Palm CalculusPart 1

The Importance of the Viewpoint

JYLeBoudec

1May2015

1. Event versus Time AveragesConsider asimulation,stateStAssumesimulationhasastationary regime

Consider anEventClock:timesTn atwhich some specific changesofstateoccurEx:arrival ofjob;Ex.queuebecomes empty

Eventaverage statistic

Timeaverage statistic

2

Example: Gatekeeper; Average execution time

3

0 90 100 190 200 290 300

50001000

Real time t (ms)

job arrival

50001000

50001000

Execution time for a job thatarrives at t (ms)

Viewpoint 1: System Designer Viewpoint 2: Customer

Example: Gatekeeper; Average execution time

4

0 90 100 190 200 290 300

50001000

Real time t (ms)

job arrival

50001000

50001000

Execution time for a job thatarrives 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 aredifferentAmetricdefinitionshouldmentionthesamplingmethod(viewpoint)Differentsamplingmethodsmayprovidedifferentvalues:thisisthesamplingbias

PalmCalculus isasetofformulasforrelatingdifferentviewpoints

CanoftenbeobtainedbymeansoftheLargeTimeHeuristic

5

Large Time Heuristic Explainedon an Example

Wewant torelate andWe apply thelargetimeheuristic

1. Howdowe evaluate these metrics inasimulation?

…

where indexofnext greenorred arrow at orafter6

Large Time Heuristic Explainedon an Example

2. Breakoneintegral into pieces that matchthe ’s:

…

7

Large Time Heuristic Explainedon an Example

2. Breakoneintegral into pieces that matchthe ’s:1

…1

1⋯

1⋯

1 ⋯

1⋯

8

Large Time Heuristic Explainedon an Example

3. Compare

9

Large Time Heuristic Explainedon an Example

3. Compare

10

This is Palm Calculus !

11

cov ,

In which case do we expect to see 

A. Sn=90,10,90,10,90;Xn=5000,1000,5000,1000,5000B. Sn =90,10,90,10,90;Xn =1000,5000,1000,5000,1000C. BothD. NoneE. Idon’tknow

12

Sn = 90, 10, 90, 10, 90; Xn 

Sn = 90, 10, 90, 10, 90; Xn 

Both

None

I don’t know

72%

0%0%

28%

0%

Solution

IncaseA, and arepositivelycorrelated(whentheintervalislong,soistheprocessingtime),i.e. .BythePalmcalculusformula:

IncaseB,thecorrelationisnegative,therefore

AnswerA

13

The Large Time Heuristic

Formallycorrectifsimulationisstationary

Itisarobustmethod,i.e.independentofassumptionsondistributions(andonindependence)

14

Other «Clocks»

15

Flow 1 Flow 2

Flow 3

Distribution of flow sizesfor an arbitrary flowfor an arbitrary packet

Which curves are for the per‐packet 

viewpoint ?

A. AB. BC. ItdependsD. Idon’tknow

16

A B

It depends

I don’t know

71%

18%

0%

12%

Solution

AnswerATherearemorepacketsinthelargeflows.Somorepacketsexperiencealargeflowsize.

17

18

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

perflow ∑

perpacket ∑where when packet belongs toflow

20

Large «Time» Heuristic

1. Howdowe evaluate these metrics inasimulation?perflow ∑

perpacket ∑where when packet belongs toflow

2. Putthepackets side byside,sorted byflow

21

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⋯

1⋯

1

Size Size Size

Large «Time» Heuristic

3. Compare

22

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

1 1

1 1 1 1var

1var

Size Size Size

Large «Time» Heuristic for PDFs of flow sizes

Putthepackets side byside,sorted byflow

1. Howdowe evaluate these metrics inasimulation?

23

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

24

25

Cyclist’s Paradox

Onaroundtriptour,thereismoreuphillthandownhill

26

The km clock vs the standard clock

ℓ speedfortheℓ kilometer1

ℓℓ

meanof ℓ

∑ 1ℓℓ

harmonicmeanof ℓ meanof ℓ

27

BorduRail claims that only 5% of trains arrivals are late

BorduKonsum claims that 30% of train users suffer from late train 

arrivals

A. AtleastoneofthemliesB. Thenumberofpassengersina

latetrain passengersinaveragetrain

C. Thenumberofpassengersinalatetrain passengersinaveragetrain

D. Idon’tknow

28

At least o

ne of the

m lie

The nu

mbe

r of p

assengers i.

The nu

mbe

r of p

assengers in

I don

’t know

0% 0%

44%

56%

29

Solution