stdevs – stochastic devs universidad de rosario | carleton university hpcs 2008 | springsim08...

STDEVS – STochastic DEVSSTDEVS – STochastic DEVS

Universidad de Rosario | Carleton University

HPCS 2008 | SpringSim08 April 15, Ottawa City.

Castro R., Kofman E., Wainer G.Castro R., Kofman E., Wainer G.

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STDEVSSTDEVS

A Formal Framework for A Formal Framework for

STSTochastic ochastic DEVSDEVS

Modeling and SimulationModeling and Simulation

Rodrigo Castro Rodrigo Castro **Ernesto Kofman Ernesto Kofman **Gabriel Wainer Gabriel Wainer ****

* Universidad Nacional de Rosario ** Carleton University* Universidad Nacional de Rosario ** Carleton University

System Dynamics and Signal Processing Lab. Advanced Real-Time Simulation Lab.System Dynamics and Signal Processing Lab. Advanced Real-Time Simulation Lab.

Argentina CanadaArgentina Canada http://www.fceia.unr.edu.ar/lsd/ http://www.sce.carleton.ca/faculty/wainer/ARS/





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AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS

Strategy. Formal Definition. Probability Spaces

Informal Idea. Formal Definition. STDEVS

The New Components. Theoretical & Practical Properties and

Implications. Example Conclusions. Next Steps.





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INTRODUCTION DEVS formalism

Developed as a general system theoretic based language.

Universal description of discrete event systems.

Stochastic models Play a fundamental role in discrete event system

theory. Any system involving uncertainties, unpredictable

human actions or system failures requires a non–deterministic treatment.

Widely adopted stochastic discrete event formalisms: Markov Chains, Queuing Networks, Stochastic Petri Nets...





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PROBLEM STATEMENT Even though most of the DEVS simulation tools have

incorporated the use of random functions...

DEVS has originally only been formally defined for deterministic systems.

Early works on mapping DEVS to stochastic systems are not completely general.

The DEVS formal framework has limited extent to

a wide family of (generalized) stochastic systems.

No previous general DEVS–based formalism for stochastic discrete event

systems.





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EARLY WORKS Previous efforts on mapping DEVS to stochastic systems

behavior have limited scope:

“DES models driven by pseudo-random sequences define DEVS models” (Aggarwal, U. of Michigan, 1975)

Problem: Not a methodology to describe DEVS stochastic models.

“Relationship established between random experiment outcomes and externally observed possible state trajectories of a DEVS simulation” (Melamed, U. of Michigan, 1976)

Problem: Limited to models described at the input/output level.

“Extended DEVS formalism taking into account internal stochastic behavior at the state transition level”(Joslyn, NASA Goddard Space Flight Center, 1996)

Problem: Limited to finite state sets models (not general sets).





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CONTRIBUTION OBJECTIVES

Provide an extension of DEVS that establishes a formal framework for modeling

and simulation of general stochastic discrete event systems.

DRIVERS

Rely on the deterministic DEVS Atomic Model definition as a starting point.

Keep the essence of the DEVS model structure, then derive from it a new stochastic model structure by

introducing the new probabilistic features needed replacing the way internal dynamics are

described.





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STDEVS STRATEGY (What can we do ?)

Define the new STochastic DEVS (STDEVS) Atomic Model structure, where internal dynamics incorporate probabilistic components, relying on the general Theory of Probability Spaces:

Think of DEVS state transitions as “Random Experiments”

Keep the general and arbitrary nature of all the original DEVS sets.

Respect the deterministic nature of the original DEVS deterministic functions.





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Keep general

Keep deterministic

Replace withProbability Spacescomponents

Think of DEVS state transitions as “Random

Experiments”

STDEVS STRATEGY

n) transitiostatelast since time Elapsed(

function advance Time :

functionOutput :

function n transitiostate External:

function n transitiostate Internal:

states possible ofSet S

uesoutput val ofSet

esinput valu ofSet

0

0ext

int

e

St

YS

SXS

SS

Y

X

a

) , ( intext

DEVS Atomic Model components:

ta

0 → e→ ta

X

Y

intext S

DEVS Atomic Model





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STDEVS FORMAL DEFINITION

),,,,,,( extint taSYXM D

),,,,,,,( extintextint taPPSYXM ST G,G

A STDEVS Model (MST) has the structure

A DEVS Model (MD) has the structure

= =Same

Functionality

Replaced

Components





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STDEVS FORMAL DEFINITION


A STDEVS Model (MST) has the structure

Components obtained from a Probability Space

construct.

Will have to answer: Given the present model state s Є S, and after the next

state transition (internal or external “random experiment”), ¿ What is the probability that the new future state s’ Є S

belongs to any given subset of S ?

Arbitrary(Finite, Infinite,

Continuous, Discrete, Hybrid...)

Can't assign probabilities to an individual s(will render always 0 for S continuous !)





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PROBABILITY SPACES INFORMAL IDEA

Arbitrary Subsets of Ssp

Sample Space : Ssp

All the possibleoutcomes (elements) of a random experiment.

We have to make sure that Ssp is measurable, given Ssp

is a totally arbitrary set.

Takes the roleof the DEVSState Space : S

Samples s Є Ssp

States s Є SAnalogous

Outcomes of a general random experiment.

Outcomes of a STDEVS state transition.





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PROBABILITY SPACES INFORMAL IDEA

F : a Sigma Field of Ssp

F is a collection of subsets (Not any collection, has special properties)

F : a member of F The F members of F

can be assigned probabilities, but not the single elements s Є F of them.

Sample Space : Ssp

P(F) : Probability Measures for

members F Є F .

Now, the structure (Ssp ,F, P ) is a Probability Space. It can fully describe random experiments on the arbitrary Sample

Space Ssp.





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PROBABILITY SPACES FORMAL DEFINITION

The pair (Ssp ,F ) is a Measurable Space if F is a Sigma Field of Ssp

1) Build a Measurable Space from the Sample Space.

F : a member of F

Now this measurable structure (Ssp ,F ) can be equipped with Probability Measures.

Recall that Ssp plays the role of the DEVS State Space.

F is a Sigma Field of Ssp if it satisfies :





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A Probability Measure P on a Measurable Space (Ssp ,F ) is an assignment of a real number P(F) to every member

F Є F, such that P obeys the following rules:

2) Build a Probability Space from the Measurable Space.

Now, the structure (Ssp ,F, P ) is a Probability Space. It can fully describe random experiments on the arbitrary Sample

Space Ssp.





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3) Make it more practical ! Sigma Fields F are theoretically essential, but not very useful in

practice.

Usually we want to pick our own collection of subsets G ЄG out of the Event Space Ssp that make some practical sense.

We are lucky:

Any arbitrarily chosen collection G ЄG of subsets always

generates a minimum Sigma Field F=M(G) . →We will

use G from now on.

The knowledge of P(G) for every G ЄG readily defines the

function P(F) for every F ЄF. → We will use P(G) from now on.

Finally: For every G ЄG , the function P(G) expresses the probability that the random experiment produces a sample s Є G as the experiment outcome.





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STDEVS THE NEW COMPONENTS


n. transitiointernal anafter )(G ' statenew a into evolve to

state in being system for the ),(y probabilita assigns that function A

]1,0[2:

. stateevery to2)( sets of collectiona assigns that function A

2:

int

int

int

int

ss

sGsP

SP

Sss

S

S

S

S

G

G

G

Let´s start with the internal transition dynamics:Power Set of S

Given a present state s , the collection Gint(s) contains all the

subsets of S that the future state s’ might belong to, with a known probability Pint(s,G).





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STDEVS THE NEW COMPONENTS


n. transitioexternal anafter ),(G ' statenew a into evolve to

),( situation at the system for the ),,( functiony Probabilit

]1,0[2:

),( tupleeach to2),( sets of collectiona assigning function A

2:

ext

0ext

ext

0ext

xqs

xqGxqP

XSP

xqxq

XS

S

S

S

G

G

G

Analogous reasoning for the external transition dynamics:

Given a present state s , an elapsed time e, and an input element x,

the collection Gext(q,x) contains all the subsets of S that the future state s’ might belong to, with a known probability Pint(s,G).

q=(s,e)





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STDEVS MAIN THEORETICAL PROPERTIES

We demonstrated the following properties of STDEVS, analogous to the DEVS main properties: (Formulas and demonstrations in our paper)

The STDEVS structure verifies Closure Under Coupling. We can couple STDEVS models in a hierarchical way,

encapsulating complex coupled models, and coupling them with other atomic ones.

The STDEVS structure is equipped with a Legitimacy Property.

We redefined the DEVS Legitimacy Property. Now, it expresses the probability of having an infinite

number of transitions in a finite interval of time.





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STDEVS THEORETICAL IMPLICATIONS

Now, with the new STDEVS formal framework we can:

Represent any stochastic system, no matter how complex the stochastic processes driving its dynamics might be.

This is true even if the system can not (or it is very difficult, expensive, etc.) be implemented in a practical simulator.

This allows to a strong theoretical probabilistic manipulation of the STDEVS structure.

(that evolves through a finite number of changes in a finite amount of time)





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STDEVS, DEVS and RND functions MAIN PRACTICAL PROPERTIES

We shall call DEVS-RND models to those DEVS models whose transition functions depend on random experiments through any random variable.

In practice, probability distributions are typically obtained by some computational manipulation of an Uniform U(0,1) random variable r obtained with a RND() pseudo-random sequence generator in most programming languages.

r can be an array of n Uniforms: r ~ U(0,1)n





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STDEVS, DEVS and RND functions MAIN PRACTICAL PROPERTIES

We demonstrated the following properties for STDEVS, of strong practical interest: (Formulas and demonstrations in our paper)

Theorem 1: A DEVS-RND model always define an equivalent STDEVS

model. Corollary 1:

A DEVS-RND model depending on n Uniforms: r ~ U(0,1) n in its transition functions always define an equivalent STDEVS model.

Corollary 2: A deterministic DEVS model always defines an equivalent STDEVS

model.(DEVS is a particular case of STDEVS)





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STDEVS, DEVS and RND functions PRACTICAL IMPLICATIONS

Now, with the new STDEVS formal framework (and its properties)we can:

Build and couple together any hierarchical system interconnecting DEVS and STDEVS models.(guaranteeing all the desired theoretical properties for doing it)

Model most of the practical situations of stochastic behavior in STDEVS without making use of probability spaces. (using the handier DEVS-RND equivalents).





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EXAMPLE Load Balancer – All components stochastics

A simple illustrative computational system : A generator offering a workload (tasks) to a two-servers cluster, with an adjustable balancer biased with a balancing factor.

LBM : Load Balancer Model LG : Load Generator CL : Cluster (Coupled) WB : Workload Balancer S1 : Server 1 S2 : Server 2

dr : Departure Rate

bf : Balancing Factor [0 to 1]

sti : Average Service Times

λi , λ’i : Average Traffic Rate

μi : Average Service Rate

Rates in [Tasks/second]

S1,S2 are M/M/1/1 queues (simplest). No buffer capacity: overflowing tasks are dropped. Service Times are Exponentially-distributed.

LG generates Poisson-distributed task workload. WB will distribute workload according a Uniform

distribution biased by a continuous factor bf .





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EXAMPLE LBM: Load Balancer Model

We want to model this system relying on the STDEVS formal framework. Then, execute the model in a DEVS simulator and verify it against

analytical results.

All the stochastic descriptions of this model are simple ones: can be readily modeled with a DEVS-RND approach.

We will show only LG with both STDEVS and DEVS-RND descriptions, for illustrative purposes.

For the rest of the components we will forget about the STDEVS description, and make use of Theorem1/Corollary1:

Concentrate only on the DEVS-RND description (much easier !)





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EXAMPLE LG: Load (tasks) Generator

Poisson discrete process (dr = λ , with dr = departure rate) ⇒

Exponentially-distributed inter-departure times σk between

task k and task k+1 : where a =dr . No inputs. Only internal state transitions. State s storages next

departure delay.STDEVS ←Equivalent→

Explicit stochastic-oriented definition

No need to be defined

Continue, real-valuedhalf-open intervalscollection

Practical Inverse Transformation method

DEVS-RND

Depends on r r ~U(0,1)

Does nothing

DETERMINISTIC





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EXAMPLE WB, S1, S2

For these components we follow an identical technique as with LG to get the DEVS-RND models:

Make the transition functions depend on a random variable r =U(0,1): and

Define them using the Inverse Transformation Method that uses r and yields the desired stochastic properties with an algorithmically programmable formula.

Note that: There is no stochastic description at the ta(s) or λ(s) functions.





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EXAMPLE WB : Workload Balancer (DEVS-RND components only)

Depends on r r ~U(0,1)

Depends on r (but doesn't “use”

it)

bf biases each port

random selection





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EXAMPLE Model verification Simulated (Marks) and Theoretical (Curves)

results

Effective Output Rate and Loss Probabilities vs. Balancing Factor bfErlang’s Formula for M/M/1/1:

LBM Model formulas:

Effective Output Rate:

Task Loss Probabilities:

Simulation is verified against theoretical expected results.





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CONCLUSIONS We presented STDEVS, a novel formalism for describing

stochastic discrete event systems.

STDEVS provides: A formal framework for modeling and simulation

of generalized stochastic discrete event systems. Shares the system theoretical approach of

DEVS. Makes use of Probability Spaces theory.

STDEVS allows for: A sound probabilistic theoretical treatment of

general stochastic DEVS. From its dynamics, not from its external

behavior. An easy practical way of implementation in

simulators. Not ‘very’different from what we were doing

so far !





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NEXT STEPS

We are developing STDEVS–based libraries for simulation tools PowerDEVS and CD++

Research area: Control Theory techniques applied to Admission Control in data networks.

QUESTIONS ?





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THANK YOU !THANK YOU !

More information:

[email protected]

[email protected]

[email protected]

stdevs – stochastic devs universidad de rosario | carleton university hpcs 2008 | springsim08...

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