Palm Calculus Made Easy The Importance of the Viewpoint

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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 CalculusMade EasyThe Importance of the Viewpoint

JY Le Boudec

1

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

2

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

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

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

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

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

7

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

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

9

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

10

Sn = 90, 10, 90, 10, 90

• Xn = 5000, 1000, 5000, 1000, 5000
• Correlation is >0
• Wc > Ws
• When do the two viewpoints coincide ?

14

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

15

Other«Clocks»

Distribution of flow sizes

for an arbitrary flow for an arbitrarypacket

Flow 2

Flow 1

Flow 3

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

17

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

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

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

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

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

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

• On a round trip tour, thereis more uphillsthandownhills

25

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

26

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
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

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
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

Intensity of a Stationary Point Process
• Intensity of selected transitions:  := expected number of transitions per time unit
Two Palm Calculus Formulae
• Intensity Formula:where by convention T0≤ 0 < T1
• Inversion Formula
• The proofs are simple in discrete time – see lecture notes
Joe’ sWaiting Time
• mean waiting time

penalty due to variability

mean time between busessystem’sviewpoint

37

40

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

t

T1

T2

T3

44

Little’s Formula

t

T1

T2

T3

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

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

49

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

54

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

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

56

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

57

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

60

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, let’s 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

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

Time = 50s

Time = 500s

Time = 100s

Time = 1000s

Time = 300s

Time = 2000s

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
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)

68

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