Title: Continuity and Change Activity Are Fundamentally Related In DEVS Simulation of Continuous Systems Be
1Continuity 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
2Outline
- 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
3Synopsis
- 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
4DEVS 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
5DEVS 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
6DEVS 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
7DEVS 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
8Activity 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.
9Mapping 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.
10PDE 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
11Activity a characteristic of continuous
functions
b
Threshold Crossings Activity/quantum
a
Activity b-a
Activity(0,T)
12DEVS 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
13Activity 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.
14DEVS Efficiency Advantage where Activity is
Heterogeneous in Time and Space
15Ratio DTSS/DEVS Transitions
16DTSS/DEVS Ratio for 1-D Diffusion
f is an increasing function of L
Alexander Muzys scalability results
17Muzys 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
18DEVS 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
19Quantization 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.
20Voice 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
21Event Set Basics
22Event set refinement sequence
23Convergence of the Sum ,Maximum variation, and
form factor
24Domain 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.
25Conclusions
- 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
26Implications
- 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.