SMS - Stochastic Petri Nets

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Stochastic ModelingStochastic Modeling

Stochastic Petri Net Models

Prof. Dr. P. Mitrevski

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Assessment of performance, reliability and

availability is a key step in the design,analysis and tuning of computer systems Example:

We have a multiprocessor system and we want to

be sure it provides enough processing power If we add a processor, how much better willperformance get?

Could additional overhead make the performance

IntroductionIntroduction

Could we get a performance improvement just bychanging the scheduling of jobs?

How would adding a processor affect the reliabilityof the system(?) Would this make the system godown more often?

If so, would an increase in performance outweighthe decrease in reliability?

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Reliability

Ability of a component or system to function correctly overa specified period of time

Availability Probability that system is working at the instant t,

regardless of the number of times it may have failed and

been repaired in the interval (0, t) Performance

Ability of the system to carry out the work it is subject to,assuming that the system (or its components) does not fail

IntroductionIntroduction

r r y A new modeling paradigm that can give combinedreliability and performance measures Systems may be able to survive the failure of one or more

of their active components and continue to provide serviceat a reduced level (gracefully degrading systems). Suchsystems cannot adequately be modeled through separatereliability/availability or performance models

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Applicable to a wide range of modeling problems Provide interesting reliability, availability,

performance and performability models Difficulties:

State space can grow much faster than the number of

components in the system being modeled, making itdifficult to specify a model correctly Far removed in shape and general feel from the system

being modeled S stem desi ners ma find it hard to translate their

Markov ModelsMarkov Models

problems into Markov models These difficulties can be overcome by using a

model with a form that is more concise andcloser to a designers intuition about what amodel should look like One such model which is quite popular is the stochastic

Petri net(!)

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A Petri net consists ofplaces, transitions,arcs and tokens

Tokens reside in places and move from oneplace to another along the arcs through the

transitions A marking is the number of tokens in each

place

Introduction to Petri net modelsIntroduction to Petri net models

untimed the interest is in studyingsequences of transition firings

A very common extension is the stochastic

Petri net (SPN), in which the transitions aretimed

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SPN model for M/M/1/k queueSPN model for M/M/1/k queue

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Ifm

is the steady-state probability for state (marking) m ofthe underlying Markov chain and n

mis the number of tokens

in queue for marking m, the average number of tokens inqueue is: mm n

ReachabilityReachability graph for SPN modelgraph for SPN modelof M/M/1/k queueof M/M/1/k queue

The steady-state probability that the queue is full is thesteady-state probability that the placejobsource is empty.We can find this by adding together the steady-stateprobabilities for all of the markings that have no tokens injobsource. There is just one such marking, (0,5).

m

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Bag (multiset) a set where members are

allowed to appear multiple times Petri net marking a bag whose elements are

place names Petri net 5-tuple (P,T,I(),O(),m0) where:

P is a set ofplaces () T is a set oftransitions () I() is the input function maps transitions to bags of

places

Petri net model definitions (1)Petri net model definitions (1)

places m0 is the initial marking of the net

A transition tis enabled by a marking m if andonly if I(t) is a subbag ofm Any transition tenabled by marking m can fire When it fires, I(t) is subtracted from the marking m and

O(t) is added to m, resulting in a new marking

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Petri net examplePetri net example10


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ReachabilityReachability graph for Petri netgraph for Petri netexampleexample

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To associate time with transitions, weconsider that a transition is enabled as soonas all required tokens are present in therequired places, but the transition does notfire right away

A transitions firing time, which is specified bya distribution function, is measured from theinstant the transition is enabled to the instant


A stochastic Petri net (SPN) is a Petri net

with timed transitions, where the firing timedistributions are assumed to be exponential

Conflicting timed transitions are viewed as

competing processes (race policy)

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A generalized stochastic Petri net(GSPN) is a PN where both immediate() and timed ()transitions are allowed

A graphical representation of a GSPN

(generally) shows immediate transitions aslines and timed transitions as thick bars orrectangular boxes


are enabled in a marking, only the immediatetransitions can fire

Conflicting immediate transitions areassigned relative firing probability values

(preselection policy)

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GSPN model for simultaneousGSPN model for simultaneous possesionpossesion of a resource:of a resource:I/O subsystem with two disks but only one I/O channelI/O subsystem with two disks but only one I/O channel

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Some of the extensions are a matterof convenience, especially regardinggraphical representation, and some

are true extensions that addmodeling power: Arc Multi licit

Petri net extensionsPetri net extensions

Inhibitor Arcs Priorities

Guards

Marking-dependent arc multiplicity Marking-dependent firing rates

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Example ofExample ofarc multiplicitiesarc multiplicities16


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An inhibitor arc from placep to transition tdisables t in any marking wherep is not empty

SPN model withSPN model with inhibitor arcsinhibitor arcs forforM/M/1/k queueM/M/1/k queue

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GSPN model for queue with twoGSPN model for queue with twojob classesjob classes

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GSPN model withGSPN model with prioritiespriorities forforqueue with four job classesqueue with four job classes

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A guard is a marking-dependent predicate that provides anadditional enabling criterion for each transition: the transition is

enabled only if the guard is satisfied

SU=server up; SD=server down; BU=buffer up; BD=buffer down

nm(Q) = number of jobs in the system

A GSPN usingA GSPN using guardsguards20


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EXAMPLE: a PN model of a communication protocol where

the arrival of a certain kind of packet causes all outgoingpackets to be flushed

MarkingMarking--dependent arc multiplicity:dependent arc multiplicity:Flushing a placeFlushing a place

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The system is operational if k-out-of-n components are working

If the failure rate for each component is , then when there arejfunctional components, the rate of occurrence of a componentfailure isj, depicted by appending the character # to the rate

MarkingMarking--dependent firing rates:dependent firing rates:GSPN model for kGSPN model for k--outout--ofof--n systemn system

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Most commonly used methods: Generation and analysis of a stochastic process

associated with the PN Let M(t) be the SPN marking at time t M(t) is a continuous time stochastic process called the

marking process or stochastic process underlyingthe SPN

The state space of the marking process is thereachability set of the SPN

The analysis of the SPN for transient or steady-state

SPN andSPN and GSPN analysis (1)GSPN analysis (1)

Continuous Time Markov Chain (CTMC) Discrete-event simulation of the PN

Avoids the generation of the reachability graph and canhelp with very large PNs

In addition, it becomes possible to handle non-Markovian nets

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Analyzing GSPNs is a bit morecomplicated because of the presence ofimmediate transitions

A marking in which at least one

immediate transition is enabled is called avanishing (, )marking; otherwise it is a tangible

SPN and GSPN analysis (2)SPN and GSPN analysis (2)

mar ng The resulting graph is called an extended

reachability graph (ERG) and can betransformed into a reduced reachability

graph that corresponds to a CTMC

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A GSPNA GSPN

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ERG and CTMC of the GSPNERG and CTMC of the GSPN

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A GSPNA GSPN

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ERGERG

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Pvv, Pvt = matrices of transition probabilities

( )

Ptt, Ptv = matrices of transition rates( )

(t=tangible; v=vanishing)

Constructing the generator matrixConstructing the generator matrixfor the underlying CTMC (1)for the underlying CTMC (1)

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The matrix describing the rates of transition from each tangible

marking to other tangible markings is: U = Ptt + Ptv (I Pvv)-1 Pvt

Constructing the generator matrixConstructing the generator matrixfor the underlying CTMC (2)for the underlying CTMC (2)

The entries in the generator matrix () for the underlying CTMC are:

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A system with two processors and three memory modules

Processors and memory modules are subject to failure and can be repaired There is only one repair facility

The system is unavailable ifppup is empty orpmup is empty

A GSPN availability modelA GSPN availability model

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A single server handles

more than one sourceof jobs (e.g. two) The system contains

stations which theserver polls one at atime

If the server finds a jobwaiting for a service ata station, the job isserved

A GSPN model for a single serviceA GSPN model for a single servicepolling systempolling system

I t ere are no jo s

waiting at a station,the server goes on topoll the next station

Mi= maximum numberof jobs for Station i

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PE = processingelement

P = processor

LM = localmemory

CM = common(shared) memory

LB = local bus GB = global bus

A singleA single--bus multiprocessorbus multiprocessorsystemsystem

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Three stages of a hypo-exponentially distributed service time

Processing efficiency = average fraction of active processors inthe system

A processor is active if it is executing instructions in its private memory

GSPN for singleGSPN for single--busbusmultiprocessor systemmultiprocessor system

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A failed node is successfully bypassedwith some probability 1-F

The network as a whole fails if any one ofthe nodes or links fails

A ring networkA ring network

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GSPN model of ring networkGSPN model of ring network

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GSPN model for queue with serverGSPN model for queue with serverfailure and repairfailure and repair

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Voice and packets arriveaccording to a Poissonprocess

The transmitter contains a

buffer to store a maximumofkdata packets A voice packet can enter

the channel only if there

ISDN channel with Poisson arrivalISDN channel with Poisson arrivalprocessprocess

be transmitted

If a voice transmission is inprogress, data packetscannot be serviced, but arebuffered

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Queuing network withQueuing network withsimultaneous resource possessionsimultaneous resource possession

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GSPN model of queuing networkGSPN model of queuing networkwith resource constraintswith resource constraints

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In a great number of real situations,

deterministic or generally (non-exponentially)distributed event times occur Timeouts in a protocol

Service times in a manufacturing systemperforming the same task on each part

Memory access or instruction execution in a low-

NonNon--MarkovianMarkovian SPN modelSPN modelextensions (1)extensions (1)

NOTE: they all have durations which areconstant or have a very low variance(!)

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Extended Stochastic Petri nets (ESPN)

General firing time distributions are allowed Under suitable hypotheses, the underlying

stochastic process is a semi-Markov process andanalytical solution methods exist

Deterministic and Stochastic Petri nets(DSPN) Allow the definition of immediate, exponential and

deterministic transitions


e s oc as c process un er y ng a s a

Markov Regenerative Process (MRGP) With the restriction that at most one deterministic

transition is enabled together with zero or moreexponentially distributed timed transitions, asteady-state and a transient solution method exist

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Markov Regenerative Stochastic Petri nets(MRSPN) A generalization of DSPNs that allows immediate,

exponential and generally distributed transitions The underlying stochastic process is still an MRGP There exist equations for the steady-state and transient

behavior for the case where every marking has at mostone generally distributed timed transition

Concurrent Generalized Petri nets (CGPN) Allow simultaneous enabling of any number of


distributed timed transitions, provided that the latter areall enabled at the same instant

The stochastic process underlying a CGPN is shown to bean MGRP

Equations for the steady-state as well transient analysisare provided

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Fluid Stochastic Petri nets (FSPN)

One or more places can hold fluid rather thandiscrete tokens

The discrete and continuous portions mayaffect each other

Able to both control the fluid flow, and have thediscrete control decisions be affected by observedfluid flow


The transient and the steady-state behavior ofFSPNs is described by a coupled system ofpartial differential equations

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SMS - Stochastic Petri Nets