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Palm Calculus The Importance of the Viewpoint

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Title: Palm Calculus The Importance of the Viewpoint


1
Palm CalculusThe Importance of the Viewpoint
  • JY Le Boudec
  • 2009

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

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

4
Example Gatekeeper
5
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

6
Large Time Heuristic Explained on an Example
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9
The Large Time Heuristic
  • We will show later that this is formally correct
    if the simulation is stationary
  • It is a robust method, i.e. independent of
    assumptions on distributions (and on independence)

10
Impact of Cross-Correlation
  • Sn 90, 10, 90, 10, 90
  • Xn 5000, 1000, 5000, 1000, 5000
  • Correlation is gt0
  • Wc gt Ws
  • When do the two viewpoints coincide ?

11
Two Event Clocks
  • Stop and Go protocol
  • Clock 0 new packets Clock a all transmissions
  • Obtain throughput as a function of t0, t1 and
    loss rate

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Throughput of Stop and Go
  • Again a robust formula

15
Other Samplings
16
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
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2. Palm Calculus Framework
  • A stationary process (simulation) with state St.
  • Some quantity Xt measured at time t. Assume that
  • (StXt) 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

20
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

21
Palm Expectation
  • Assume Xt, St are jointly stationary, Tn is a
    stationary point process associated with St
  • Definition the Palm Expectation is Et(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

22
  • 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

Xn1
Xn
23
Formal Definition
  • In discrete time, we have an elementary
    conditional probability
  • In continuous time, the definition is a little
    more sophisticated
  • uses Radon Nikodym derivative see lecture note
    for details
  • Also see BaccelliBremaud87 for a formal
    treatment
  • Palm probability is defined similarly

24
Ergodic Interpretation
  • Assume simulation is stationary ergodic, i.e.
    sample path averages converge to expectations
    then we can estimate time and event averages
    by
  • In terms of probabilities

25
Intensity of a Stationary Point Process
  • Intensity of selected transitions ? expected
    number of transitions per time unit

26
Two Palm Calculus Formulae
  • Intensity Formulawhere by convention T0
    0 lt T1
  • Inversion Formula
  • The proofs are simple in discrete time see
    lecture notes

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3. Other Palm Calculus Formulae
29
Fellers Paradox
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Campbells 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 ?V with L bits per
    second, V total bits per flow and ? flows per
    sec

32
Littles Formula
Total load
t
T1
T2
T3
33
Two Event Clocks
  • Two event clocks, A and B, intensities ?(A) and
    ?(B)
  • We can measure the intensity of process B with
    As clock?A(B) number of B-points per tick of
    A clock
  • Same as inversion formula but with A replacing
    the standard clock

34
Stop and Go
A
A
A
B
B
B
B
35
4. RWP and Freezing Simulations
  • Modulator Model

36
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)

37
Application to RWP
38
Time Average Speed, Averaged over n independent
mobiles
  • Blue line is one sample
  • Red line is estimate of E(V(t))

39
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 gt 0
  • Mean trip duration (mean trip distance)
  • Mean trip duration is infinite !
  • Was often used in practice
  • Speed decay considered harmful YLN03

40
What happens when the model does not have a
stationary regime ?
  • The simulation becomes old

41
Stationary Distribution of Speed(For model with
stationary regime)
42
Closed Form
  • Assume a stationary regime exists and simulation
    is run long enough
  • Apply inversion formula and obtain distribution
    of instantaneous speed V(t)

43
Removing Transient Matters
  • A. In the mobile case, the nodes are more often
    towards the center, distance between nodes is
    shorter, performance is better
  • The comparison is flawed. Should use for static
    case the same distribution of node location as
    random waypoint. Is there such a distribution to
    compare against ?
  • A (true) example Compare impact of mobility on a
    protocol
  • Experimenter places nodes uniformly for static
    case, according to random waypoint for mobile
    case
  • Finds that static is better
  • Q. Find the bug !

Random waypoint
Static
43
44
A Fair Comparison
  • We revisit the comparison by sampling the static
    case from the stationary regime of the random
    waypoint

Static, same node location as RWP
Random waypoint
Static, from uniform
45
  • Is it possible to have the time distribution of
    speed uniformly distributed in 0 vmax ?

46
5. PASTA
  • There is an important case where Event average
    Time average
  • Poisson Arrivals See Time Averages
  • More exactly, should be Poisson Arrivals
    independent of simulation state See Time Averages

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49
Exercise
50
Exercise
51
6. Perfect Simulation
  • An alternative to removing transients
  • Possible when inversion formula is tractable
  • Example random waypoint
  • Same applies to a large class of mobility models

52
Removing Transients May Take Long
  • If model is stable and initial state is drawn
    from distribution other than time-stationary
    distribution
  • The distribution of node state converges to the
    time-stationary distribution
  • Naïve so, lets simply truncate an initial
    simulation duration
  • The problem is that initial transience can last
    very long
  • Example space graph node speed 1.25
    m/sbounding area 1km x 1km

53
Perfect simulation is highly desirable (2)
  • Distribution of path

Time 50s
Time 500s
Time 100s
Time 1000s
Time 300s
Time 2000s
54
Solution Perfect Simulation
  • Def a simulation that starts with stationary
    distribution
  • Usually difficult except for specific models
  • Possible if we know the stationary distribution

Sample Prev and Next waypoints from their joint
stationary distribution Sample M uniformly on
segment Prev,Next Sample speed V from
stationary distribution
55
Stationary Distrib of Prev and Next
56
Stationary Distribution of Location Is also
Obtained By Inversion Formula
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58
No Speed Decay
59
Perfect Simulation Algorithm
  • Sample a speed V(t) from the time stationary
    distributionHow ?
  • A inversion of cdf
  • Sample Prev(t), Next(t)How ?
  • Sample M(t)

60
Conclusions
  • A metric should specify the sampling method
  • Different sampling methods may give very
    different values
  • Palm calculus contains a few important formulas
  • Which ones ?
  • Freezing simulations are a pattern to be aware of
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