By M. Almone, G. Conte Presented by Yinglei Song

11

A class of Generalized A class of Generalized Stochastic Petri Nets for the Stochastic Petri Nets for the performance Evaluation of performance Evaluation of

Mulitprocessor SystemsMulitprocessor SystemsBy M. Almone, G. ConteBy M. Almone, G. Conte

Presented by Yinglei SongPresented by Yinglei Song

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OutlineOutline

BackgroundBackground Modeling concurrent systems with Petri NetsModeling concurrent systems with Petri Nets Stochastic Petri Nets (SPN)Stochastic Petri Nets (SPN) Markov Chains (MC)Markov Chains (MC) Generalized Stochastic Petri Nets (GSPN)Generalized Stochastic Petri Nets (GSPN)

The steady state distribution in GSPNThe steady state distribution in GSPN Computing the steady state distribution Computing the steady state distribution

more efficiently.more efficiently. Examples.Examples. Numerical results.Numerical results.

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Petri NetsPetri Nets

A model that consists ofA model that consists of P, a set of placesP, a set of places T, a set of transitionsT, a set of transitions A, a set of directed arcsA, a set of directed arcs M, a vector that stands for the number of M, a vector that stands for the number of

tokens in each place. (referred to as a tokens in each place. (referred to as a marking). marking).

The The reachability setreachability set of a marking. of a marking. k-boundedk-bounded Petri Nets. Petri Nets.

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An ExampleAn Example

A Petri Net for modeling bisexual A Petri Net for modeling bisexual populationpopulation

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Stochastic Petri NetsStochastic Petri Nets

The modeling ability of a PN is limitedThe modeling ability of a PN is limited transition occurs with different transition occurs with different

probabilities in real systems.probabilities in real systems. New parameter sets are needed for New parameter sets are needed for

modeling different transition rates or modeling different transition rates or probabilities.probabilities.

A new parameter set R is thus added to A new parameter set R is thus added to the definition of Petri Nets.the definition of Petri Nets.

A Stochastic Petri Net (SPN) is defined as a A Stochastic Petri Net (SPN) is defined as a five-tuple (P, T, A, M, R).five-tuple (P, T, A, M, R).

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A Markov ChainA Markov Chain

A Markov Model (MM) is comprised ofA Markov Model (MM) is comprised of A Markov Chain (MC) is a sequence A Markov Chain (MC) is a sequence

states generated following states generated following transitions in an MM.transitions in an MM. S: a set of statesS: a set of states T: a set of transitionsT: a set of transitions P: a set of probabilities associated with P: a set of probabilities associated with

each transitioneach transition

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SPN and MCSPN and MC

It can be proved that SPN is equivalent to a It can be proved that SPN is equivalent to a MCMC

The set of states in MC is equivalent to the The set of states in MC is equivalent to the set of all possible markings in the set of all possible markings in the corresponding SPNcorresponding SPN

The transition probabilities in the MC can be The transition probabilities in the MC can be computed with transition rates in the computed with transition rates in the corresponding SPNcorresponding SPN

The transition probability matrix can thus be The transition probability matrix can thus be determined from the transition rates in SPNdetermined from the transition rates in SPN

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SPN and MCSPN and MC

The sojourn time in each marking is an The sojourn time in each marking is an exponentially distributed random exponentially distributed random variable with average:variable with average:

1][

Hi

ia rT

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SPN and MC SPN and MC

The transition probabilities in the The transition probabilities in the corresponding MC is determined by:corresponding MC is determined by:

iHmm

sij r

rP

1010

The transition matrix of MCThe transition matrix of MC

The transition matrix of a MC is defined The transition matrix of a MC is defined as:as:

nnnn

ij

n

n

PPP

P

PPP

PPP

...

.........

...

...

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22221

11211

1111

The dynamics of MCThe dynamics of MC

The dynamical equation of a MC can be The dynamical equation of a MC can be written as:written as:

Ptptp

PtptpSi

ijij

)()1(

)()1(

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The steady state distribution of The steady state distribution of MCMC

The steady state distribution of the MC is The steady state distribution of the MC is a fixed point of the dynamical equation:a fixed point of the dynamical equation:

0

0)(

Qp

PIp

Ppp

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Generalized Stochastic Petri Generalized Stochastic Petri NetsNets

Neither PN nor SPN is able to perfectly Neither PN nor SPN is able to perfectly model all the real systems.model all the real systems.

Transition rates in real systems may span Transition rates in real systems may span a wide range including a few orders of a wide range including a few orders of magnitude.magnitude.

GSPN is a model that allows both GSPN is a model that allows both timedtimed transitions and transitions and immediateimmediate transitions. transitions.

GSPN is able to model real systems with GSPN is able to model real systems with an appropriate granularity of time.an appropriate granularity of time.

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An example of GSPNAn example of GSPN

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Switching probabilities of the Switching probabilities of the GSPNGSPN

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The reachability set of the The reachability set of the GSPNGSPN

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Timed and intermediate Timed and intermediate transitions may be correlatedtransitions may be correlated

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Time vs. State in GSPNTime vs. State in GSPN

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OutlineOutline




2020

GSPN steady state GSPN steady state distributiondistribution

Two types of markings (states) exist Two types of markings (states) exist in a GSPN:in a GSPN: Tangible statesTangible states are markings that are are markings that are

associated with only timed transitions.associated with only timed transitions. Vanishing statesVanishing states are markings that are are markings that are

associated with at least on immediate associated with at least on immediate transition.transition.

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AssumptionsAssumptions

The reachability set of GSPN is finite.The reachability set of GSPN is finite. Transition rates remain constant and Transition rates remain constant and

do not evolve with time.do not evolve with time. The initial marking is reachable with The initial marking is reachable with

a nonzero probability from any a nonzero probability from any marking in the reachability set. marking in the reachability set.

No marking exists that “absorbs” the No marking exists that “absorbs” the process.process.

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NotationsNotations

Following notations are used to derive Following notations are used to derive the steady state distribution:the steady state distribution:

S: the set of states in the SPP.S: the set of states in the SPP. T: the set of tangible states in S.T: the set of tangible states in S. V: the set of vanishing states in S.V: the set of vanishing states in S.

VTS

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The steady state The steady state distribution distribution

The steady state distribution must The steady state distribution must satisfy:satisfy:

0)( UIY

YUY

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OutlineOutline




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Efficient computation of steady Efficient computation of steady state distributionstate distribution

The inverse of the transition matrix The inverse of the transition matrix needs to be computed in time needs to be computed in time

The dimensionality of the transition The dimensionality of the transition matrix can become very big.matrix can become very big.

The computation of the inverse of the The computation of the inverse of the transition matrix can become very transition matrix can become very inefficient.inefficient.

More efficient approaches are needed for More efficient approaches are needed for computing the steady state distribution.computing the steady state distribution.

)||(|| 3UO

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The approachThe approach

The dimensionality of the transition The dimensionality of the transition matrix can be reduced by observing matrix can be reduced by observing the figure:the figure:

t1i r

j

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The effective transition The effective transition matrixmatrix

If we only consider the tangible states, If we only consider the tangible states, the transition matrix can be computed the transition matrix can be computed with:with:

Vr

irijij jrefu )Pr(

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OutlineOutline




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An exampleAn example

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A GSPN for the systemA GSPN for the system

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A simplified modelA simplified model

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Another simplified modelAnother simplified model

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A third simplified exampleA third simplified example

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Interesting questionsInteresting questions

Can we further simplify the GSPN Can we further simplify the GSPN used such that all resources can be used such that all resources can be abstracted as tokens?abstracted as tokens?

If the answer is “no”, what actually If the answer is “no”, what actually determines that, the topology of the determines that, the topology of the system? system?

Is a mathematical proof possible?Is a mathematical proof possible?

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OutlineOutline




3636

Numerical resultsNumerical results

The upper bound (M is infinitely large)The upper bound (M is infinitely large) The lower bound (M is equal to b)The lower bound (M is equal to b) To understand the dependence of the To understand the dependence of the

throughput on M, further investigation throughput on M, further investigation is needed.is needed.

GSPN provides a convenient way for GSPN provides a convenient way for this purpose. this purpose.

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ResultsResults

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ConclusionConclusion

Extended from SPN and PN, the GSPN model Extended from SPN and PN, the GSPN model can provide a finer description of the real can provide a finer description of the real system.system.

The GSPN is mathematically equivalent to a The GSPN is mathematically equivalent to a MC.MC.

The steady state distribution of GSPN can be The steady state distribution of GSPN can be efficiently computed.efficiently computed.

Real system can be analyzed to deeper level Real system can be analyzed to deeper level if GSPN is adopted. Exact solutions can be if GSPN is adopted. Exact solutions can be obtained for some complicated situations.obtained for some complicated situations.

By M. Almone, G. Conte Presented by Yinglei Song

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