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Palm Calculus Made Easy The Importance of the Viewpoint

Palm Calculus Made Easy The Importance of the Viewpoint. JY Le Boudec. Contents. Informal Introduction Palm Calculus Other Palm Calculus Formulae Application to RWP Other Examples Perfect Simulation. 1. Event versus Time Averages. Consider a simulation, state S t

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Palm Calculus Made Easy The Importance of the Viewpoint

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  1. Palm CalculusMade EasyThe Importance of the Viewpoint JY Le Boudec 1

  2. Contents • Informal Introduction • Palm Calculus • Other Palm Calculus Formulae • Application to RWP • Other Examples • Perfect Simulation 2

  3. 1. Event versus Time Averages • Consider 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 3

  4. Example: Gatekeeper; Averageexecution time job arrival 0 90 100 190 200 290 300 Real time t (ms) Execution time for a job that arrives at t (ms) 5000 5000 5000 1000 1000 1000 Viewpoint 2: Customer Viewpoint 1: System Designer 4

  5. Example: Gatekeeper; Averageexecution time job arrival 0 90 100 190 200 290 300 Real time t (ms) Execution time for a job that arrives at t (ms) 5000 5000 5000 1000 1000 1000 Viewpoint 2: Customer Viewpoint 1: System Designer Two processes, withexecution times 5000 and 1000 Inspector arrives at a random timered processor isusedwithproba 5

  6. Sampling Bias • Ws and Wc are different • A metricdefinitionshould mention the samplingmethod (viewpoint) • Differentsamplingmethodsmayprovidedifferent values: thisis the samplingbias • Palm Calculusis a set of formulas for relatingdifferentviewpoints • Can oftenbeobtained by means of the Large Time Heuristic 6

  7. Large Time HeuristicExplained on an Example • We want to relate and Weapply the large time heuristic • 1. How do weevaluatethesemetrics in a simulation ? 7

  8. Large Time HeuristicExplained on an Example • We want to relate and Weapply the large time heuristic • How do weevaluatethesemetrics in a simulation ?where index of next green or redarrowat or after 8

  9. Large Time HeuristicExplained on an Example • Break one integralintopiecesthat match the ’s: 9

  10. Large Time HeuristicExplained on an Example • Break one integralintopiecesthat match the ’s: 10

  11. Large Time HeuristicExplained on an Example • Compare 11

  12. Large Time HeuristicExplained on an Example • Compare 12

  13. This is Palm Calculus ! 13

  14. Sn = 90, 10, 90, 10, 90 • Xn = 5000, 1000, 5000, 1000, 5000 • Correlation is >0 • Wc > Ws • When do the two viewpoints coincide ? 14

  15. The Large Time Heuristic • Formally correct ifsimulationisstationary • It is a robustmethod, i.e. independent of assumptions on distributions (and on independence) 15

  16. Other«Clocks» Distribution of flow sizes for an arbitrary flow for an arbitrarypacket Flow 2 Flow 1 Flow 3 16

  17. Load Sensitive Routing of Long-Lived IP FlowsAnees Shaikh, Jennifer Rexford and Kang G. ShinProceedings of Sigcomm'99 ECDF, per packet viewpoint ECDF, per flow viewpoint 17

  18. Distribution of flow sizes for an arbitrary flow for an arbitrarypacket Mean flow size:per flow per packet Flow 2 Flow 1 Flow 3 18

  19. Large «Time» Heuristic • How do weevaluatethesemetrics in a simulation ? • Put the packetsside by side, sorted by flow Flow n=3 Flow n=1 Flow n=2 p=7 p=8 p=9 p=4 p=3 p=2 p=5 p=6 p=1 19

  20. Large «Time» Heuristic • How do weevaluatethesemetrics in a simulation ?per flow per packetwhere whenpacketbelongs to flow • Put the packetsside by side, sorted by flow Flow n=3 Flow n=1 Flow n=2 p=7 p=8 p=9 p=4 p=3 p=2 p=5 p=6 p=1 20

  21. Large «Time» Heuristic Flow n=3 Flow n=1 Flow n=2 • Compare p=7 p=8 p=9 p=4 p=3 p=2 p=5 p=6 p=1 21

  22. Large «Time» Heuristic Flow n=3 Flow n=1 Flow n=2 • Compare p=7 p=8 p=9 p=4 p=3 p=2 p=5 p=6 p=1 22

  23. Large «Time» Heuristic for PDFs of flow sizes • Put the packetsside by side, sorted by flow • How do weevaluatethesemetrics in a simulation ? Flow n=3 Flow n=1 Flow n=2 23

  24. 24

  25. Cyclist’sParadox • On a round trip tour, thereis more uphillsthandownhills 25

  26. The km clock vs the standard clock • speed for the kilometer 26

  27. 2. Palm Calculus : Framework • A stationary process (simulation) with state St. • Some quantity Xt measured at time t. Assume that (St;Xt) is jointly stationary I.e., St is in a stationary regime and Xt depends on the past, present and future state of the simulation in a way that is invariant by shift of time origin. • Examples • St = current position of mobile, speed, and next waypoint • Jointly stationary with St: Xt = current speed at time t; Xt = time to be run until next waypoint • Not jointly stationary with St: Xt = time at which last waypoint occurred

  28. Stationary Point Process • Consider some selected transitions of the simulation, occurring at times Tn. • Example: Tn = time of nth trip end • Tn is a called a stationary point process associated to St • Stationary because St is stationary • Jointly stationary with St • Time 0 is the arbitrary point in time 28

  29. Palm Expectation • Assume: Xt, St are jointly stationary, Tn is a stationary point process associated with St • Definition: the Palm Expectation isEt(Xt) = E(Xt | a selected transition occurred at time t) • By stationarity: Et(Xt) = E0(X0) • Example: • Tn = time of nth trip end, Xt = instant speed at time t • Et(Xt) = E0(X0) = average speed observed at a waypoint

  30. E(Xt) = E(X0) expresses the time average viewpoint. • Et(Xt) = E0(X0) expresses the event average viewpoint. • Example for random waypoint: • Tn = time of nth trip end, Xt = instant speed at time t • Et(Xt) = E0(X0) = average speed observed at trip end • E(Xt)=E(X0) = average speed observed at an arbitrary point in time Xn+1 Xn

  31. Intensity of a Stationary Point Process • Intensity of selected transitions:  := expected number of transitions per time unit

  32. Two Palm Calculus Formulae • Intensity Formula:where by convention T0≤ 0 < T1 • Inversion Formula • The proofs are simple in discrete time – see lecture notes

  33. 35

  34. 3. Other Palm Calculus Formulae 36

  35. Joe’ sWaiting Time • mean waiting time penalty due to variability mean time between busessystem’sviewpoint 37

  36. Feller’s Paradox

  37. WeencounteredFeller’sParadoxAlready 39

  38. For a Poisson process, what is the mean length of an interval ? 40

  39. Rate Conservation Law 41

  40. 42

  41. 43

  42. Campbell’s Formula • Shot noise model: customer n adds a load h(t-Tn,Zn) where Zn is some attribute and Tn is arrival time • Example: TCP flow: L = λV with L = bits per second, V = total bits per flow and λ= flows per sec Total load t T1 T2 T3 44

  43. Little’s Formula Total load t T1 T2 T3 45

  44. 4. RWP and Freezing Simulations • Modulator Model: 46

  45. Is the previous simulation stationary ? • Seems like a superfluous question, however there is a difference in viewpoint between the epoch n and time • Let Sn be the length of the nth epoch • If there is a stationary regime, then by the inversion formulaso the mean of Sn must be finite • This is in fact sufficient (and necessary) 47

  46. Application to RWP 48

  47. Time Average Speed, Averaged over n independent mobiles • Blue line is one sample • Red line is estimate of E(V(t)) 49

  48. A Random waypoint model that has no stationary regime ! • Assume that at trip transitions, node speed is sampled uniformly on [vmin,vmax] • Take vmin = 0 and vmax > 0 • Mean trip duration = (mean trip distance) • Mean trip duration is infinite ! • Was often used in practice • Speed decay: “considered harmful” [YLN03]

  49. What happens when the model does not have a stationary regime ? • The simulation becomes old

  50. Stationary Distribution of Speed(For model with stationary regime)

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