Continuity and Change Activity Are Fundamentally Related In DEVS Simulation of Continuous Systems Be

1
Continuity and Change (Activity) Are
Fundamentally Related In DEVS Simulation of
Continuous SystemsBernard P. Zeigler

• Arizona Center for Integrative Modeling and
  Simulation(ACIMS)
•         University of ArizonaTucson, Arizona
  85721, USAzeigler_at_ece.arizona.eduwww.acims.arizo
  na.edu

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Outline

• Review DEVS Framework for MS
• Brief History of Activity Concept Development
• Summary of Recent Results
• Theory of Event Sets Basis for Activity Theory
• Conclusions and Implications

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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 sets most
  economical representation
• DEVS quantization can achieve this optimal
  representation

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DEVS Background

• DEVS Discrete Event System Specification
• Based on formal MS 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

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DEVS Hierarchical Modular Composition

• Atomic lowest level model, contains structural
  dynamics -- model level modularity

Coupled composed of one or more atomic and/or
coupled models
Hierarchical construction
coupling
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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

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DEVS Expressability
Coupled Models
Atomic Models
Partial Differential Equations
can be components in a coupled model
Ordinary Differential Equation Models
Processing/ Queuing/ Coordinating
Networks, Collaborations
Physical Space
Spiking Neuron Networks
Spiking Neuron Models
Processing Networks
Petri Net Models
n-Dim Cell Space
Discrete Time/ StateChart Models
Stochastic Models
Cellular Automata
Quantized Integrator Models
Self Organized Criticality Models
Fuzzy Logic Models
Reactive Agent Models
Multi Agent Systems
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Activity Theory unifies continuous and discrete
paradigms
DEVS can represents all decision making and
continuous dynamic elements
Heterogeneous activity in time and space
Quantization allows DEVS to naturally focus
computing resources on high activity regions
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.
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Mapping Ordinary Differential Equation Systems
into DEVS Quantized Integration
DEVS Integrator
DEVS instantaneous function
Theory of Modeling and Simulation, 2nd Edition,
Bernard P. Zeigler , Herbert Praehofer , Tag Gon
Kim , Academic Press, 2000.
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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
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Activity a characteristic of continuous
functions
b
Threshold Crossings Activity/quantum
a
Activity b-a
Activity(0,T)
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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
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Activity Calculations for 1-D Diffusion
This shows that the activity per cell in all the
three cases goes to a constant as N (number of
cells) tends to infinity.
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DEVS Efficiency Advantage where Activity is
Heterogeneous in Time and Space
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Ratio DTSS/DEVS Transitions
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DTSS/DEVS Ratio for 1-D Diffusion

f is an increasing function of L
Alexander Muzys scalability results
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Muzys Fire Front model
Accumulated Activity
Instantaneous Activity
Region Of Imminence
Peak Bars
S. R. Akerkar, Analysis and Visualization of
Time-varying data using the concept of 'Activity
Modeling', M.S. Thesis, University of
Arizona,2004
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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
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Quantization in Digital Processing
Transmit to next stage only when quantum exceeded
Harsha Gopalakrishnan, DEVS Scalable Modeling of
a High performance pipelined DIF FFT core with
Quantization, MS Thesis U. Arizona.
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Voice 300 Hz - 3000 Hz
At q .06
Music 300 Hz - 3000 Hz
At q .02
Reduction (at q0 0.06) 52
Reduction (at q0 0.02) 30.8
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Event Set Basics
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Event set refinement sequence
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Convergence of the Sum ,Maximum variation, and
form factor
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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.
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Conclusions

• Activity Theory 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

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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 continuityquantization 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 papers variation measures.
• education -- revamp teach of the calculus to
  dispense with its mysterious foundations (limits,
  continuity) that are too difficult to convey to
  learners.