Palm Calculus The Importance of the Viewpoint

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Palm CalculusThe Importance of the Viewpoint

• JY Le Boudec
• 2009

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Contents

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

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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

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Example Gatekeeper
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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
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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)

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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 ?

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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

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Other Samplings
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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

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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

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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

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• 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
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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

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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

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Intensity of a Stationary Point Process

• Intensity of selected transitions ? expected
  number of transitions per time unit

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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
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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

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Littles Formula
Total load
t
T1
T2
T3
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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

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Stop and Go
A
A
A
B
B
B
B
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4. RWP and Freezing Simulations

• Modulator Model

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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)

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Application to RWP
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Time Average Speed, Averaged over n independent
mobiles

• Blue line is one sample
• Red line is estimate of E(V(t))

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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

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What happens when the model does not have a
stationary regime ?

• The simulation becomes old

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Stationary Distribution of Speed(For model with
stationary regime)
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Closed Form

• Assume a stationary regime exists and simulation
  is run long enough
• Apply inversion formula and obtain distribution
  of instantaneous speed V(t)

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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
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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
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• Is it possible to have the time distribution of
  speed uniformly distributed in 0 vmax ?

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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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Exercise
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Exercise
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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

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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

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Perfect simulation is highly desirable (2)

• Distribution of path

Time 50s
Time 500s
Time 100s
Time 1000s
Time 300s
Time 2000s
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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
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Stationary Distrib of Prev and Next
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Stationary Distribution of Location Is also
Obtained By Inversion Formula
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No Speed Decay
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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)

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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