65E+04I
4.65E+04
0.OOE+00I
0.00E+00
Number
Blue
Aircraft
Killed
(out
of
100)
I16+0
Latency=
1
hr
3.26E+01
1
2.75E+01
21.57E+01
1
1166E+01
1.16E+01lI
1.16E+011.600.5E0
Latency=
0.5
hr
4.78E+00I
1.23E+00,
1.19E+001
8.07E-01
6.42E-01
I6.38E-01
7.70E-02!
6.70E-02
Number
Redr
Weapons
Expended
Latency=
I hr
1.09E+04j
9.17E+03
8.56E+03
5.52E+03
3.88E+031
3.88E+03
3.85E+02~
3.85E+02
,Latency=
0.5
hr
1.59E+031 4.1IOE+02
3.96E+02i
2.69E+021
2.14E+021
2.13E+021
2.56E+01' 2.23E+01
Parametric
Values
Constant
Input
Rate:
500
H
tgts,
500
S
tgts
per
day
pHjH=pHiS=pSjS=0.7;
pSIH=0.
1;
100
shooters
Time
until
tracked
target
is
lost=1
hr.
100
aircraft;
1
hr.
on
station;
I11
hrs.
off
station
Discussion
of
Tables
6.1
and
6.2
If
target
classification
and
BDA
are
perfect,
then
both
Architectures
result
in
the
attrition
of
most
of
the
Red
targets
for
both
shooter
latencies.
When
the
shooter
latency
time
is
0.5
hour
there
is
little
difference
in
attrition
for
the
two
Architectures
if
either
the
probability
of
correct
target
classification
or
the
probability
of
correct
BDA
is
equal
to
1.
If
the
shooter
latency
is
1
hour,
then
more
targets
are
attrited
when
one
increases
the
15

probability
of
correct
BDA
from
0.5
to
1,
than
if
the
probability
of
correct
classification
is
increased
from
0.5
to
1.
More
weapons
are
expended
for
the
case
of
the
probability
of
correct
classification
equal
to
1
and
probability
of
correct
BDA
is
0.5,
than the
case
in
which
the
probability
of
correct
classification
is equal
to
0.5
and
the
probability
of
correct
BDA
is
1.
If
the
shooter
latency
is
1
hour
and
the
probability
of
correct
BDA
and
probability
of
correct
classification
is
0.5,
then
decreasing
the
shooter
latency
to
0.5
hour
results
in
greater
attrition
than
increasing
either
probability
by
itself.
However,
this
improvement
is
at
the
expense
of
greater
weapon
expenditure
per
target
killed.
16

TABLE
6.3
Summary
of
Combat
Outcomes
(Measures
of
Effectiveness)
Combat
Duration
=
25
days
For
accelerated
input
to
a
max
of
3E4
Hard
targets
and
I1E4
Soft
targets
C=0.5;
BDA=O.5
C=1.0;
BDA=O0.5
C=0.5;
BDA=1.O
C=1.0;
BDAI1.O
Arch.
I
Archi11
Arch.
I
Archi11
Arch.
I
Archi11
Arch.
I
I
Arch
11
Number
Undetected
I
Latency=
1
hr 2.37E+041
2.86E+04
2.37E+04i
2.04E+04
2.13E+04!
2.12E+04
1.43E+04
1
1.43E+04
Latency=
0.5
hr
1.89E+04!
1.68E+04
1.37E+041
9.88E+03
1.13E+041
1.12E+04 4.63E+021
4.27E+02
Number
weapons
expended
Latency=
1
hr
4.09E+041;
4.30E+04
5.16E+04
4.45E+04
3.55E+041
3.55E+04
3.57E+041
3.57E+04
Latency= 0.5
hr
8.06E+04j
7.
16E+041
8.54E+0411
7.31E+04
5.81E+041
5.822=+04
5.60E+041
5.60E+041
Number
Reds
killed
(out
of 4.002+04)
Latency=
1
hr
1.59E+041
1.142+04
1.60E+04I
1.93E+04
1.81E+041
1.81E+04
2.522+'04 2.52E+04
Latency=
0.5
hr
2.09E+04!
2.29E+04
2.61
E+041
2.99E+04
2.82E+041
2.82E+04
3.93E+04
3.93E+04
Number
weapons-
__
_
_
expended
per
target
I
killed
Latency=
1
hr
2.57E+O0j
3.772+001
3.23E+00i
2.31E2+00
1.96E+001
1.96E+00
1.46E+00
1.46E+00
Latency= 0.5
hr
3.86E+00,
3.12E+001
3.27E+0011
2.44E+00
2.06E+00,;
2.06E+00
1.42E+00
1.42E+00
Number erroneous
weapons
expended170+40.00000+0
Latency=
1
hr
2.23E+04'
2.65E+04
1.44E+04i
8.59E+03
1.70E+04117E0
.E0
.OE0
Latency= 0.5
hr
4.93E+04
4.13E+04
2.42E+041
1.53E+04
2.82E+041
2.82E+04
0.002+00
0.002+00
Number
Blue
Aircraft
Killed
(out
of
100)T
Latency=
1
hr
1
.28E+01
1.
62E+01
1.33E+011
1.10E+01
1.14E+01j
1.14E+01
8.16E+00
8.15E+00
Latency=
0.5
hr
9.92E+001
8.42E+001
7.58E+001
5.302+00
5.67E+001
5.672+00
1.34E+00
1.33E+00
Number
RedIi1
Weapons
Expended
OlOi28+0130+2
Latency=
1
hr
4.52E+03
5.68E+03
4.662+03;1
3.872=+03
4.02E+03
4.02E+032.703.OE2
L1atency=
0.5
hr
I3.50E+031
2.982+03
2.672+031.82+320+3
4.5320113+1
Parametric
Values
Accelerated
Input
Rate:
100
H
tgts,
100
S
tgts
per
day
to
maximum
of
30K
H
tgts
and
10OK
S
tgts
pHIH~pS
IS=0.7;
pSIH1-=0.
1;
100
shooters
Time
until tracked target
is
lost=1
hr.
100
aircraft;
1
hr.
on
station;
I11
hrs.
off
station
17

TABLE
6.4
Summary
of
Combat
Outcomes
(Measures
of Effectiveness)
Combat
Duration
= 50
days
For
accelerated
input
to
a
max
of
3E4
Hard
targets
and
1E4
Soft
targets
C=0.5;
BDAO0.5 C=1.0;
BDA=O.5
C=0.5;
BDA7-1.O
C=1.0;
BDA--1.0
NmeUneetd
Arch.
I
IArch
11
Arch.
I
Arch
11
-Arch.
I
jArch
11
Arch.
1
Arch
11
Latency=
1
hr
1.67E+04
1.90E+04
1.62E+04
1.22E+04
1.34E+04
1.34E+04
1.42E+03
1.39E+03
Latency=
0.5 hr
8.91E+03
6.38E+03
3.23E+02
0.00E+00
0.OOE+00
0.00E+00
O.O+0-
.0E+00
Number
weapons
expended
Latency=
1
hr
6.16E+04
8.30E+04
7.80E+04
6.96E+04
5.38E+04 5.38E+04
5.44E+04
5.44E+04
Latency=
0.5
hr
1.25E+05
1.14E+051
1.35E+051
1.02E+05
8.66E+04
8.66E+04
5.70E+04
5.70E+041
Number
Reds
killed
(out
of
4.O0E+04)
Latency=
1
hr
2.30E+04
2.10E+04
2.37E+04
2.77E+04
2.63E+04 2.63E+04
3.83E+04
3.83E+04
Latency=
0.5
hr
3.09E+041
3.35E+04
3.96E4-04
4.OOE+04
4.OOE+04
41.00E+04
4.00E+041
4.OOE+04
Number
weapons
expended
per
target
killed
Latency=
1
hr
2.68E+00
3.95E+00
3.29E+00
2.51E+00
2.05E+001
2.05E+00
1.42E+00
1.42E+00
Latency=
0.5
hr
4.04E+00
3.40E+001
3.41E+001
2.55E+00
2.17E+001
2.17E+00
1.43E+00~
1.43E+001
Number
erroneous
weapons
expended
Latency=
1
hr
3.36E+04
4.73E+04
2.24E+04
1.16E+04
2.58E+04' 2.58E+04
0.OOE+00
0.OOE+00
Latency=
0.5
hr
7.67E+04
6.68E+04
3.94E+04
2.24E+04
4.20E+04~
4.20E+04
0.00E+00
0.00E+001
Number
Blue
Aircraft
Killed
(out
of
100)
Latency--
I
hr
4.40E+01
5.19E+01
4.38E+01
3.60E+01 3.83E+01
3.84E+01
2.06E+01
2.06E+01
Latency=
0.5
hr
3.11E+01
2.63E+011
1.83E+01
1.03E+01
1.31E+01 1.31E+01
1.37E+00
1.36E+00
Number
Red
Weapons
Expended
Latency=
1
hr
1.47E+04
1.73E+04
1.46E+04
1.20E+04 1.28E+04
1.28E+04
6.86E+03
6.86E+03
Latency=
0.5
hr
1.04E+04
8.76E+03
6.09E+03
3.42E+03
4.36E+031
4.36E+03
4.57E+021
4.55E+021
Parametric
Values
Accelerated Input
Rate:
100
H tgts,
100
S
tgts
per day
to
maximum
of
30OK
H
tgts
and
10OK
S
tgts
pHIH-pSIS=O.7;
pSfH=0.1;
100
shooters
Time
until
tracked
target
is
lost=1
hr.
100
aircraft;
1
hr.
on
station;
I11
hrs.
off
station
Discussion
of Tables
6.3
and
6.4
If
the
shooter
latency
is
1
hour
and
the
probabilities
of
correct
classification
and
correct
BDA
are all
0.5,
then
decreasing
the
shooter
latency
to
0.5
hour
results
in greater
attrition
than
increasing
either
the probabilities
of
correct
classification
or
probabilities
of
correct
BDA
to
1.
However,
decreasing
the
latency
to
0.5
hour
results
in
a
larger number
18

of
weapons
required
for
every
target
killed,
while
increasing
the
probabilities
of
correct
classification
(respectively
the
probabilities
of
correct
BDA)
to
1
results
in
fewer
weapons
required
for
every
target
killed.
Increasing
the
probabilities
of
correct
BDA
results
in
the
smallest
number
of
weapons
per
target
killed.
If
the
probabilities
of
correct
BDA
are
equal
to
1,
both
Architectures
result
in
the.
same
number
of
kills
and
number
of
weapons
per
kill.
The
number
of
targets
killed
and
number
of
weapons
expended
per
kill
are
smallest
when
the
probabilities
of
correct
BDA
are
equal
to
1.
Tables
6.5
-
6.7
present
further
results
concerning
the
scenario
with
linear
input.
TABLE
6.5
Summary
of
Combat
Outcomes
Linear
Input
C=0.5,
BDA=0.5
Sensor
Rate=2500
Sensor
Rate=1250
Sensor
Rate=2500
Sensor
Rate=1250
Combat
Duration=25
Combat
Duration=25
Combat
Duration=50
Combat
Duration=50
days
days
days
days
2.5E+04
Reds
entered
2.5E+04
Reds
entered
5.0E+04
Reds
entered
5.0E+04
Reds
entered
region
region
region
region
Arch
I
Arch2
Arch
I
Arch2
Arch
i
Arch2
Arch1
Arch2
Number
weapons
expended
Latency=1
hr
5.23E+04
4.89E+04
5.08E+04
5.09E+04
10.3E+04
9.75E+04
10.OE+04
10.11E+04
(erroneous)
(3.27E+04)
(2.93E+04)
(3.16E+04)
(3.16E+04)
(6.43E+04)
(5.88E+04)
(6.21E+04)
(6.26E+04)
Latency=0.5
hr
9.05E+04
6.93E+04
8.OOE+04
8.59E+04
18.3E+04
14.3E+04
16.2E+04
17.5E+04
(erroneous)
(5.66E+04)
(4.03E+04)
(4.98E+04)
(5.27E+04)
(11.4E+04)
(8.30E+04)
(10.1E+04)
(10.8E+04)
Number
Reds
killed
Latency=1
hr
1.42E+04
1.60E+04
1.54E+04
1.44E+04
2.79E+04
3.10E+04
3.OOE+04
2.81E+04
Latency=O.5
hr
2.34E+04
2.41E+04
2.13E+04
'
2.38E+04
4.71E+04
'
4.96E+04
4.30E+04
,
4.76E+04
Number
weapons
expended
per
target
killed
Latency=1
hr
3.68E+00
3.06E+00
3.29E+00
3.53E+00
3.69E+00
3.08E+00
3.33E+00
3.59E+00
Latency=0.5
hr
3.87E+00
2.87E+00
3.76E+00
3.60E+00
3.89E+00
2.88E+00
3.77E+00
3.67E+00
Discussion
of
Table
6.5
Table
6.5
presents
results
for
total
number
of
weapons
expended,
the
number
of
erroneous
weapons
expended,
the
number
of
Reds
killed,
and
the
number
of
weapons
expended
per
target
killed
for two
shooter
latencies
(1
hour
and
0.5
hour)
and
two
19

regional
sensor
rates
(4=5000
and
4=2500).
The
probabilities
of
correct
classification
and
the
probabilities
of
correct
BDA
are
0.5.
First
consider
a
shooter
latency
of
1
hour.
Note
that
Architecture
1
has
more
kills
when
the
regional
sensor
rate
is
smaller
(2500)
than
for
the
larger
(5000)
sensor rate.
Recall
that
for
Architecture
1
the
regional
sensor
is
providing
all
of
the
BDA. Dead
targets
that
are
(mis)classified
as
live
by
the regional sensor
are
returned
to the
targeting
list for
further
action.
The
larger
sensor
rate
is
increasing
the rate
at
which these
misclassified
dead
targets
are
returned
to
the
targeting
list.
These
misclassified
dead
targets
decrease
the
amount
of
effort
the shooters
devotes
to
the
live
targets
on the
list.
Additionally,
for Architecture
1
the
slower
sensor
rate
results
in
fewer
expended
weapons
per
target
killed
than
the
higher
sensor
rate.
Recall
that
Architecture
2
has additional
immediate
BDA capability.
In
this
case
the
larger
regional
sensor
rate results
in
the
removal
of
more
misclassified
dead targets resulting
from
the
immediate
BDA
from
the
targeting list than
that
resulting
from
the
slower
regional sensor rate.
Thus,
Architecture
2
has
more Red
targets
killed
for the
faster
regional
sensor
rate
than
for
the
slower regional
sensor
rate.
Further,
the faster
sensor
rate
results
in
fewer weapons
expended
per target
killed
than
the
slower
sensor
rate.
Next consider
a
shooter
latency
of
0.5
hour. The
shorter
shooter
latency
allows
Architecture
1
to
prosecute
more
misclassified
dead
targets
in
addition
to
the
live
targets
on
the
targeting
list.
Thus,
the
number
of
targets
killed
for
Architecture
1
is
greater
for
the
larger
sensor
rate
than
the
smaller
sensor
rate.
However,
the
number
of
weapons
expended
per
killed
target
is
also
greater
for the faster sensor rate
than
the slower
sensor
rate.
The
smaller shooter
latency
results
in
fewer
misclassified dead
targets
being
removed
by
the regional
sensor
from
the
target
list
in
Architecture
2.
However,
Architecture
2
still has
a
slightly
larger
number
of
targets
killed
for
the
larger
sensor rate.
Further,
the
faster
sensor
rate
results
in fewer weapons
expended
per
target killed
than
the
slower sensor rate.
20

For
Architecture
1,
the
larger
sensor
rate
results
in
more
weapons
expended
per
target
killed
for
a
shooter
latency
of
0.5
hour
as
compared
to
that
for
a
shooter
latency
of
1
hour.
For
Architecture
2,
the
larger
sensor
rate
results
in
fewer
weapons
expended
per
target
killed
for
a
shooter
latency
of
0.5
hour
as
compared
to
that
for
a
shooter
latency
of
1
hour.
For
Architecture
1,
the
smaller
sensor
rate
results
in
more
weapons
expended
per
target
killed
for
a
shooter
latency
of
0.5
hour
as
compared
to
that
for
a
shooter
latency
of
1
hour.
For
Architecture
2,
the
smaller
sensor
rate
results
in
more
weapons
expended
per
target
killed
for
a
shooter
latency
of
0.5
hour
as
compared
to
that
for
a
shooter
latency
of
1
hour.
TABLE
6.6
Summary
of
Combat
Outcomes
Linear
Input
C=1.0,
BDA=O.5
Sensor
Rate=2500
Sensor
Rate=1
250
Sensor
Rate=2500
Sensor Ratel125O
Combat
Duration=25
Combat
Duration=25
Combat
Duration=50
Combat
Duration=50
days
days
days
days
2.5E+04
Reds
entered
2.5E+04
Reds entered
5.OE+04
Reds
entered
5.OE+04
Reds
entered
region
region
region
region
Arch
1
Arch
2
Arch
I
Arch
2
Archl1
Arch
2
ArchlI
Arch
2
Number
weapons
expended
Latency=1
hr
5.43E+04
4.95E+04
5.28E+04
5.25E+04
10.7E+04
1O.OE+04
10.4E+04
10.5E+04
(erroneous)
(1
.55E+04)
(1
.09E+04)
(2.95E+04)
(1
.42E+04)
(3.07E+04)
(2.33E+04)
(2.92E+04)
(2.87E+04)
Latency=O.5
hr
8.38E+04
5.89E+04
8.26E+04
7.06E+04
16.9E+04
11.9E+04
16.8E+04
14.2E+04
(erroneous)
(2.44E+04)
(1.19E+04)
(2.40E+04)
(1.78E+04)
(4.94E+04)
(2.40E+04)
(4.90E+04)
(3.56E+04)
Number
Reds
killed
Latency=1
hr
1.65E+04
1.97E+04
1.65E+04
1.70E+04
3.26E+04
3.81
E+04
3.26E+04
3.34E+04
Latency=0.5
hr
2.46E+04
j2.47E+04
2.44E+04
j2.47E+04
4.96E+04
4.97E+04
4.93E+04
j4.96E+04
Number weapons
expended
per
target
killed
Latency=1
hr
3.29E+00
2.50E+00
3.20E+00
3.09E+00
3.28E+00
2.62E+00
3.19E+00
3.14E+00
Latency=O.5
hr
,3.41E+00
2.38E+00
3.39E+00 2.86E+00
3.41
E+00 2.39E+00
3.41E+00
2.86E+00
Discussion
of
Table
6.6
Table
6.6
presents
results
for
the
case
in
which
there
is
perfect
target
classification
and
the
probabilities
of
correct
BDA
are
0.5.
Perfect
target
classification
implies
that
the
most
efficient
weapon
is
expended
on
each target
type,
resulting
in
fewer
weapons
expended
per
target killed
than
if
target
classification
were
not
perfect.
21

For
Architecture
1,
the
regional
sensor
is
performing
all
the
BDA.
If
the
regional
sensor
misclassifies
a
dead
target
as
live, the
dead target
is
placed
on
the
targeting
list
for
further
prosecution.
A
larger
regional
sensor
rate
hastens
the
placement
of
misclassified
dead
targets
to
the
targeting
list.
For
a
shooter
latency
of
1
hour,
the
number
of
targets
killed
using
Architecture
1
for
both
regional
sensor
rates
are
equal.
However,
the
number,
of
weapons
expended
per
target
killed
is
larger
for
the
higher
regional
sensor
rate
than
for
the
smaller
sensor
rate.
For
the
shorter
shooter
latency
of
0.5
hours,
the
number
of
targets
killed
is
slightly
less
for the higher
regional
sensor
rate
than
the
lower
sensor
rate.
The
number
of
weapons
expended
per
target
killed
is
less
for
the smaller
sensor
rate than
the
larger
sensor
rate.
Architecture
2
has
immediate
BDA
capability
in
addition
to
the
regional
sensor
BDA.
The
regional
sensor
can
remove
dead
targets
that
have
been
misclassified
by
the
immediate
BDA
capability
from
the targeting
list.
For
the
shooter
latency
of
1
hour,
the
number
of
targets
killed
using
Architecture
2
is
greater
for
the
higher
regional
sensor
rate
than
the
lower
one.
The
number
of
weapons
expended
per killed
target
for
the
higher
sensor
rate
is
less
than
that
for
the
smaller
sensor rate.
For
the
shooter
latency
of
0.5
hours, the
number
of
targets
killed
is
about the
same
for
both
regional
sensor
rates.
However,
the
number
of
weapons
expended
per
target
killed
is
less
for
the
larger regional
sensor
rate
than
the
smaller
sensor
rate.
For
Architecture
1,
the
shorter
shooter
latency
results
in
more
weapons
expended
per
target
killed
than
the
larger
shooter
latency.
For
Architecture
2,
the
shorter
shooter
latency
results
in
more
weapons
expended
per
target
killed
than
the
larger
shooter
latency.
22

TABLE
6.7
Summary
of
Combat
Outcomes
Linear
Input
C=0.5,
BDA=1
.0
Sensor
Rate=2500
Sensor
Rate=1
250
Sensor
Rate=2500
Sensor
Ratel
250
Combat
Duration=25
Combat
Duration=25
Combat
Duration=50
Combat Duration=50
days
days
days
days
2.5E+04
Reds
entered
2.5E+04
Reds
entered
5.OE+04
Reds
entered
5.OE+04
Reds
entered
region
region
region
region
Arch
I
Arch
2
Arch
1
Arch
2
Archi1
Arch
2
Arch
I
Arch
2
Number weapons
expended
Latency=1
hr
3.92E+04
3.92E+04
3.88E+04
3.90E+04
7.82E+04
7.83E+04
7.74E+04
7.77E+04
(erroneous)
(1.93E+04) (1.93E+04)
(1.91
E+04)
(1.91
E+04)
(3.85E+04)
(3.85E+04)
(3.81
E+04)
(3.82E+04)
Latency=O.5
hr
4.70E+04
4.70E+04
4.70E+04
4.70E+04
9.45E+04
9.45E+04
9.45E+04
9.45E+04
(erroneous)
(2.32E+04)
(2.31
E+04)
_(231E+04)_
(2.31E+04)
(4.65E+04)
(4.65E+04)
(4.65E+04):
(4.65E+04)
Number
Reds
killed
Latency=1
hr
2.12E+04
2.12E+04
2.09E+04
2.
1OE+04
4.22E+04 4.23E+04
4.18E+04
4.20E+04
Latency=0.5
hr
2.48E+04
2.48E+04
2.48E+04
2.48E+04
4.98E+04
4.98E+04
4.98E+04
4.98E+04
Number weapons
expended
per
target killed
Latency=1
hr
1
.85E+00
1.85E+00
1
.86E+00
1.86E+00
1
.85E+00
1.85E+00
1
.85E+00
1.85E+00
Latency=O.5
hr
1
.90E+00
1.90E+00
1
.90E+0O0
:-1.90E+00O
1
.90E+00
1.89E+00
1
.90E+00
1.90E+00
Discussion
of
Table
6.7
Table
6.7
presents
results
for
the
case
in
which
the
BDA
is
perfect
and
the
probabilities
of
correct
target classification
are
0.5.
For
the
particular
scenario
being
considered,
the
main
effect
of
target
misclassification
is
to
reduce
the
probability
of
killing
a
hard
target
when
the
hard
target is
misclassified
as
soft.
Since
the
BDA
is
perfect,
the
perfect
BDA
will
inform
the
shooters
that
a
missed
target
is
alive
and
that
target
will
be
re-targeted.
The
difference
between
Architectures
1
and
2
in
this
case
is
when
the
BDA
informnation
becomes
available.
Architecture
1
has
all
of
its
BDA
done
by
the
regional
sensors.
As
a
result,
the
outcome
of
the
BDA
is
delayed.
The
additional
immediate
BDA
capability
of
Architecture
2
implies
that the
result
of
BDA
is
known
instantaneously.
Thus,
missed
targets
are
always
available
for
re-targeting
with
Architecture
2.
Missed
targets
are
available
for
re-targeting
after
a
delay
in
Architeetture
1.
23

7.
Summary
and Conclusions
This
paper
describes
and
exploits
a
low-resolution,
high-level
modeling
methodology
for the
study
of
the
effectiveness
of
a
system
of
systems.
The
methodology
facilitates
quick
turnaround
efficient
exploration
of
sensitivities
of
measures
of
Blue combat
success
to
realistically imperfect
Blue ISR
and
IW
capabilities:
limited
and
imperfect
sensor
surveillance
and
reconnaissance,
particularly
BDA,
and
finite,
hence
saturable,
communications
and
weapons delivery
capability.
The
model
explicitly,
but purposefully
skeletally,
represents aircraft
sorties,
fires,
sensor/shooter
latencies,
target losses,
imperfect
target
type
classification,
imperfect
weapon
assignment, and
BDA.
The
model
is
deterministic/expected
value
or
"fluid"
in
style,
although
it
represents
time-dependent,
non-linear stochastic
features such
as
system
saturability
by
means
of
an
analytical
device.
The
methodology
is
illustrated
by
the
comparison
of
two
sensor-shooter
architectures
in
a
strike
scenario.
In
Architecture
1,
battle damage
assessment
is
performed
by
regional
sensors,
and
is
delayed.
In
Architecture
2,
an
additional capability
of
immediate
BDA
is
added
to
that
of
regional sensors. Model
experimentation suggests
the following
conclusions.
"*
Decreasing
the
Blue
sensor/shooter
latency
can
result
in
greater
Red
attrition
than
increasing
the
probabilities
of
correct
target
classification
and
BDA.
However,
the
improvement
is at
the
expense
of
greater weapon expenditure
per
target killed.
"*
One
possible
effect
of
erroneous
BDA
is
to
return
dead
targets
to
the shooter
targeting
list.
These
dead
targets
require
shooter
resources, and
so
decrease
the
amount
of
effort
the
shooters devote
to live
targets.
Thus,
dead targets
that
are
returned
to
the
targeting
list
not
only
result
in
wasted
weapon
expenditure
but
also
in
additional
delay
to
prosecute
live
targets.
The re-attacks
may
actually
place
manned aircraft
at
extra
risk;
see
Table
6.2.
A
dead
target
returned
to
the
targeting
list
through
erroneous
BDA
has
a
similar
effect
to
that
of
a
decoy.
If
the
shooter
24

latency
is
long,
then
increasing
the
probability
of
correct
BDA
can
result
in
more
targets
being
killed
than
increasing
the
probability
of
correct
target
classification.
*
It
is
important
to
balance
the resources
of
the
sensors
and
the
shooters.
If
the BDA
is
imperfect,
then
myopically
increasing
the
regional
sensor
rate
can
undesirably
increase
the
number
of
dead targets
that
are
erroneously
returned
to the
targeting
list.
If
the shooter
latency
is
large,
then
resources
erroneously
allocated
to the
dead
targets
on the
targeting
list
result
in
more
live targets
being
lost
before
a
weapon
reaches
them.
Thus, for
Architecture
1,
the
effect
of
the
increase
in
sensor
detection
rate
can
be
a
decrease
in
the
number
of
targets
killed
during
a
time
period;
see
Table
6.5.
*
The effect
of
the additional
immediate
BDA
capability
of
Architecture
2
depends
on
the
regional sensor
rate.
The
regional
sensor
can
remove
dead
targets
from the
targeting
list
that have been
placed
there
by
erroneous
instantaneous
BDA.
Thus,
increasing
the
regional
sensor
rate
can
increase
the
number
of
targets
killed
for
Architecture
2.
25

References
Agnew,
C.E.,
"Dynamic
Modelling
and
Control
of
Congestion Prone
Systems,"
Operations
Research,
24,
409-419
(1976).
Anderson,
L.B.,
"Attrition
Formulas
for
Deterministic
Models
of
Large-Scale
Combat,"
Naval
Research
Logistics,
42,
345-373
(1995).
Dockery,
J.T.
and
Woodcock,
A.E.R., The
Military
Landscape:
Mathematical
Models
of
Combat,
Woodland
Publishing,
Ltd.,
Cambridge,
England,
1993.
Filipiak,
J.,
Modeling
and
Control
of
Dynamic
Flows
in
Communication
Network,
Springer-Verlag,
Berlin, Germany,
1988.
Gaver, D.P.
and
Jacobs,
P.A.,
"Probability
Models
for
Battle
Damage
Assessment
(Simple
Shoot-Look-Shoot
and
Beyond),"
Naval
Postgraduate
School
Technical
Report,
NPS-OR-97-014,
Monterey,
CA,
1997.
Gaver,
D.P.
and
Jacobs,
P.A.,
"Stochastic and
Deterministic
Models
of
Targeting
with
Dynamic
and
Error-Prone
BDA,"
Naval
Postgraduate
School
Technical
Report,
NPS-
OR-97-018,
Monterey,
CA,
1997.
Gaver,
D.P.
and
Jacobs,
P.A.,
"Attrition
Modeling
in
the
Presence
of
Decoys:
An
Operations-Other-Than-War
Motivation,"
Naval
Postgraduate
School
Technical
Report,
NPS-OR-96-01
1,
Monterey,
CA,
1996.
Gaver,
D.P.,
Jacobs,
P.A.,
and
Youngren,
M.A.,
"Analytical Models
for
Battlespace
Information
War
(BAT-IW)
Part
1,"
Naval
Postgraduate
School
Technical
Report,
NPS-OR-98-001,
Monterey, CA,
1998.
Gaver,
D.P.,
Jacobs,
P.A.,
and
Youngren,
M.A.,
"Analytical
Models
for
Battlespace
Information
War (BAT-IW)
Part
2,"
Naval
Postgraduate
School
Technical
Report
NPS-OR-99-002,
Monterey,
CA,
1999.
Ilachinski,
A.,
Land
Warfare
and
Complexity,
Part
I.:
An
Assessment
of
the
Applicability
of
Nonlinear
Dynamic
and
Complex
Systems
Theory
to
the
Study
of
Land
Warfare,
Center
for
Naval
Analyses,
Alexandria,
VA,
1996.
Kleinrock,
L.,
Queueing
Systems, Vols.
1
and
2,
Wiley
Interscience,
1975.
Munson,
K.H.
Jr.,
"Toward
Assessment
of
Dominant
Battlespace
Awareness: A
Remote
Sensor
System
Model,"
MS
Thesis,
Naval Postgraduate
School,
Monterey,
CA,
March
1996.
Murray,
J.D.,
Mathematical
Biology,
Springer-Verlag,
Berlin,
Germany,
1989.
Osmundson,
J.S.,
"A
systems
engineering
methodology
for
information systems"
Systems
Engineering,
Vol.
3,
No.
2,
July
2000, pp.
68-81.
Rider,
K.L.,
"A Simple
Approximation
to
the
Average
Queue
With
Time-Dependent
Queue,"
Journal
of
the
A
CM,
23,
361-367
(1976).
The Mathworks,
Inc.,
MATLAB
Reference
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Version
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The
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Natick,
MA
01760,
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26

Taylor,
J.G.,
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Research
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of
America,
Arlington,
VA,
1980.
27

APPENDIX
I
In
this
appendix
all
mathematical
development
is
recorded.
Al.
State
Variables
Below
are
listed
the
state
variables
that
describe
the
system
at
any
time
t:
Rj(t)
=
Number
of
detected,
hence
targetable,
alive
Reds
that
are
of
type
j
in
(sub)region
i
at
time
t.
Note:
j
can
denote
a
false
target
or
decoy
(which
includes
those
previously
targeted,
missed
and
incorrectly
left
for
dead).
Ry(t)
=
Number
of
alive
Reds
undetected
of
type
j
that
are
in
(sub)region
i
at
time
t.
Note:
these
include
some
of
those
previously
targeted.
M.(t)
=
Number
of
dead
Red
targets
of
type
j
in
(sub)region
i
unclassified
at
time
t
(required
in
Architecture
Model
1;
BDA
classifies
these
(possibly
incorrectly)).
Dy(t)
=
Number
of
dead
Red
targets
of
type
j
in
(sub)region
i
that
are
still
counted
as
alive
at
time
t.
Ko.(t)
=
Number
of
Red
targets
of
typej
in
(sub)region
i
killed
during
(0,
t].
Si(.,t)
=
Sensor
effort
"looking"
at (sub)region
i
at
time
t.
The
°
indicates
that
this
may
be
modified
by'Blue
in
accordance
with
perceived
Red
state
conditions.
Bi(.,t)
=
Number
of
Blue
(missile)
Shooters
prosecuting
targets
in
(sub)region
i
at
time
t.
Again,
the
dot
e
signifies
possible
feedback-driven
or
scripted
shooter
allocations.
A
i
(°,t) =
Number
of
aircraft
in
region
i
at
time
t.
Same
comment
concerning.
;i(t)
=
Number
of
live
aircraft
at
time
t.
Wij(t)
=
Number
of
Weapons
expended
by
non-aircraft
assets
in
(sub)region
i
at
targets
perceived
to
be
of
typej
during
[0,
t].
W,.(a;t)
=
Number
of
Weapons
expended
by
aircraft
assets
in
(sub)region
i
at
targets
perceived
to
be
of
typej
during
[0,
t].
Wi)(d;t)
=
Number
of
Weapons
expended
all
assets
in
(sub)region
i
at
dead
targets
of
type
j
perceived
to
be
alive
during
[0,
t].
Ri.(t)
-
E
RU
(t);
total
number
of
detected
Red
targets
in
(sub)regiyn
i
ut
time
i
.
28

R
=.
(t)=y
ký(t);
total
number
of
undetected
Red
targets
in
(sub)region
i
at
time
t.
I
Dio(t)
=
JD.(t);
total
number
of
dead
Red
targets
in
(sub)region
i
that
are
perceived
to
i
be
alive
at
time
t.
Next
there
follows
a
listing
of
current
parameters
that
enter the
equations
for
state
development.
These
are
initially
constants,
but may be
made
time,
or
state
dependent;
in
the latter
case
dynamic
adaptation
can
be
modeled.
A2.
Parameters
The
following parameters
are
required
to
specify
the
dynamic
evolution
of
the
process
of
states:
ýi
=
(sweep)
rate
at
which
Red
targets
are
processed
by
one
sensor
system
in
(sub)region
i;
this
can
be
generalized
to
account
for
different
sensor
types,
but
has
not
been.
VV
=
rate
at
which Red
targets
of
typej
are
lost
by
the
sensor
system
in (sub)region
i.
ay
=
rate
at
which
Red
targets
of
type
j
in
(sub)region
i
are
active
(shoot)
and are
detected
by
Blue.
ao.(a)
=
probability
a
Red
target
of
typej
is
detected
while
firing
at
a
Blue
aircraft
pikj(a) =
probability
a
target
of
typej
in (sub)region
i
is
killed
by
a
Blue
aircraft
when
the
Blue
aircraft
is
targeting
it
as
a
type
k
target.
pj(R,B)
=
probability
a
Red
of
typej
kills
a Blue
aircraft.
cyk(a;a)
=
probability
an
alive
Red
target
of
typej
in
(sub)region
i is
classified
as
a
type
k
when
it
is
being
prosecuted
by
a
Blue
aircraft.
c
11
k(d;a)
=
probability
a
dead
Red
target
of
typej
in
(sub)region
i
that
is
perceived
to
be
alive
is
classified
as a
type
k
when
it is
being prosecuted
by
a
Blue
aircraft.
=
rate at
which
a
detected
Red
target
in
(sub)region
i
is
prosecuted
by
a
shooter.
Wik
=
fraction
of
a
Blue
shooter's
effort that
is
used
to
prosecute
targets
of
type
k
in
(sub)region
i.
wik(a)
=
fraction
of
a
Blue
aircraft's
effort
in
(sub)region
i
that
is
used
to
prosecute
targets
of
type
k.
29

cijk(a)
=
probability
an
alive
Red
target
of
typej
in
(sub)region
i is
classified
as
a
type
k
when
it
is
being
prosecuted
by
a
Blue
shooter.
cUk(d)
=
probability
a
dead
Red target
of
typej
in
(sub)region
i
that
is
not
perceived
to
be
dead
is
classified
as a
type
k
when
it
is
being
prosecuted
by
a
Blue
shooter.
my(d)
=
probability
a
dead
target
of
typej
in
(sub)region
i
is
declared
dead
by
the
field
sensor system.
moj(a)
=
probability
a
live
target
of
typej
in
(sub)region
i
is
declared
live
by
the
field
sensor
system.
Pikj
=
probability
a
target
of
typej
in
(sub)region
i
is
killed
by
a
Blue
shooter
when
the
Blue
shooter
is
targeting
it
as
a
type
k
target.
r
i
kj(dla)
=
probability
a
live
target
of
typej
in
(sub)region
i
that
has
been
prosecuted
by
Blue
shooters
as
a
type
k
and
is
still
alive,
but
is
classified
as
dead
by
the
shooter
sensor
system;
rikj(ala)
=
1 -
rikj(dla).
rgkj(dld)
=
probability
a
dead
target
of
typej
in
(sub)region
i
that has
been
prosecuted
by
Blue
shooters
as
a
type
k
is
classified
as
dead
by
the
shooter
sensor
system;
rikj(
ald)
=
1 -
rikj(dld).
rikj(dld;a)
=
probability
a
dead
targef
of
typej
in
(sub)region
i
that
has
been
prosecuted
by
Blue
aircraft
as
a
type
k
is
classified
as
dead
by
the
aircraft;
rikj(ald;a)
=
1 -
rikj(dld;a).
rikj(dla;a)
=
probability
a
live
target
of
typej
in
(sub)region
i
that
has
been
prosecuted
by
Blue
aircraft
as a
type
k
is
still
alive,
but
is
classified
as
dead
by
the aircraft;
rikj(ala;a)
=
1
-
rikj
1
dja;a).
I/fl.(a)
=
mean
on-station
time
for
an
aircraft
in
(sub)region
i.
I/,pi(a)
=
mean
time
between
an
aircraft's
departure
from
region
i
and
its
next
arrival.
4j(R,B)
=
rate
of
fire
at a
Blue aircraft
by
a
Red
of
typej
in
(sub)region
i.
5i,(B,R)
=
rate
of
fire
at a
Red
target
by
a
Blue
aircraft
in
(sub)region
i.
,y
=
arrival
rate
of
targets
of
typej
into
(sub)region
i
from
outside
the
region.
7
iv
=
rate
at
which
a
target
of
typej
in
(sub)region
i
moves
into
(sub)region
L
H(x)
=
amount
of
effort
each
server expends
when
there
are
x
customers
waiting
or
being served;
H(x)
=
x/(l
+
x).
See
Filipiak
(1988),
and
Gaver,
Jacobs,
and
Youngren
(1998).
30

A.3
Architecture
I
(Deferred
BDA):
ModeI
Defender
Dynamics
dR-
/(t))
dt
S~Migration
Rate
of
change
of
Mration
undetected
Red,
from exterior
typej,
in
9?qi
into
i
- .((t) +
R.(t)
+
D.(t)
+
mi.
()
Si(t)Rj.(t)mo.(a)
-D.it))
Ro(t)
+
R
1
.(t)
+
+i.
(t).-+
mi.(t)
Sensor detection
of
undetected
Red,
typej,
in
qR
1
.
Detected
unit
declared
alive.
+VyRo.(t)-E",Yijfjkj(t)+
Eye~y(Rej(t)+-Rj(t))
Detected
s
st
Migration
Migration
into
R..
unet
(detected
becomes
undetected
if
out
of
J-i
leave Jq, in
which
detected)
Wi.AR.
(
t)cu
(a
)(1--
Pij,)
+,tiH(Ri.
(t)
+
Di.
(t))Bi(t)~
WX"
Ri.(t)+
(a
)
"Attrition":
an
alive
Red
target
is
missed
-
ajo.Rj
(t) -
a
.(a)
8i(R,
B)I(Ai(t)>
O)
.(t)
Detection
from
Red
action
Detection
from
Red
firing
at
aircraft
(e.g.,
SCUD
fired)
+,5j(B,
R)
"t)E
•
wik(a)
0.1+Rijt)
D(t)
c0.k
(a;
a)(1
-
Pikj
(
a
))r,.j
(dja;
a)
k
0.0O1
+
R.(t)
+
D,.(t)
~
"Attrition":
a
miss
by
Blue
aircraft
and
misclassification
as
dead
31

dRo'(t)
iH((t
R(D i. ( ))
Si
(t)kRi(t)my.(a)
dt
=.
(t)
+ R
1
.(t)
+
D,.(t)
+
Mi.(t)
Rate
of
change
Senso
detection
Undetected
Red,
typej
of
detected
Red,
typej,
in
ai
-
vo.R.(t)
-
itjR#
(t)
Detected
lost
Migration
out
of
91i
PuH(R
1
.
(t)
+ Dw.
(t))B,
WikRW
(t)cik(a)
+
auij(t)
Missiles
-
__H(_(t+Di())_
t)_k
Ri.(t)+
Di.(t)
W
,R ( (
Detection
from
"Attrition"
Red
detected
(dead
disappear,
alive
join
ij
(t))
Red
action
"+ao.
(a)i8o.(
R,
B)I(
Ai
(t)
>
O)RO.
(t)
Detection from Red
firing
at
aircraft
Aircraft-,(B,R)A(
wk(a)
Ri'(t)
D
,
(
a)
a)[Pikj(a)
+
-
pi•
(alrkj(
a)]
k
0.01
+
Ri.
Di.+
Cqk
(a;
Oa[.(a)
(V
(d~a
"Attrition":
a
shot
by
Blue
aircraft
dM
(t)
=
AitH(Ri.
(t)
+
Di.
(t))Bi
(t)E
wikRi
(t)c#k
(a)pikj
S,ý
k
Ri.
(t)
+
Di.
(t)
Rate
change
"Attrition": Alive
Red,
typej,
killed
in
qi
unclassif.
dead
Red,
typej,
in
Jqi
S+
Ri.
+
Di.
+
M
1
.)S
1
(t)
-Mo
(t)
-•XgH(•.,
~~ki
+R.+D.+M.St)
.(t)
+
Ri.
(t)
+
Di.
(t)
+
Mi.
(t)
Detection
of
unclassified dead
Red
(with
prob.
my
(d)
declared
dead
and ignored
after;
with prob.
1-
.(d)
declared
alive
and retargeted)
+,u
H(i.(t)+
D.
t))i
()E
w
k#
Di(
t)cik
(d
)
+l'tH(R'(t+,
i'())B~t)k
Ri-.(t)'+D-
i.t)•
Shot
at
("Attrition"
of)
Red dead,
misclassif.
as
alive,
and retargeted
in Jqi
32

dDy(t)
(
+ ( + Mmu(t)(1
-mQ(d))
dt
=iH(R°(t)
(
Di(t)+Mi°(t))Si(t)-i.(t)
+
Ri.(t)
+
Di.(t)
+Mi(t)
Rate
change
dead
Sensor
detection
of
unclassified
dead
Red,
Red,
typej,
class.
as
live
m
Ci
typej,
that
classifies
alive;
9ti
Missile
-
j.tiH(R.
(t)+D.(t))Bi(t)j
wkDij
(t)+
Di.(d)
\k
R,.(t)+
Di(t)
"Attrition"
of
dead,
classified
as
alive
Red,typej,
in
9
1i
- 9i(BR) i (~j
ik-(a)
D#(t)
co
k(d;
a)
rok(dld;
a)
-¢•iB'R)i(I)k
wi~
)0.0
1
+
Ri
(t)
+
OA.(t)
Blue
aircraft
retargeting
a
dead Red
and
class,
as
dead
k,5i
00
+)
AR.(t)
+
D,.(t')
cuk(a;
a)pikj(a)r
k(ald;a)
+
•iB')A(IZ
ik
a
0.0
1
+
Ri.
(t)
+D.(t
"Attrition":
a
hit
by
Blue
aircraft
and
target
is
misclassified
as
alive
dA(t)
W
-,i(a)Ai(t)
- • U
(R,B)(RU.(t)
+
ku.(t))p
1
(R,B)I(Ai(t)
>
O)
dt
•
c
endurance
,
on
site
Attrition
of
Blue
by
Red
+
Ai(a)I((A(t)-
Ai(t))
>
0)
[
1
;(t)[A-(t
-
Ai(t)]
Arrival
rate
of
aircraft
to
ai
dAt)
d
=
-
,"
j
0(R'B)(Ro.(t)
+kii(t))pj(R,B)I(Ai(t)
>O)
Attrition
of
Blue
aircraft
by
Red
dW#(t)_
H(Ri.(t)
+
Di.(t))Bi(t)wij,
Rik(t)cikj(a)+Dik(t)cikj(d)
dt
k
R,.(t)+
Di.(t)
dWij(t;
a)
=
6(B,R)Aj(t)wi(a)_
Rik(t)cikj(a;
a)
+
Dik(t)cikj(d;a)
dt
k
0.Ol
+
Ri.(t)
+
A.3(t)
33

dWU (;
d)wik
Dy.(t)cijk
(d)
dWj(t;d)dt
/iH(R.I(t)+
Di.(t))Bi(t)Xk
Ri.(t)+
Di.(t)
Rate
change
wpns
fired
at
dead
"Attrition"
of
dead,
classified
as
alive
Red,typej,
in
g-i
Red,
atej,
class.
as
live
m
Ri
+
9i(BR)Ai(t)Ewik(a)
D#D(t)
k
0.01
+
Ri.(t)
+
Di.(t)
cok
(d;a)
Blue
aircraft
retargeting
a
dead
Red
and
classified
as
dead
dKi(t)
A
mH(Ri.
(t)
+ Di.
(t))B,
(t),
Wik
R,
(t)c+k
(a)pi.
k
Rate
of
change
of
dead
Red,
"Attrition"
Red
detected
(dead
disappear,
alive
join
-R
(t))
typej,
in
l
+
S(BR)j~tJUjka)
Rij
(
t)
cijk
(a;
a)pikj
(a)
+
6,(B,R)AI(t)X.,vwk(a)0.1
R.(t)+D.t
k
0.0
1
+
Ri.
(t)
+
Di.
(t)
C
1
aapk
a
"Attrition"
by
Blue
aircraft
34

A4.
Architecture
II
(Immediate
BDA):
Model
Defender Dynamics
dR-o.(t)
=R
1
4(0)
+
Z'Vyki~e(t)
dt
i
Migration
Rate
of
change
of
into
a.
Migration undet.
undetected
Red,
Red into
JRi
typej,
in
ai
-ýj~
kj
()
+Ri.(t
+
i.
t)Si
t)
ij(t)my
(a)
R-.
(t)
+
R,.
(t)
+
Di.
(t)
Sensor
detection
undetected
Red,
typej,
in
cki
+
vURU(t)
-1yiejij(t)+
Er
yRej(t)
Detected lost
-
Migration
Migration
detected
into
qRi;
becomes
out
of
Ri
undetected
upon
transit
+/2in(Rio(t)
+
Di.(t))Bi(t)Z
WikRij
(t)cjk
(a)(1
-
Pikj
)rikj
(dl
a)
k
Ri.
(t)
+DA.(t)
"Attrition":
a
miss;
alive
classif.
dead
-
aijL
j(t)
-
a.(a)So.(R,B)I(Ai(t)
>
O)ky(t)
Detection
from Red
action
Detection
from Red
firing at
aircraft
+,i
(B,
R)Ai
(t)Z
wik
(a)
Ry
(t)
cyk
(a;
a)(1
-
pj
(a))?kj
(dI
a;
a)
k
0.01
+
Ri.(t)+
Di.(t)
"Attrition":
a
miss by
Blue
aircraft
35

Multi-Region,
Multitype
Targets
dR
(td
=
4iH(k,.
(t) +
Ri.
(t)
+
Di.
(t))Si(t)
k.
(t)mi.
(a)
dt
i(t+
.(t+D.(t
Rate
increase
Sensor
detection
of
Red
typej
in
9qi
detected
Red,
typej,
in
9ti
-
vijRU(t)
--
Z
,ijRy(t)
I ýi
Lost/
undetected
Migration
out
of
ai
-,ui
H(Ri.
(t) +
Di.
(t))Bi
(t)_
wEkRo'(t)c'k(a)[pi
+(1-pik)rkI(dla)]
k
Ri.(t)+Di.(t)
"Attrition"
by
Shooter
(missiles/gunnery)
+
aj
k(t)
+
ai(a)So.(R,
B)I(Ai(t)
>
0)ky
(t)
Detection
from
Red
action
Detection
from Red
firing
at
aircraft
(e.g.
SCUD
shot)
-.
j(B,
R)A,(t)Xwjk(a)
R#R(t)
c~ik(a;
a)[pii
(a)
+
(1-
pikj(a))rikj
(di
a;a)]
k
0.01+
Ri.(t)+
D,.(t)
"Attrition":
a
shot
by
Blue
aircraft
dDij(t)
=
H(R.
(t)
+
Ri.
(t)
+
Di. (t))S
)
Di
(t)mo
(d)
dt
= i
-
t.
(t)
+
Ri.
(t)
+
Di.
(t)
Rate
increase
Sensor
detection;
classified
dead
(slow
BDA)
Red
dead
misclassified
as
alive
+/piH(Ri,.
(t)
+
Di.
(t))Bi
(t)E
wikRo
R(t)cijk(a)pikJ
rkj
(aid)
k
Ri.
(t)
+
Di.°(t)
"Attrition"
misclassified
as
dead
-ui
H(Ri.
(t)
+
Di.
(t))Bi
(t)E_
wikOD
(t)cjk
(d)rij
(did)
k
Ri.
(t)
+DA.(t)
"Attrition"
delayed
classif.
dead
as
dead
+i±
(B,
R)Ai
(t)Z
Wik
(a)
Rio
(t)
Cijk
(a;
a)piV
(a)ryk
(ald;
a)
"Attrition":
a
hit
by Blue
aircraft
-6D
(B,
R)A,
(1)1
Wk
(a)
Dii
(t)
cik(d;a)ryk(ala;a)
k
0.01
+
Ri.
(t)
+
Di.
(t)
Blue aircraft retargeting
a
dead
Red
36

dW
t
(t)
tHR
) +
D
.
Rik(t)cikj(a)
+
Dik(t)cikj
(d)
dt
k
Ri.
(t)
+
Di.
(t)
dWij
(t;a)
-
6i
(B,
R)Ai
(t)wo
(a)_
Rik
(t)cikj(a;a)
+
Dik
(t)cikj
(d;
a)
dt
k
O.O1
+
Ri.(t)+
Di.(t)
dAijt
W
)=
-fOi(a)Ai(t)
-
1
,ij
(R,
B)(RU
(t)
+
Ru
(t))pj(R,
B)I(Ai
(t)
>
0)
dt
endurance
,
on
site
Attrition
of
blue
by
Red
+
pi
(a)I(-A(t)
-
Ai
(t))1RR,
(t)
[At)-
Ai
(t)]
S1+
R
i.W
Arrival
rate
of
aircraft
to
SIi
dA(t)
1.5u
_-6.(R,
B)(RU
(t)
+ k.
(t))pj
(R,
B)I(Ai
(t)>
0)
dt
j
Attrition
of
Blue
aircraft
by
Red
dWi
(dt
- p
1
iH(Ri.(t)
+
DA.(t))Bi(t)
Rik.
(t)c
i.(d)
Rate
increase
"Attrition"
delayed
classif.
dead
as
dead
of
wpns fired
at
type
J
Red
ead
misclassified
as
alive
+
.5(B,
R)
A
(t)wk(a)
DU
(t)
Cijk(d;a)
k
i(BR
i.0
w1+R.(t)
+
Di.(t)
Blue
aircraft
retargeting
a
dead
Red
My t)
piH(R.
()
+Di.(t)Bi
t~j
wikR
Ri(t)Co'k(a)pikj
dK'
(t)
+
DR+D..(t))Bi
t)L
WR
(t
(t)
Rate
increase
"Attrition"
by
Shooter
(missiles/gunnery)
dead
Red
typej,
in
4i
+
9i(BR)Ai(t)Z
wik(a)
Rio(t)+
k
0.0
1
+
Ri.
(t)
+
Di.
(t)Ck(aapj()
"Attrition":
a
shot
by
Blue
aircraft
37

APPENDIX
II
Number
undetected
tgts
Architecture
1:
Delayed
BDA
Constant
Input
Rate
: 500
H
tgts,
500
S
tgts
per
day
pHIH=pHIS:pSIS=0.7;
pSjH=0.1;
100
shooters
missile
firing
latency-l
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
25000
20000
200
Prob(
correct
cdass.)=Prob(correct
BDA)=
0.50
15000
--
Prob(correct
class.)=1;
Prob(c'orrect
BDA)=0.5
1 0
--
0--0Prob(correct
class.)=0.5;
Prob(correct
_ _BDA)=1
50000
E
100-'-*
Prob(correct
class.)=P(correct
B3DA)=1
5000
0
10
20 30 40
50
Time
Figure
4(b)
4(b)
With
100
(missile)
shooters
having
basic latency
of
1
hour,
allowing
attacks
by
100
aircraft,
and
using Architecture
1,
the
number
of
undetected targets
(b-i)
remains
about
0
if
all
classification
and
BDA
capabilities
are
perfect
(100%),
but
(b-2)
continue
to
grow
if
classification
and
BDA
skills
both
drop
to
50%.
In
a
tradeoff
between
(b-3)
perfect
classification
and
50%
BDA,
or
(b-4)
50%
classification
and
perfect
BDA,
the
latter
is
the
more
effective.
But
the
system
cannot
control
the
number
of
undetected
targets,
which
grows
nearly
linearly.
38

Number
undetected
tgts
Architecture
I:
Delayed
BDA
Constant
Input
Rate:
500
H
tgts,
500
S
tgts
per
day
pHjH=pHjS=pS!S=0.7;
pSIH=0.1;
100
shooters
missile
firing
latency=0.50
hr.
100
aircraft;
1
hr. on
station;
11
hrs.
off
station
3000
2500
---
Prob(correct class.)=Prob(correct
S2000BDA)=0.50
--
rrob~correct
class.)=1; Prob(correct
___________________
BDA)=0.5
1500
Prob(correct
class.)=0.5;
Prob(correct
BDA)=1
S1000
-
•
----
Prob(correct
class.)=Prob(correct
BDA)=1
500
--------.. . .
0
-
0
10
20
30 40
50
Time
Figure
4(c)
4(c)
Reducing
latency.to
0.5
hour
from
1
hour
improves/reduces
the undetected
target
backlog
considerably when
classification
and
BDA
are
imperfect:
(c-i)
if
classification
and
BDA
skills
are
50%
then
undetected
backlog
continues
to
increase,
but
at
-
10%
the
rate
obtained when
latency
is
1
hour;
(c-2)
if
probability
of
correct
classification
is
increased
to
100%,
but
BDA
is
50%
then
the backlog
quickly
reaches
a
steady-state
value
of
around
400;
if
(c-3)
classification skill
if
50%
while BDA
is
100%
that
steady-state
backlog
is
-
200,
and
(c-4)
if
both
skills
are
100%
then
there
is
virtually
no
backlog.
Clearly
the
latency
has
a
powerful
effect
on
this
measure
of
total
system
capability.
39

Total
Number of Wpns
shot
Architecture
I:
Delayed
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
/
day
pHJH=pHJS=pSJS=0.7; pSIH=0.1;
100
shooters
missile
firing
latency=1
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
160000
------------..
-
.......
. .. . .
. .
. .
.
....
.......
.-....-..............................................
-
140000
120000
Prob(correct
dass.)=1;
Prob(correct
=
100000
BDA)=0.5
-"
Prob(
correct
class.)=Prob(correct
S
8BDA)=
0.50
""8-
Prob(correct
class.)=0.5;
E
Prob(correct
BDA)=1
z 60000
-.-
Prob(correct
class.)=P(correct
BDA)=4
40000
20000
0
0
5
10
15
20 25 30 35 40
45
50
Time
Figure
4(d)
4(d)
If
latency
is
1
hour
then
in
all
cases
the
total
number
of
weapons expended/shot
increases
nearly
linearly
in
time,
but
at
considerably
different
rates.
(d-1)
For perfect
(100%)
classification
and
BDA
the
number
after
50
days
is
-
70,000. (d-2)
For
perfect
(100%)
classification
and
50%
BDA
the
number
after
50
days
is
-
110,000,
an
increase
of
nearly
60%,
while
(d-3)
if
classification
is
100%
and
BDA
is 50%
the
corresponding
number
of
shots
is
-
80,000. (d-4)
If
both
classification
and
BDA
are
at
the
50%
level
the
number
of
shots
is
-
100,000.
Consequently,
if
the
system
is
initially operating
at
the
classification
50%,
BDA
50%
level,
by
far
the
greatest
payoff
is
increasing
BDA.
40

Total Number
of
Wpns
shot
Architecture
1:
Delayed
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
/
day
pHIH~pHIS=pSIS=0.7;
pSIH0O.i;
100
shooters
missile
firing
latency=0.50
hr.
100
aircraft;
1
hr.
on
station;
I1I
hrs.
off
station
160000
-________________
140000
___
120000
--
Prob(carrect
class.)=Prob(correct
S100000
BDA)=0.50
)KE
Prob(correct class.)=1;
Prob(correct
_______
DA)=0.5
I 6-
Prob(correct
class.)=0.5;
Prob(correct
E
BDA)=1
z 60000
Prob(correct class.)=Prob(correct
40000
20000
__
0
0
10
20
30
40
50
Time
Figure
4(e)
4(e)
Suppose
latency
is
reduced
to
0.5
hour.
The
pattern
of
(d)
above
is
qualitatively
followed
in
the
same
order,
but
the
total
number
of
weapons
expended
is
much
greater
case
by
case.
It
remains
preferable
to
improve
BDA
than
to
improve
classification
capability,
starting
them
both
at
the
50%
level.
41

Total
number
tgts
killed
Architecture
1:
Delayed
BDA
Constant
Input
Rate
:
500
H
tgts;
500
S
tgts
per
day
Activity
rate=0.02;
pHIH~pHIS~pSIS=0.7;
pSIH=0.1; 100
shooters
missile
firing
latencyl1
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off station
60000
......... ..... ... . ..........
...... ....................... ...............
40000
.---
Prab(correct
class.)
=P(correct
BDA)=l
-.-
Prob(correct ciass.)=0.5; Prob(correct
30000
_______
BDA)=1
E
-0--000
Prob(correct
class.)=i;
Prob(correct
BDA)=0.5
z
20000
_____
~-Prob(
correct
class.)=Prob(correct
BDA)=
0.50
0
0
10
20
30 40
50
Tim.i
Figure
4(f)
4(f) Consider
the
cumulative
number killed
if
latency
is
1
hour.
Here
50,000
Red
targets
are
killed
after
50
days
if
classification
and
BDA
are
both perfect
(100%).
If
both
are
50%
capable
this
number
drops
to
-28,000;
if
classification improves
to
100%,
with
BDA
still
50%
the
number
killed
increases
to
-'
33,000,
while
if
classification
remains
at
50%,
but
BDA
increases
to
perfection
(100%),
the
number
killed
by
time
50
increases
to
-43,000.
Once
again
the
relative
advantage
of
increasing
BDA
is
suggested.
42

Total
number
tgts
killed
Architecture
I:
Delayed
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
per
day
pHIH=pHIS=pSIS=0.7;
pSIH=0.1;
100
shooters
missile
firing
latency=0.5
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
60000-
-
................
.
- - - - -
- - - -
- --................................- .
50000
40000
_
--
Prob(correct
class.)=Prob(correct
BDA)=1
---
Prob(correct
class.)=1;
Prob(correct
30000
_
BDA)=0.5
EProb(correct
class.)=0.5;
Prob(correct
z
20000
Prob(correct
class.)=Prob(correct
BDA)=0.50
10000
0
0
10
20
30
40
50
Time
Figure
4(g)
4(g)
Next
let
latency
be
reduced
to
0.5
hour.
Then
all
degrees
of
classification-BDA
quality
(examined)
are
essentially
equivalent,
achieving
50,000
kills
after
50
days.
Of
course,
the
choice
to
improve
BDA
from
50%
(along
with
50%
classification)
obtains
this
kill
rate
far
more
economically
than
do
the
other
choices.
43

Number of
erroneous
Wpns
shot
Architecture
1:
Delayed
BDA
Constant
Input
Rate
:
500
H
tgts;
500
S
tgts
per
day
pHJH=pHJS=pSjS=0.7;
pSIH=0.1; 100
shooters
missile
firing
latency=1
hr.
100
aircraft;
1
hr. on
station;
11
hrs.
off
station
140000
.
120000
100000
SProb(
correct
cdass.)=Prob(correct
BDA)=
0.50
"
80000
0- -
Prob(correct
class.)=0.5;
Prob(correct
BDA)=1
Prob(correct
class.)=1;
S60000
Prob(correct
BDA)=0.5
u
.----
Prob(correct
class.)=P(correct
BDA)=I
40000
20000
0
0
5 10 15
20
25 30 35 40
45
50
Time
Figure
4(h)
4(h)
For
latency
of
1
hour
we
see
that
the
number
of
erroneous
weapons
fired
increases
almost
linearly.
Erroneous
here
includes Soft
target
weapons
fired
at
Hard
targets
(minimal
effect),
Hard
target
weapons
fired
at
Soft
targets
(maximal
expense,
and
in
some
cases
small,
direct effect)
and
weapons
fired
at
dead
targets.
(h-i)
For perfect
classification
and
BDA
there
are
no
errors;
however,
(h-2)
for
50%
capability
for
both
classification
and
BDA
the total
number
reaches
-
60,000.
(h-3)
If
probability
of
correct
classification
is
50%,
but
BDA
is
improved
to
perfect
(100%),
the
expenditure
drops
to
-
40,000,
while
(h-4)
if
classification
increases
to
perfection
(100%),
and
BDA remains
at
50%,
the
erroneous shots expended
fall
to
-
17,000.
Thus, this
measure
is,
in this
case,
actually
improved more
by
increasing
classification
capability
than
by
improving
BDA.
44

Number
of
erroneous
Wpns
shot
Architecture
1:
Delayed
BDA
Constant
Input
Rate: 500
H
tgts;
500
S
tgts
pHIH=pHjS=pSjS=0.7;
pSIHO0.1;
100
shooters
missile
firing
latency=-0.50
hr.
100
aircraft;
I
hr. on
station;
11
hrs.
off
station
140000
120000
-____________________
_________
100000
___
________________
--
Prob(correct
class.)=Prob(correct
C
BEDA)=0.50
80000
---
Prob(correct
class.)=0.5:
Prob(correct
BDA
=
C
)K
Prob(correct
class.)=1;
Prob(correct
60000
-
____________
I
BDA)=0.5
-0-
Prob(correct
class.)=Prob(correct
BDA)=1
20000
0
0
10
20
30
40
50
Time
Figure
4(i)
4(i)
For
latency
of
0.5
hour
the
number
of
erroneous
weapons
fired
tends
to
increase
linearly,
but
more rapidly,
from
case
to
case
because
of
the
increased
raw
shooting
rate.
In
this case
as
well,
it
seems
functionally
better
to
improve
classification
than
BDA.
45

Number
of
Red
Wpns
Expended
Architecture
1:
Delayed
BDA
Constant Input
Rate: 500
H
tgts;
500
S
tgts/day
pHIH~pHIS~pSISO.7;
pSIHO0.1;
100
shooters
missile
firing
Iatency=1
hr.
100
aircraft;
1
hr.
on
station;
I11
hrs.
off
station
12000
....
- -.
-
.....-
. -.
*-.......--
..
10000
S8000
-j--Prob(
correct class.)=Prob(correct
0
~BDA)=
0.50
i. I
.---
Prob(correct
class.)1I;
S6000
etBA=.
O
-*-Prob(correct
class.)=0.5;
Prob(carrect
BDA)=l
4000
_______________
--
*--Prob(correct
class.)=P(correct
E
z
200
0
5
10 15
20
25
30
35
40 45
50
Time,
Figure
4(j)
4(j)
With latency
of 1
hour
there
is
dramatic
reduction
in
Red
weapons
fired
by
improving BDA
(50%
to
100%,
classification
at
50%)
instead
of
improving
classification
(from
50%
to
100%,
BDA
at
50%).
46

Number
of
Red
Wpns Expended
Architecture
1:
Delayed
BDA
Constant Input
Rate:
500
H
tgts;
500
S
tgts/day
pHIH=pHIS~pSISO0.7;
pSIH=0.1;
100
shooters
missile
firing
Iatencyo-.50
hr.
100
aircraft;
I
hr.
on
station;
I I
hrs.
off
station
1800
....
.
.. ..--
- -
- - - - -
- - -
1600
1200-
Prob(correct
class.)=Prob(correct
0
BDA)=0.50
~1000
--
-_____________
4Prob(correct
class.)1I;
Prob(correct
BDA)=0.5
________________________-~-Prob(correct
class.)=0.5;
Ix80
Prob(correct
BDA)1l
I
4
Prob(correctclass.)=Prob(correct
z
400
0
10
20
30
40
50
Time
Figure
4(k)
4(k) With
latency
of
0.5
hour
there
is an
effect
in
improving
classification,
but
more
of
an
effect
from
improving
BDA
quality.
47

Figure(s)
5:
Architecture
2
The basic
Figure
4(a)
applies
here
first,
showing
the
assumed
arrival
of
Red
targets
into
region
9Jt
and
the
number
undetected.
Number
undetected
tgts
Architecture
I1:
Immediate
BDA
Constant
Input
Rate
:
500
H
tgts,
500
S
tgts
per
day
pHIH=pHIS=pSIS=0.7;
pSIH=0.1;
100
shooters
missile
firing
latencyrl
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
25000
20000
Prob(correct
class.)=Prob(correct
BDA)=0.50
S15000-
"-4,-Prob(correct
class.)=1;
Prob(correct
BDA)=0.5
---
Prob(correct
class.)=0.5;
E10000
Prob(correct
BDA)=1
z
-.-
Prob(correct
class.)=Prob(correct
BDA)=1
5000
09
0
10
20
30
40
50
Time
Figure
5(b)
5(b)
Same
general
pattern
as
4(b):
using
Architecture
2
with
1
hour
latency
the
number
of
undetected
targets
(b-1)
remains
small
if
classification
and
BDA
are
perfect,
but
continue
to
grow
(but
at
a
slower
rate
than
for
Architecture
1)
if
classification
and
BDA
skills
are
50%.
Again,
(b-3)
the
number
of
undetected
targets
is
reduced
more
by
improving
BDA
than
by
improving
classification.
48

Number
undetected
tgts
Architecture
II:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts,
500
S
tgts
per
day
pHIH=pHIS=pSIS=0.7;
pSIH=0.1;
100
shooters
missile
firing
latency-0.50
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
400
.. .
... ......
.. .. .. .. .
.......
350
300
7
_________
Prob(correct
class.)=Prob(correct
250
_BDA)=0.50
-
Prob(correct
class.)=1;
Prob(correct
" ____BDA)=0.5
200
---
4,-
Prob(correct
class.)=0.5;
Prob(correct
BDA)=1
E10--
Prob(correct
class)=Prob(correct
BDA)=1
z
100
-_______________________
50
o0
0
10
20
30
40
50
Time
Figure
5(c)
5(c)
Latency
of
0.5
hour
leads
as
before
to
steady-state
backlogs
of
undetected
targets:
(c-i)
perfect
classification
and
BDA
yield
very
small
such
backlogs;
(c-2)
50%
classification
and
BDA
is
essentially
the
same
(about
370)
as
in
Architecture
1;
(c-3)
improvement
of
BDA
to
perfect
(100%)
here
reduces
backlog
to
-
130,
an
improvement
over
Architecture
1.
49

Total
Number of
Wpns
shot
Architecture
Ii:
Immediate
SDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
/
day
pHjH=pHIS=pSIS=0.7;
pSIH=0.1;
100
shooters
missile
firing
latency=l
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
160000
-. -.--. --
----
*----*--*--**--**----**- -- *-. . . ..
140000
120000
-
-4-
Prob(correct class.)=1;
Prob(correct
100000-
BDA)=0.5
I
-a-
Prob(correct
cdass.)=Prob(correct
BDA)=0.50
---
Prob(correct class.)=0.5; Prob(correct
E
BDA)=1
Z
60000
-- 4--
Prob(correct
class.)=Prob(correct
BDA)=1
40000
20000
0
0
10
20
30 40
50
Time
Figure
5(d)
5(d)
When latency
is
1
hour
the
total number
of
weapons
shot
is
reduced
quite
marginally,
to
about
100,000
from
Architecture
1
when
classification
and
BDA
are
(d-1)
both
50%,
and
(d-2)
when
classification
is
improved
to 100%,
BDA
remaining
at
100%.
Improving BDA
to
100%
leads
to
improvement
with
classification
at
50%,
but
gives
nearly
the
same
result
as
in
Architecture
1.
50

Total Number
of Wpns
shot
Architecture
II:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
I
day
pHIH=pHISpSISO.7;
pSIHO.1;
100
shooters
missile
firing
latency=.50
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
160000-------------------------------------------------
140000
120000
I
--
Prob(correct
class.)Prob(correct

100000
BDA)=0.50
-4---
Prab(coITect
class)=1;
Prob(correct
o
BDA)=0.5
S
.0
I
-4--
Prab(conect
dass.)=0.5;
Prob(correct
E
-i -4-
Prob(correct
class)=Prob(Correct
40000
0
0 10
20
30
40
50
Time
Figure
5(e)
5(e)
If
latency
is
reduced
to
0.5
hour the
number
of
weapons
shot
is
noticeably
reduced,
case-by-case,
from
expenditures
in
Architecture
1.
51

Total
number
tgts
killed
Architecture
Ih:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
per
day
pHIH=pHIS=pSIS=0.7;
pSJH=0.1;
100
shooters
missile
firing
latency-l
hr.
100
aircraft;
I
hr.
on
station;
11
hrs.
off
station
50000
-
40000
---
Prob(correct
class.)=Prob(correct
BDA)=1
---
Prob(correct
class.)=0.5;
Prob(correct
30
_
_
_
_
_
BDA)=1
o
300-0-0
Prob(correct
class.)=1;
Prob(correct
"I
BDA)=0.5
-*
Prob(correct
class.)=Prob(correct
20000
i
BDA)=0.50
10000
0
0 5
10
15
20 25
30 35
40
45
50
Time
Figure
5(f)
5(f)
When
the
latency
is
one
hour
the
number
of
targets
killed
is
-
50,000
when
classification
and
BDA
is
perfect;
this
is
the
same
as
Architecture
1.
When
BDA
is
perfect,
but
classification
is
50%,
the
result
is
the same
as
for
Architecture
1.
Architecture
2
results
in
slightly
more
kills than
Architecture
1
in
the
other
cases.
52

Total
number
tgts
killed
Architecture
II:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
per
day
pHIH=pHjS=pSIS=0.7;
pSIH=0.1;
100
shooters
missile
firing
latency=0.5
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
60000
50000
-
40000
--
Prob(correct
class)=Prob(correct
BDA)=1
-4--
Prob(correct
class.)=1;
Prob(correct
"300
BDA)=0.5
.0
.--
Prob(correct
dass.)=0.5;
Prob(correct
E
BDA)=1
z
20000
Prob(correct class.)=Prob(correct
10000
BDA)=0.50
10000
0
10
20
30
40
50
Time
Figure
5(g)
5(g)
Decreasing
the
latency
to
0.5
hour
results
in
all
degrees
of
classification-BDA
quality
(examined)
being
equivalent,
achieving
50,000
kills
after
50
days.
53

Number
of
Erroneous
Wpns
Shot
Architecture
II:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgtslday
pH
H~pHIS~pSISO0.7;
pSIHO0.1;
100
shooters
missile firing
Iatencyl-
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
140000
..........
...
----------
-
----
**
.
--
120000
10~0 ]
-~Prab(correct
cdass.)=Prob(correct
]
BDA)=0.50
S80000
Pmob(correct
dass.)=0.5;
Prob(carrect
BDA)=1
C-*
Pmb(correct
class.)=1
Prob(correct
260000
BDA)=0.5
W
--
Prab(correct
class.)=Prob(carrect
40000BDA)=1
40000
0
1
0
10
20
30
40
50
Time
Figure
5(h)
5(h)
For
latency
1
hour:
(h-i)
the
number
of
erroneous
weapons
shot
increases
nearly
linearly
to
-60,000
in
50
days,
when
classification
and
BDA qualities
are
both
50%,
very
little
less
than
in
Architecture
1.
Increasing
BDA
alone
to
100%
drops
the
expenditure
to
-
40,000,
but
instead increasing classification
alone
to
100%
reduces
expenditures
to
slightly
less
than
the
corresponding
Architecture
1
figure.
54

Number
of
Erroneous
Wpns
shot
Architecture
11:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts
per
day
pHIH=pHIS~pSIS=0.7;
pSjH=0.1;
100
shooters
missile
firing
Iatency=.50
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
140000...................................-
120000
-_______
100000
--
Prob(correct
class.)=Prob(correct
r
BDA)=0.50
180000
-4-Prob(correct
dass.)=0.5; Prob(correct
C
BDA)=1
0
A-
Prob(correct
class.)1I;
Prob(correct
o60000
OPOOOO00
1
DA)=0.5
W
-0-Prob(correct
class)=Prob(correct
40000BDA)=1
20000
______
0i
0
10
20
30
40
80
Time
Figure
5(i)
5(i)
Decreasing
latency
to
0.5
hour
makes
negligible
difference
in
the
cross-architecture
erroneous
weapon
expenditure
except
when
classification
is
100%,
and
BDA
quality
50%:
then
the shorter
latency
is
associated with close
to
a
50%
decrease
in
erroneous
shots.
55

Number
of
Red
Wpns
Expended
Architecture
fl:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgtslday
pHIH~pHIS~pSISO.7;
pSIHO0.1; 100
shooters
missile
firing
Iatency--1
hr.
100
aircraft;
1
hr.
on
station:
11
hrs.
off
station
12000
.
....
.
......... ------ ----------
* - - - - -
10000
____________________
_
800
0h
Prob(correct
class.)=Prob(correct
c
BDA)=0.50
4-Prob(correct
class.)=1; Prob(correct
_______________________________BDA)=0.5
X*
Prob(correct
class.)=0.5;
Prob(correct
I
-Prob(correct
class.)=Prbb(correct
BDA)=1
z
2000
0
1W
0
10
20
30
40
50
Time
Figure
5(j)
Number
of
Red
wpns
Expended
Architecture
II:
Immediate
BDA
Constant
Input
Rate:
500
H
tgts;
500
S
tgts/day
pHIH~pHIS=pSISO0.7;
pSIH=0.1;
100
shooters
missile
firing
latency--.50
hr.
100
aircraft;
1
hr.
on
station;
11
hrs.
off
station
4000
-----------------
- - -
3500
03000
-~Prob(correct
class.)=Prob(correct
S2500
BDA)=0.50
-4-Prob(carrect
class.)=l;
Prob(correct
I
BDA)=0.5
~ 200
-W
Prob(correct
class.)=0.5;
Prob(correct
0
1500
--
Prob(correct
class)=Prob(correct
BOA)=1
E
z
1000
500
0
0
10
20
30
40
50
Time
Figure
5(k)
56

APPENDIX
III
Expected
Number
of
Times
Through
the
Shooting
Server
for
Architecture
1
There
are
j
=
1, ..., J
types
of
targets. When
a
live
target
of
type
j
is
acquired
it
is
classified
as a
type
k
with probability
cjk(a).
The
classification
of
the
target
influences
the
weapon
that
will
be
shot
at
it.
The
probability
a
shot
kills
a
target
of
type
j
that
is
classified
as
type
k
is
pjk.
After
each
shot,
battle
damage
assessment
(BDA)
is
performed.
The
probability
a
dead
target
of
type
j
is
correctly assessed
as
dead
is mj(d).
If
a
dead
target
is
misclassified
as
live,
it is
returned
to
the
shooter
server.
The probability
the
shooter
server
classifies
a
dead
target
of
typej
as a
type
k
is
cjk(d).
Once
a
dead target
is
classified
as
dead,
it is
removed
from the
list
of
targets.
The
target
is
reclassified
independently
each
time the target
passes
through
the
shooter
server.
A
target
of
type]
can
pass
through
the
shooter
server
more
than
once
while
it
is
alive
and
may
return
to
the
shooter
server
when it
is
dead
if
it
is
misclassified
as
live.
Let
NA(j)
=
Number
of
times through
the
shooter server
for
a
typej
target while
it
is
alive
NDo()
=
Number
of
times
through shooter
server
for
a
typej
target
while
it is
dead
Ny(j)
=
Total
number
of
times through
the
shooter
server
(both
alive
and
dead)
for
a
typej
target
E[NA
(j)]
=
1 + I
cjk
(a)(1
-
Pjk
)E[NA
(j)]
(111.1)
k
Solving,
E[NA
(j)]=
1-
•_cjk
(a)(1
-
Pjk)
k
(111.2)
__ 1
cjk
(a)pjk
k
E[
ND(j)]
( 1-
mj
(d))E[N°5
(j)]
(111.3)
57

where
E[N°(j)]
=1
+Z
cjk
(d)(l-
mj(d))E[ND
(j)]
(IliA)
k
Solving,
E[NDO(j)]=1
1
C-
cjk(d)[l
-mj(d)]
k
(111.5)
1
mj(d)
Thus,
E[NT(j)]=
E[NA(j)]+[1-mj(d)]E[N°(J)]
1
I
[l-mj(d)]l_1c
(d11.m6d)
l-_
cj•k
(al(l
-pjk
)
c-
(d111.6)
d)
k
k
_cjk(a)pjk
[1-
mj(d)]
mjld)
k
Example
Suppose
there
are
two
target types: Hard
(H)
and
Soft
(S).
Let
the probability
of
correct
target
classification
be
cHH
= css =
c.
Let
the
probability
of
correct
BDA
of
a
dead
target
as
dead be mj(d)
=
m.
E[NT(H)]=
1
1
-[C(1-PHH)
+(1
-c00
-
PHS)
1
+(1-m)-
[l-m]
(111.7)
1_____
1-rn
cpHH
+ (1 -
c)PHS
m
1
1-rn
f(c)
m
where
f
(c)
=
CPHH
+
(1-c)pPs
(111.8)
58

Note
that,
ac
E[Nr(H)]
=
PHH-PHS
(III.9)
amE[N,(H)]
=
(m11.10)
Note
that
(111.9)
and
(111.
10)
are
both
nonpositive.
If
(111.9)
>
(111.
10)
then
increasing
the
probability
of
correct
BDA,
m,
will result
in
a
greater
decrease
in
the
expected
number
of
shots
required
to
kill and
correctly
assess
a
dead
target
as
dead
than
increasing
the
probability
of
correct
classification,
c.
The
figures
below
display
regions
in
which
it
is
better
to
increase
the
probability
of
correct
BDA
assessment,
m,
than
to
increase
the
probability
of
correct
target
classification,
c,
and
vice-versa. Values
of
the
probability
of
correct target
classification,
c,
are
plotted
on
the
y-axis
and
the values
for
the
probability
of
correct
BDA,
m,
are
plotted
on the
x-axis. The
plotted
line
are
those values
of
(m,
c)
such
that
(111.9)
is
equal
to
(III.10).
If
the
value
of
(m,
c)
is
above
the line, then
increasing
m
will
result
in
a
greater decrease
in
the
expected
number
of
shots required to
kill
a
hard
target
and
correctly
assess
the
dead
target
is
dead than increasing
the
probability
of
correct target
classification,
c.
If
the
value
of
(m,
c)
is
below
the
line,
then
increasing
the
probability
of
correct
target
classification,
c,
will
result
in
a
larger
decrease
in
the
expected
number
of
shots
required
to
kill
a
target
and
correctly
assess
it is
killed
than increasing
the
probability
of
correct
BDA.
For
fixed
value
of
m,
let
c(m)
be that
value
of
c
for
which
(111.9)
equals
(111.10).
If
c
>
c(m),
increasing
the
probability
of
correct
BDA,
m,
results in
a
larger
decrease
in
the
expected
number
of
shots
required
to
kill
and
correctly
assess
that
a
dead target
is
dead
than
increasing
the
probability
of
correct
classification
c.
Comparison
of
the
four
figures
shows
the
following
about
the
behavior
of
c(m). The
value
of
c(m)
is
nondecreasing
as
the
value
of
m
increases.
If
the
probability
of
kill
is
the
same
for
a
correctly
classified
59

target
and
an
incorrectly
classified
target,
then
(111.9)
equals
0
and
it
is
always
better
to
increase
the
probability
of
correct
BDA,
m.
The
largest
value
of
m
such
that
c(m)=
0
(increasing
the
probability
of
correct
BDA
is
always
better)
decreases
as
the
difference
between
the
probability
of
kill
for
a
correctly
classified
target
and
the
probability
of
kill
for
an
incorrectly
classified
target
increases.
The
smallest
value
of
m
such
that
c(m)
=
1
(increasing
the
probability
of
correct
target
classification
is
always
better)
increases
as
the
difference
between
the
probability
of
kill
for
a
correctly
classified
target
and
the
probability
of
kill
for
an
incorrectly
classified
target
increases.
If
for
some
m,
0
<
c(m)
<
1,
then
increasing
the
difference
between
the
probability
of
kill
for
a
correctly
classified
target
and
an
incorrectly
classified
target
will
decrease
the
value
of
c(m).
Hence,
increasing
the
probability
of
kill
for
a
correctly
classified
target
while
keeping
the
probability
of
kill
for
an
incorrectly
classified
target
the
same
may
make
it
more
advantageous
to
increase
the
probability
of
correct
BDA.
60

Architecture
I
If
c
is
above
the
line,
increasing
the
Prob.
of
correct
BDA,
m,
will
result
in
a
greater
decrease
in
the
expected
number
of
weapons
fired
at
a
target
pHH=0.5,
pHS=0.1
0.9
--
____________
__________
o
0.8
S0.7
i04
a-
S0.5
0
0.1
0.2 0.3
0.4
0.5
0.6
0.7
0.8
0.9
m=Prob,
of
correct
BDA
Figure
0.1.1
Architecture
I
If
c
is
above
the
line,
increasing
the
Prob.
of
correct
BDA, m,
will
result
in
a
greater
decrease
in
the
expected
number
of
weapons
fired
at
a
target
pHH=0.7,
pHS=0.1
1----------.............................................
.... . .....-.--.-.....
-- -
0.9
0.8
S0.
E
0.7
0.6
0.3
L
0.2
O'0
0.1
0.2
0.3 0.4 0.5
0.6
0.7 0.8
0.9
m=Prob,
of
correct
BDA
Figure
0.1.2
61

Architecture
I
If
c
is
above
the
line,
increasing the
Prob.
of
correct
BDA,
m,
will
result
in
a
greater
decrease
in
the
expected number
of
weapons
fired
at a
target
pHH=0.9, pHS=0.1
0.9
0.8
-
0
!E
0.7
0
(U
0.
EP
•,0.5
80.4
-______________
0
.F
0.3III.
0.
p
IH0.2
0.1
0,
0.
1
0
0.1
0.2
0.3 0.4
0.5 0.6
0.7
0.8
0.9
m=Prob.
of
correct
BDA
Figure
111.3
Architecture
I
If c is
above
the
line,
increasing
the
Prob.
of
correct
BOA,
m,
will result
in a
greater decrease
in
the
expected
number
of
weapons
fired
at
a
target
pHH=1,
pHS=0.l
0.9
g0.8
~E0.7
(0.
(0.
S0.6
0.2
S0.41
0
0~ 0.3.
.
04 05
06
. . .
~ 0.2

INITIAL
DISTRIBUTION
LIST
1.
Research
Offi
ce
(Code
09)
.............................................................................................
1
Naval
Postgraduate
School
Monterey,
CA
93943-5000
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Knox
Library
(Code
013)
.............................................................................
2
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CA
93943-5002
3.
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....................................................................
2
8725
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J.
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Rd.,
STE
0944
Ft.
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VA
22060-6218
4.
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M
astowski
(Editorial
Assistant)
.................................................................
2
Department
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Research
Naval
Postgraduate
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Monterey,
CA
93943-5000
5.
Distinguished
Professor
Donald
P.
Gaver
(Code
OR/Gv)
.......................................
4
Department
of
Operations
Research
Naval
Postgraduate
School
Monterey,
CA
93943-5000
6.
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Patricia
A.
Jacobs
(Code OR/Jc)
...............................................................
4
Department
of
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Research
Naval
Postgraduate
School
Monterey,
CA
93943-5000
7.
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John
Osmundson
..................................................................
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copy)
Department
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josmundson@nps.navy.mil
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..................................................................
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gsschacher@monterey.nps.navy.mil
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...........................................................................
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.......................................................
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MOVES
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kemple@monterey.nps.navy.mil
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Institute
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cdmarash@nps.navy.mil
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64