Analytic Prediction of
Emergent Dynamics for ANTS
James Powell, Todd Moon and
Dan Watson
Utah State University
APED Project Goals:


Construct models reflecting the dynamics
of multiple agents collaborating
autonomously to accomplish missions
Use dynamical systems techniques to
characterize emergent behaviors in terms
of design inputs and scenarios
APED Techniques
(continuous setting)



Use rate-equation modelling to develop
differential equations for task completion
Determine critical solutions which finish
`just in time,’ bounding space into
satisficing and non-satisficing regions
Examine stability of critical solutions for
desirable design traits
APED Techniques
(discrete setting)



Characterize discrete decisions in terms of
costs (PR) and benefits (PS)
Determine convergence to satisficing
behavior using praxeic analysis (PS > b PR)
Evaluate emergent behaviors for
desirability
Continuous Modelling –
Digging Ditches


A number of tasks (ditches) need to be
accomplished, with differing start times
(tj), deadlines (Dj), work densities (rj(s) –
man-hours required per foot), and
projected lengths (Lj)
Overseers negotiate among themselves
for the services of a common pool of (M)
`men’ to do the digging.
Rate Completion Modelling
Man-hours required to dig a short distance,
rj ( s )  s
must equal fraction of men allocated to task for the
amount of time the work took,
f j M t 1  a j 
fraction of total
resources allocated to
task j
fraction of time
necessary for negotiation
and communication
Deriving a general model
rj ( s ) s  f j M t (1  a j )
s M f j

(1  a j )
t rj ( s )
Passing to the continuous limit,
ds M f j

(1  a j )
dt rj ( s )
If Fj is the fraction of ditch j remaining un dug,
dF j
d

dt
dt

M fj
s
(1  a j )
1    L j rj ( s )
 L j 
Illustrating a particular scenario…


Constant work density – Rj man-hours required, uniform distribution
over entire ditch
The end result of negotiation is to devote resources to those tasks
nearest to completion:
fj 
1 /( R j F j )
1 /( R F )
k
k
all tasks

Real-time effect of negotiation is to add an overhead per allocated
resource + a constant communcation overhead:

aj 
M fj  
t
aj   M
1/ R j F j 
1/ R F 
k
all tasks
k

The differential model:
M

Fj  
Rj
1 / R j Fj 
1/R F 
k
all tasks
weighted toward tasks
closest to completion
k

1    



1 / R j F j  
1/ Rk Fk 


all tasks
communication
overhead
per-resource cost of
negotiation
Some dynamics consequences
Summing all equations…
2
1
/(
R
F
)
 k k
d
Rk Fk   M (1   )  

2
dt
1 /( Rk Fk )
total rate
of task
accomplishment


  M 1    
N

can be no greater than this,
accounting for real-time negotiation
and communication overhead
Critical solutions – two dominant
competitors
Fj
Task 2 starts
Critical Trajectory
Task 1 Fails
Task 1 Succeeds
t
Deadline for
Task 1
Deadline for
Task 2
We hope to assess the efficacy of
negotiation goals by:


Characterizing the probability of
successful completions in random
environments
Determining the sensitivity of task
completions to additional tasks,
unforseen difficulties, and changing
deadlines/constraints
The efficacy of negotiation
implementations can be predicted from


How the likely negotiation overheads in
real time affect task completions
How overheads create barriers to overall
accomplishment as tasks are added
We are testing the theory using
simulations in random environments




Differing allocation goals: democratic,
socialistic, opportunistic, just in time…
Random start times for new tasks
Tasks are assigned with random difficulty
Deadlines are assigned after start with
random (not necessarily reasonable)
frequency
Opportunistic Allocation

Tasks with small
remaining work loads
get priority
1
RjFj
fj 
1
1
1
1



R1F 1 R 2 F 2 R 3 F 3
RNFN

(inverse of task' s work left)
(sum of all inverses of tasks' work)
Outcomes...



Yardstick for success =?
One meaningful measure involves the
relative number of tasks completed
successfully and the number that failed
The “success ratio” is fairly consistent for
most strategies over several runs:
(# of successful ly completed tasks)
Sm 
(# of failed tasks)
Outcomes...