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Palm Calculus Part 1 The Importance of the Viewpoint JY Le Boudec 1 May 2015
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1. Event versus Time Averages Consider a simulation, state S t Assume simulation has a stationary regime Consider an Event Clock: times T n at which some specific changes of state occur Ex: arrival of job; Ex. queue becomes empty Event average statistic Time average statistic 2
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Example: Gatekeeper; Average execution time 3 090100190200290300 5000 1000 Real time t (ms) job arrival 5000 1000 5000 1000 Execution time for a job that arrives at t (ms) Viewpoint 1: System Designer Viewpoint 2: Customer
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Example: Gatekeeper; Average execution time 4 090100190200290300 5000 1000 Real time t (ms) job arrival 5000 1000 5000 1000 Execution time for a job that arrives at t (ms) Viewpoint 1: System Designer Viewpoint 2: Customer
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Sampling Bias W s and W c 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
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Large Time Heuristic Explained on an Example 6
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This is Palm Calculus ! 11
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A.Sn = 90, 10, 90, 10, 90; Xn = 5000, 1000, 5000, 1000, 5000 B.S n = 90, 10, 90, 10, 90; X n = 1000, 5000, 1000, 5000, 1000 C.Both D.None E.I don’t know 12
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Solution 13
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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
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Other «Clocks» 15 Flow 1 Flow 2 Flow 3 Distribution of flow sizes for an arbitrary flow for an arbitrary packet
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Which curves are for the per-packet viewpoint ? A.A B.B C.It depends D.I don’t know 16
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Solution Answer A There are more packets in the large flows. So more packets experience a large flow size. 17
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18 Load Sensitive Routing of Long-Lived IP Flows Anees Shaikh, Jennifer Rexford and Kang G. Shin Proceedings of Sigcomm'99 ECDF, per flow viewpoint ECDF, per packet viewpoint
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19 Flow 1 Flow 2 Flow 3 Distribution of flow sizes for an arbitrary flow for an arbitrary packet
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Large «Time» Heuristic 20
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Large «Time» Heuristic 21 Flow n=1 Flow n=2 Flow n=3 p=1 p=2 p=3 p=4 p=5p=6 p=7 p=8 p=9
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Large «Time» Heuristic 3.Compare 22 Flow n=1 Flow n=2 Flow n=3 p=1 p=2 p=3 p=4 p=5p=6 p=7 p=8 p=9
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Large «Time» Heuristic for PDFs of flow sizes Put 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
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Cyclist’s Paradox On a round trip tour, there is more uphill than downhill 26
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The km clock vs the standard clock 27
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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 28
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29 Solution
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