Continuity and Change (Activity) Are Fundamentally Related In DEVS Simulation of Continuous Systems Bernard P. Zeigler Arizona Center for Integrative Modeling and Simulation (ACIMS) University of Arizona Tucson, Arizona 85721, USA

Outline Review DEVS Framework for M&S Brief History of Activity Concept Development Summary of Recent Results Theory of Event Sets – Basis for Activity Theory Conclusions and Implications

Synopsis A continuous curve can be represented by a sequence of finite events sets whose points get closer together at just the right rate We can measure the amount of change in such a continuous curve – this is its activity The activity divided by the largest change in an event set gives the size of this set’s most economical representation DEVS quantization can achieve this optimal representation

DEVS = Discrete Event System Specification Based on formal M&S framework Derived from mathematical dynamical system theory Supports hierarchical, modular composition Object oriented implementation Supports discrete and continuous paradigms Exploits efficient parallel and distributed simulation techniques DEVS Background

DEVS Hierarchical Modular Composition Atomic: lowest level model, contains structural dynamics -- model level modularity + coupling Coupled: composed of one or more atomic and/or coupled models Hierarchical construction

DEVS Theoretical Properties Closure Under Coupling Universality for Discrete Event Systems Representation of Continuous Systems –quantization integrator approximation –pulse representation of wave equations Simulator Correctness, Efficiency

Atomic Models Ordinary Differential Equation Models Spiking Neuron Models Coupled Models Petri Net Models Cellular Automata n-Dim Cell Space Partial Differential Equations Self Organized Criticality Models Processing/ Queuing/ Coordinating Processing Networks Networks, Collaborations Physical Space DEVS Expressability can be components in a coupled model Multi Agent Systems Discrete Time/ StateChart Models Quantized Integrator Models Spiking Neuron Networks Stochasti c Models Reactive Agent Models Fuzzy Logic Models

Activity Theory unifies continuous and discrete paradigms Heterogeneous activity in time and space Quantization allows DEVS to naturally focus computing resources on high activity regions DEVS can represents all decision making and continuous dynamic elements DEVS concentrates its computational resources at the regions of high activity. While DEVS uses smaller time advance (similar to time step in DTSS) in regions of high activity. DTSS uses the same time step regardless of the activity.

Mapping Ordinary Differential Equation Systems into DEVS Quantized Integration DEVS instantaneous function DEVS Integrator  d s 1 /dt s 1 f 1 x  d s 2 /dt s 2 f 2  d s n /dt s n f n s x s x s x...  d s 1 /dt s 1 f 1 x  d s 2 /dt s 2 f 2  d s n /dt s n f n s x s x s x... DEVS S F F F Theory of Modeling and Simulation, 2nd Edition, Bernard P. Zeigler, Herbert Praehofer, Tag Gon Kim, Academic Press, 2000.

PDE Stability Requirements Courant Condition requires smaller time step for smaller grid spacing for partial differential equation solution This is a necessary stability condition for discrete time methods but not for quantized state methods Ernesto Kofman, Discrete Event Based Simulation and Control of Hybrid Systems, Ph.D. Dissertation: Faculty of Exact Sciences, National University of Rosario, Argentina

Activity – a characteristic of continuous functions Activity(0,T) = b a Activity = |b-a| #Threshold Crossings = Activity/quantum

#DEVS Transitions = #Threshold Crossings R. Jammalamadaka,, Activity Characterization of Spatial Models: Application to the Discrete Event Solution of Partial Differential Equations, M.S. Thesis: Fall 2003, Electrical and Computer Engineering Dept., University of Arizona

Activity Calculations for 1-D Diffusion Initial state ActivityActivity/N as N  ∞ Rectangular pulse 2HN(W/L)(1 –W/L)2H(W/L) (1 –W/L) Triangular pulse (N-1)*H/4H/4 Gaussian pulse Constant/L This shows that the activity per cell in all the three cases goes to a constant as N (number of cells) tends to infinity.

DEVS Efficiency Advantage where Activity is Heterogeneous in Time and Space Time Period T time step size # time steps =T/ activity A quantum q # crossings =A/q Potential Speed Up = #time steps / # crossings X number of cells

Ratio: DTSS/DEVS Transitions

#DTSS/#DEVS Ratio for 1-D Diffusion initial stateDTSS/DEVS Rectangular Pulse where w is the width to the length ratio Triangular Pulse Gaussian Pulse *f is an increasing function of L Alexander Muzy’s scalability results

Muzy’s Fire Front model Instantaneous Activity Peak Bars Accumulated Activity Region Of Imminence S. R. Akerkar, Analysis and Visualization of Time-varying data using the concept of 'Activity Modeling', M.S. Thesis, University of Arizona,2004

DEVS vs DTSS in Parallel Distributed Simulation J. Nutaro, Parallel Discrete Event Simulation with Application to Continuous Systems, Ph. D. Dissertation Fall 2003,, Univerisity of Arizona

N-1 N QB QB QB N/2 Units N-1 N N 2 QB QB QB N/2 Units N-1 N 2 QB QB QB N/2 Units n = log 2 N Stages Q = QuantizerB = Butterfly N = 1024 (powers of 2) n = 10 (log 2 N) q = Quantum for respective stage QFFT Module q = q 0 /Nq = q 0 /(N-1)q = q 0 / Harsha Gopalakrishnan, DEVS Scalable Modeling of a High performance pipelined DIF FFT core with Quantization, MS Thesis U. Arizona. Quantization in Digital Processing Transmit to next stage only when quantum exceeded

Music 300 Hz Hz Voice 300 Hz Hz Reduction (at q 0 = 0.06) = 52% Input 300 – 3000 Hz QFFT Module (Forward Transform) Output QFFT Module (Backward Transform) Analog LPF 3000 Hz cut-off QFFT System Reduction (at q 0 = 0.02) = 30.8% At q =.02 At q =.06

Event Set Basics

Event set refinement sequence size = 5 size = 9 size = 17 size = 33

Convergence of the Sum, Maximum variation, and form factor

Domain and Range Based Event Sets domain-based event set with equally spaced domain points separated by step. denote a range-based event set with equally spaced range values, separated by a quantum For an n-th degree polynomial we have. So that potential gains of the order of are possible.

Conclusions Activity T heory confirms that where there is heterogeneity of activity in space and time, DEVS will have significant advantage over conventional numerical methods This lead us to try reformulating the math foundations of continuity in discrete event terms

Implications sensing– most sensors are currently driven at high sampling rates to obviate missing critical events. Quantization-based approaches require less energy and produce less irrelevant data. data compression – even though data might be produced by fixed interval sampling, it can be quantized and communicated with less bandwidth by employing domain-based to range-based mapping. reduced communication in multi-stage computations, e.g., in digital filters and fuzzy logic is possible using quantized inter-stage coupling. spatial continuity–quantization of state variables saves computation and our theory provides a test for the smallest quantum size needed in the time domain; a similar approach can be taken in space to determine the smallest cell size needed, namely, when further resolution does not materially affect the observed spatial form factor. coherence detection in organizations – formations of large numbers of entities such as robotic collectives, ants, etc. can be judged for coherence and maintenance of coherence over time using this paper’s variation measures. education -- revamp teach of the calculus to dispense with its mysterious foundations (limits, continuity) that are too difficult to convey to learners.