Palm Calculus Made Easy - École Polytechnique Fédérale ...perfeval.epfl.ch/printMe/palm.pdf ·...

Palm CalculusMade Easy

The Importance of the ViewpointJYLeBoudec

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Contents

1. InformalIntroduction2. PalmCalculus

3. OtherPalmCalculusFormulae4. ApplicationtoRWP5. OtherExamples6. PerfectSimulation

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1. Event versus Time Averages

Considerasimulation,stateStAssumesimulationhasastationaryregime

ConsideranEventClock:timesTnatwhichsomespecificchangesofstateoccur

Ex:arrivalofjob;Ex.queuebecomesempty

Eventaveragestatistic

Timeaveragestatistic

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Example: Gatekeeper; Average execution time

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0 90 100 190 200 290 300

50001000

Real time t (ms)

job arrival

50001000

50001000

Execution time for a job that

arrives at t (ms)

Viewpoint 1: System Designer Viewpoint 2: Customer

Example: Gatekeeper; Average execution time

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0 90 100 190 200 290 300

50001000

Real time t (ms)

job arrival

50001000

50001000

Execution time for a job that

arrives at t (ms)

Viewpoint 1: System Designer Viewpoint 2: Customer

Two processes, with execution times 5000 and 1000

5000 10002 3000

Inspector arrives at a random timered processor is used with proba

90100 5000

10100 1000

4600

Sampling Bias

Ws andWc aredifferentAmetric definition should mentionthesampling method (viewpoint)Different sampling methods may provide different values:this is thesampling bias

PalmCalculus is asetofformulasforrelating different viewpoints

Canoften be obtained bymeans oftheLargeTimeHeuristic

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Large Time Heuristic Explainedon an Example

Wewant torelate andWe apply thelargetimeheuristic

1.Howdowe evaluate these metrics inasimulation?

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Wewant torelate andWe apply thelargetimeheuristic

1. Howdowe evaluate these metrics inasimulation?1

…1

where indexofnext greenorred arrow at orafter1

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2. Breakoneintegral into pieces that matchthe ’s:1

…1

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2. Breakoneintegral into pieces that matchthe ’s:1

…1

1⋯

1⋯

1 ⋯

1⋯

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3. Compare1

⋯

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3. Compare1

⋯1

⋯

cov , ̅ cov ,1

cov ,

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This is Palm Calculus !

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cov ,

Sn =90,10,90,10,90Xn =5000,1000,5000,1000,5000

Correlationis>0

Wc >Ws

Whendothetwoviewpointscoincide?

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The Large Time Heuristic

Formally correctifsimulation is stationary

Itis arobust method,i.e.independent ofassumptions ondistributions(andonindependence)

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Other «Clocks»

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Flow 1 Flow 2

Flow 3

Distribution of flow sizesfor an arbitrary flowfor an arbitrary packet

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Load Sensitive Routing of Long-Lived IP FlowsAnees Shaikh, Jennifer Rexford and Kang G. Shin

Proceedings of Sigcomm'99

ECDF, per flow viewpoint

ECDF, per packet viewpoint

Meanflowsize:perflowperpacket

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Flow 1 Flow 2

Flow 3

Distribution of flow sizesfor an arbitrary flowfor an arbitrary packet

Large «Time» Heuristic

1. Howdowe evaluate these metrics inasimulation?

2. Putthepackets side byside,sorted byflow

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Flow n=1 Flow n=2 Flow n=3

p=1 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=9


1. Howdowe evaluate these metrics inasimulation?perflow ∑

perpacket ∑where when packet belongs toflow

2. Putthepackets side byside,sorted byflow

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p=1 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=9

1⋯

1⋯

1


3. Compare

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p=1 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=9


3. Compare

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p=1 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=9

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1 1

1 1 1 1var

1var

Large «Time» Heuristic for PDFs of flow sizes

Putthepackets side byside,sorted byflow

1. Howdowe evaluate these metrics inasimulation?

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Cyclist’s Paradox

Onaroundtriptour,there is moreuphillsthan downhills

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The km clock vs the standard clock

ℓ speedfortheℓ kilometer1

ℓℓ

meanof ℓ

∑ 1ℓℓ

harmonicmeanof ℓ meanof ℓ

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2. Palm Calculus : Framework

Astationaryprocess(simulation)withstateSt.SomequantityXt measuredattimet.Assumethat

(St;Xt)isjointlystationary

I.e.,St isinastationaryregimeandXt dependsonthepast,presentandfuturestateofthesimulationinawaythatisinvariantbyshiftoftimeorigin.Examples

St =currentpositionofmobile,speed,andnextwaypointJointlystationarywithSt:Xt =currentspeedattimet;Xt =timetoberununtilnextwaypointNotjointlystationarywithSt:Xt =timeatwhichlastwaypointoccurred

Stationary Point Process

Considersomeselectedtransitions ofthesimulation,occurringattimesTn.

Example:Tn =timeofnth tripend

Tn isacalledastationarypointprocessassociatedtoStStationarybecauseSt isstationaryJointlystationarywithSt

Time0isthearbitrary pointintime

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Palm ExpectationAssume:Xt,St arejointlystationary,Tn isastationarypointprocessassociatedwithStDefinition :thePalmExpectation is

Et(Xt)=E(Xt |aselectedtransitionoccurredattimet)

Bystationarity:

Et(Xt)=E0(X0)

Example:Tn =timeofnth tripend,Xt =instantspeedattimetEt(Xt)=E0(X0)=averagespeedobservedatawaypoint

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E(Xt)=E(X0)expressesthetimeaverageviewpoint.Et(Xt)=E0(X0)expressestheeventaverageviewpoint.Exampleforrandomwaypoint:

Tn =timeofnth tripend,Xt =instantspeedattimetEt(Xt)=E0(X0)=averagespeedobservedattripendE(Xt)=E(X0)=averagespeedobservedatanarbitrarypointintime

Xn

Xn+1

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Formal DefinitionIndiscretetime,wehaveanelementaryconditionalprobability

Incontinuoustime,thedefinitionisalittlemoresophisticated

usesRadonNikodymderivative– seelecturenotefordetailsAlsosee[BaccelliBremaud87]foraformaltreatment

Palmprobability isdefinedsimilarly

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Ergodic InterpretationAssumesimulationisstationary+ergodic,i.e.samplepathaveragesconvergetoexpectations;thenwecanestimatetimeandeventaveragesby:

Intermsofprobabilities:

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Intensity of a Stationary Point ProcessIntensity ofselectedtransitions: :=expectednumberoftransitionspertimeunit

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Two Palm Calculus FormulaeIntensityFormula:

wherebyconventionT0 ≤ 0<T1

InversionFormula

Theproofs aresimpleindiscretetime– seelecturenotes

3. Other Palm Calculus Formulae

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Joe’ sWaitingTime

var

meanwaitingtime var

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0.5 mean time between busessystem’s viewpoint

penalty due to variability

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Feller’s Paradox

We encountered Feller’s Paradox Already

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For a Poisson process, what is the mean length of an interval ?

Rate Conservation Law

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Campbell’s Formula

Shotnoisemodel:customern addsaloadh(t‐Tn,Zn)whereZn issomeattributeandTn isarrivaltime

Example:TCPflow:L=λVwithL=bitspersecond,V=totalbitsperflowandλ=flowspersec

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t

Total load

T1 T2 T3

Little’s Formula

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t

Total load

T1 T2 T3

4. RWP and Freezing Simulations

ModulatorModel:

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Is the previous simulation stationary ?

Seemslikeasuperfluousquestion,howeverthereisadifferenceinviewpointbetweentheepochn andtime

LetSn bethelengthofthenth epochIfthereisastationaryregime,thenbytheinversionformula

sothemeanofSn mustbefinite

Thisisinfactsufficient(andnecessary)

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Application to RWP

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Time Average Speed, Averaged over nindependent mobiles

BluelineisonesampleRedlineisestimateofE(V(t))

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A Random waypoint model that has no stationary regime !

Assumethatattriptransitions,nodespeedissampleduniformlyon[vmin,vmax]Takevmin =0 andvmax >0

Meantripduration=(meantripdistance)

Meantripdurationisinfinite!

Wasoftenusedinpractice

Speeddecay:“consideredharmful”[YLN03]

max

0max

1 v

vdv

v

What happens when the model does not have a stationary regime ?

Thesimulationbecomesold

Stationary Distribution of Speed(For model with stationary regime)

Closed Form AssumeastationaryregimeexistsandsimulationisrunlongenoughApplyinversionformula andobtaindistributionofinstantaneousspeedV(t)

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Removing Transient MattersA. Inthemobilecase,thenodesaremoreoftentowardsthecenter,distancebetweennodesisshorter,performanceisbetterThecomparisonisflawed.Shoulduseforstaticcasethesamedistributionofnodelocationasrandomwaypoint.Istheresuchadistributiontocompareagainst ?

Random waypoint

Static

A(true)example:Compareimpactofmobilityonaprotocol:

Experimenterplacesnodesuniformlyforstaticcase,accordingtorandomwaypointformobilecaseFindsthatstaticisbetter

Q. Findthebug!

A Fair Comparison

Werevisitthecomparisonbysamplingthestaticcasefromthestationaryregimeoftherandomwaypoint

Random waypoint

Static, from uniform

Static, same node location as RWP

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Isitpossibletohavethetimedistributionofspeeduniformlydistributedin[0;vmax]?

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5. PASTAThereisanimportantcasewhereEventaverage=Timeaverage“PoissonArrivalsSeeTimeAverages”

Moreexactly,shouldbe:PoissonArrivalsindependentofsimulationstateSeeTimeAverages

6. Perfect Simulation

AnalternativetoremovingtransientsPossiblewheninversionformulaistractableExample:randomwaypoint

Sameappliestoalargeclassofmobilitymodels

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Removing Transients May Take Long

Ifmodelisstableandinitialstateisdrawnfromdistributionotherthantime‐stationarydistribution

Thedistributionofnodestateconvergestothetime‐stationarydistribution

Naïve:so,let’ssimplytruncateaninitialsimulationduration

Theproblemisthatinitialtransiencecanlastverylong

Example[spacegraph]:nodespeed=1.25m/sboundingarea=1kmx1km

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Perfect simulation is highly desirable (2)

Distributionofpath:

Time = 100s

Time = 50s

Time = 300s

Time = 500s

Time = 1000s

Time = 2000s

Solution: Perfect Simulation

Def:asimulationthatstartswithstationarydistributionUsuallydifficultexceptforspecificmodelsPossibleifweknowthestationarydistribution

Sample Prev and Next waypoints from their joint stationary distributionSample M uniformly on segment [Prev,Next]Sample speed V from stationary distribution

Stationary Distrib of Prev and Next

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Stationary Distribution of Location Is also Obtained By Inversion Formula

No Speed Decay

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Perfect Simulation Algorithm

SampleaspeedV(t)fromthetimestationarydistributionHow?A:inversionofcdf

SamplePrev(t),Next(t)How?

SampleM(t)

ConclusionsAmetricshouldspecifythesamplingmethodDifferentsamplingmethodsmaygiveverydifferentvaluesPalmcalculuscontainsafewimportantformulas

Whichones?

Freezingsimulationsareapatterntobeawareof

Palm Calculus Made Easy - École Polytechnique Fédérale ...perfeval.epfl.ch/printMe/palm.pdf ·...

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