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