Title: Palm Calculus The Importance of the Viewpoint
1Palm CalculusThe Importance of the Viewpoint
2Contents
- Informal Introduction
- Palm Calculus
- Other Palm Calculus Formulae
- Application to RWP
- Other Examples
- Perfect Simulation
31. 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
4Example Gatekeeper
5Sampling 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
6Large Time Heuristic Explained on an Example
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9The 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)
10Impact 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 ?
11Two 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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14Throughput of Stop and Go
15Other Samplings
16Load 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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192. 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
20Stationary 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
21Palm 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
23Formal 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
24Ergodic 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
25Intensity of a Stationary Point Process
- Intensity of selected transitions ? expected
number of transitions per time unit
26Two 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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283. Other Palm Calculus Formulae
29Fellers Paradox
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31Campbells 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
32Littles Formula
Total load
t
T1
T2
T3
33Two 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
34Stop and Go
A
A
A
B
B
B
B
354. RWP and Freezing Simulations
36Is 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)
37Application to RWP
38Time Average Speed, Averaged over n independent
mobiles
- Blue line is one sample
- Red line is estimate of E(V(t))
39A 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
40What happens when the model does not have a
stationary regime ?
- The simulation becomes old
41Stationary Distribution of Speed(For model with
stationary regime)
42Closed Form
- Assume a stationary regime exists and simulation
is run long enough - Apply inversion formula and obtain distribution
of instantaneous speed V(t)
43Removing 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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44A 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 ?
465. 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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49Exercise
50Exercise
516. 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
52Removing 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
53Perfect simulation is highly desirable (2)
Time 50s
Time 500s
Time 100s
Time 1000s
Time 300s
Time 2000s
54Solution 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
55Stationary Distrib of Prev and Next
56Stationary Distribution of Location Is also
Obtained By Inversion Formula
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58No Speed Decay
59Perfect 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)
60Conclusions
- 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