Palm CalculusPart 1

The Importance of the Viewpoint

JY Le Boudec

1May 2015

1. Event versus Time AveragesConsider a simulation, state St

Assume simulation has a stationary regime

Consider an Event Clock: times Tn at which some specific changes of state occur

Ex: arrival of job; Ex. queue becomes empty

Event average statistic

Time average 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 that arrives at t (ms)

Viewpoint 1: System Designer Viewpoint 2: Customer

Sampling Bias

Ws and Wc are different

A metric definition should mention the sampling method (viewpoint)Different sampling methods may provide different values: this is the sampling bias

Palm Calculus is a set of formulas for relating different viewpoints

Can often be obtained by means of the Large Time Heuristic

5

Large Time Heuristic Explained

on an Example

We want to relate and We apply the large time heuristic

1. How do we evaluate these metrics in a simulation ?

where index of next green or red arrow at or after

6

𝑋 1𝑋 2𝑋 3𝑋 3

𝑋 5𝑋 6

𝑆3

Large Time Heuristic Explained

on an Example

2. Break one integral into pieces that match the ’s:

7

𝑋 1𝑋 2𝑋 3𝑋 3

𝑋 5𝑋 6

𝑆3

Large Time Heuristic Explained

on an Example

3. Compare

9

𝑋 1𝑋 2𝑋 3𝑋 3

𝑋 5𝑋 6

𝑆3

This is Palm Calculus !

11

𝑊 𝑐=𝜆cov (𝑆 ,𝑋 )+𝑊 𝑠

𝑺𝒏

𝑿𝒏

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, 1000

C. BothD. NoneE. I don’t know

12

72%

0%0%

28%

0%

The Large Time Heuristic

Formally correct if simulation is stationary

It is a robust method, i.e. independent of assumptions on distributions (and on independence)

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. It dependsD. I don’t know

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71%

18%

0%

12%

Mean flow size:per flow per packet

19

Flow 1 Flow 2

Flow 3

Distribution of flow sizesfor an arbitrary flowfor an arbitrary packet

Large «Time» Heuristic1. How do we evaluate these metrics in a simulation ?per flow per packet where when packet belongs to flow

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Large «Time» Heuristic1. How do we evaluate these metrics in a simulation ?

per flow per packet where when packet belongs to flow

2. Put the packets side by side, sorted by flow

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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

𝑆𝑃=1𝑃 (𝑆1+𝑆1+𝑆2+𝑆2+𝑆3+𝑆3+𝑆3+𝑆3+𝑆3+…)

Size Size Size

Large «Time» Heuristic

3. Compare

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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

Size Size Size

Large «Time» Heuristic for PDFs of flow sizesPut the packets side by side, sorted by flow

1. How do we evaluate these metrics in a simulation ?

23

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

24

25

Cyclist’s Paradox

On a round trip tour, there is more uphill than downhill

26

The km clock vs the standard clock

speed for the kilometer

27

BorduRail claims that only 5% of trains arrivals are late

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

arrivals

A. At least one of them liesB. The number of passengers in a

late train passengers in average train

C. The number of passengers in a late train passengers in average train

D. I don’t know

28

0% 0%

44%

56%