Post on 08-Jan-2018
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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
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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
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
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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
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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:
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𝑋 1𝑋 2𝑋 3𝑋 3
𝑋 5𝑋 6
𝑆3
Large Time Heuristic Explained
on an Example
3. Compare
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𝑋 1𝑋 2𝑋 3𝑋 3
𝑋 5𝑋 6
𝑆3
This is Palm Calculus !
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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
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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)
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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
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
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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 ?
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Flow n=1 Flow n=2 Flow n=3
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Cyclist’s Paradox
On a round trip tour, there is more uphill than downhill
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The km clock vs the standard clock
speed for the kilometer
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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
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0% 0%
44%
56%