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 (P R ) and benefits (P S ) Determine convergence to satisficing behavior using praxeic analysis (P S > b P R ) Evaluate emergent behaviors for desirability

Continuous Modelling – Digging Ditches A number of tasks (ditches) need to be accomplished, with differing start times (t j ), deadlines (D j ), work densities (r j (s) – man-hours required per foot), and projected lengths (L j ) 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, must equal fraction of men allocated to task for the amount of time the work took, fraction of time necessary for negotiation and communication fraction of total resources allocated to task j

Deriving a general model Passing to the continuous limit, If F j is the fraction of ditch j remaining un dug,

Illustrating a particular scenario… Constant work density – R j man-hours required, uniform distribution over entire ditch The end result of negotiation is to devote resources to those tasks nearest to completion: Real-time effect of negotiation is to add an overhead per allocated resource + a constant communcation overhead:

The differential model: weighted toward tasks closest to completion communication overhead per-resource cost of negotiation

Some dynamics consequences Summing all equations… total rate of task accomplishment can be no greater than this, accounting for real-time negotiation and communication overhead

Critical solutions – two dominant competitors Deadline for Task 1 Deadline for Task 2 Task 2 starts FjFj t Critical Trajectory Task 1 Fails Task 1 Succeeds

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

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:

Outcomes...