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OutlineOutlineOutline

Estimation and Removal of Clock Skew from Network Delay MeasurementsSue B. Moon, Paul Skelly ,Don TowsleyProceedings of IEEE INFOCOM 1999, New York, NY, March1999.

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Outline

- Motivation
- Clock Terminology
- Desirable properties of skew estimation algorithms
- Algorithms
- Comparison
- Conclusion

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Outline

- Motivation
- Clock Terminology
- Desirable properties of skew estimation algorithms
- Algorithms
- Comparison
- Conclusion

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Motivation

- Network transit times are used to infer fundamental network properties – delay, bottleneck link speed, available bandwidth, queuing.
- For delay measurements ,a sender needs to add timestamps to packets for a receiver to gather delay information
- Echo based techniques versus receiver based techniques

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Since the clocks at both end systems are involved in measuring delay, the synchronization of the two clocks becomes an issue in the accuracy of the delay measurement.

- When two clocks run at different frequencies (i.e. have a clock skew) inaccuracies are introduced in the measurement. We focus on filtering out the effects of clock skew, specifically in one-way delay measurements

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The measured delay is not the actual delay but includes the clock offset between the two clocks plus the end-to-end delay

- End-to-end delay consists of transmission and propagation delays plus variable queuing delay

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Why does the delay show an increasing trend (100ms over the duration of 70 minutes) at the receiver?

- Increasing congestion and queuing delay?However, minimum observed delay increases over time.
- Speed difference between the sender and receiver clocks?
- The linear increase in delay attests to a constant speed difference between the sender and receiver clocks
- If the clocks have a non-zero skew, not only is the end-to-end delay measurement off by an amount equal to the offset, but it also gradually increases or decreases over time depending on whether the sender clock runs slower or faster than the receiver clock.

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Outline

- Motivation
- Clock Terminology
- Desirable properties of skew estimation algorithms
- Algorithms
- Comparison
- Conclusion

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Clock Terminology

Ct(t) = t

Resolution: The smallest unit by which time is updated (a “tick)

Offset: The difference between the time reported by a clock and the “true” time;

The offset of Ca is (Ca(t)-t).

The offset of the clock Ca relative to Cb at time t ≥ 0 is Ca(t)-Cb(t)

Frequency: The rate at which the clock progresses. The frequency at time t of Ca is C’a(t)

Skew: The difference in the frequencies of a clock and the “true” clock. The skew of Ca relative to Cb at time t (C’a(t) – C’b(t))

Drift: The drift of clock Ca is C”(t). The drift of Ca relative to Cb at time t ≥ 0 is (C”a(t) – C”b(t))

Accuracy : How close the absolute value of offset (at a particular moment) is to zero

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Clock Terminology (Contd.)

- Two clocks are said to be synchronizedat a particular moment if both the relative offset and skew are zero.
- When it is clear that we refer to two clocks, neither of which is the true clock in our discussion, we simply refer to relative offset and relative skew as offset and skew, respectively.
- We assume that the sender and receiver clockshave constant frequencies, and their skew and clock ratio areconstant over time (drift = 0)

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Let α and β be the estimates for α and d1.

- Then the delay after skew removal is

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- Motivation
- Clock Terminology
- Desirable properties of skew estimation algorithms
- Algorithms
- Comparison
- Conclusion

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Desirable properties of skew estimation algorithms

- The time and space complexity of algorithm should be linear in N.
- We will compare the time complexity of skew estimation algorithms as a function of the number of delay measurements.

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Desirable properties of skew estimation algorithms (Contd.)

- Since the purpose of the skew estimation is to remove the skew from delay measurements, it is desirable that the delays be non-negative after the skew is removed

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Desirable properties of skew estimation algorithms (Contd.)

- The skew estimation algorithm should be robust in the sense that it is not affected by the magnitude of the actual skew.
- That is, the difference between the estimate and the actual skew should be independent of the actual skew.

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- Motivation
- Clock Terminology
- Desirable properties of skew estimation algorithms
- Algorithms
- Comparison
- Conclusion

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LINEAR PROGRAMMING ALGORITHM

- To fit a line that lies under all the data points, but as closely to them as possible.
- Feasible region : the line should lie under all the data points
- Objective Function: To minimize the sum of the distances between the line and all the data points on the y-axis.

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There are infinitely many pairs of D and that satisfy the condition above, if the feasible region defined above is not trivial.

- Our objective function to minimize the distance between the line
- and all the delay measurements is stated as

Objective Function :

Constraint:

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Other Algorithms

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PAXSON’S ALGORITHM

- Step 1. Partition di ’s into segments, and pick the minimum delay measurement from each segment. The selected measurements are called the “de-noised” one-way transit times (OTTs).
- Step 2. Pick the median of the slopes of all possible pairs of the “de-noised” OTTs. If the median slope is negative, assume that the OTTs have a decreasing trend (here we assume a decreasing trend is detected).
- Step 3. Select the cumulative minima test from the “denoised” OTTs and test if the number of cumulative minima is large enough to show that the decreasing trend found in Step 2 is probabilistically not likely, if there is no trend
- Step 4. If it passes the cumulative minima test, pick the median from the slopes of all possible pairs of the cumulative minima: output it as the estimate of α-1. Otherwise, the algorithm concludes that there is no skew, and outputs α = 0.

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PIECEWISE MINIMUM ALGORITHM

- Partitions the delay measurements into segments
- Pick a minimum from each segment
- Connect them to obtain a concatenation of line segments.
- The minima are the same as the “de-noised” OTTs in Paxson’s algorithm. The resulting concatenation of line segments is the estimate of the skew, and is very unlikely to be a straight line.
- When the skew is as obvious as in our figure the resulting concatenation of line segments is close to a straight line, and can be used as a rough estimate.

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LINEAR REGRESSION ALGORITHM

- Linear regression is a standard technique for fitting a line to a set of data points.
- It is optimal in the mean square sense if the network delays are normally distributed, but is not robust in the presence of outliers.
- It is not a good choice for a skew estimation, even when applied to the “de-noised” OTTs above.
- It can be used only as a reference algorithm that requires no knowledge of the underlying behavior of delay measurements.

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- Motivation
- Clock Terminology
- Desirable properties of skew estimation algorithms
- Algorithms
- Comparison
- Conclusion

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Computational Complexity

- Time complexity of a two variable linear programming problem is proven to be O(N). The algorithm, exploits the fact that ti’s are sorted
- The other three algorithms have the complexity of O(N).

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Non-negative delay after the skew removal

- In order to guarantee that the delay remains positive after the after the skew is removed, a skew estimation algorithms must estimate d1 correctly.
- The linear programming algorithm ,however, is the only one that estimates d1..
- Paxson’s original algorithm for skew estimation is for two-way measurements after the clock offset has been removed.
- The linear regression algorithm provides an estimate of β. However, this is just the y-intercept of the regression line which bears no relevance to the correct estimation of d1
- The piecewise minimum algorithm outputs a concatenation of line segments, and the slopes of those line segments are skew estimates. The algorithm does not have any provision to guarantee that all the data points lie above the concatenation of line segments.

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Robustness

- Linear Programming algorithm satisfies robustness .i.e. the estimated skew doesn't depend on the magnitude of clock skew.
- Linear Regression Algorithm satisfies this property.
- Piecewise minimum is effected by the magnitude of skew since it depends on the calculation of minima.
- Simulations of Paxson's algorithm also show that it doesn’t satisfy this property.

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Simulation on Paxson’s algorithm

Purpose :To show the variability of the difference in the actual and the estimated

skew over a range of clock skews.

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Measurements

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Trace I Measurements

Linear Programming Algorithm

- Linear Programming Algorithm works well with Trace I readings

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Trace II Measurements

Linear Programming Algorithm

- Takes into account the excessive congestion due to multicasting

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Trace II Measurements (continued)

Paxsons Algorithm

- Paxsons algorithm works well with the high loss rate too

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Trace II Measurements (continued)

Linear Regression Algorithm

- Delay has a decreasing trend which is not true.
- Doesn’t work well in the presence of outliers

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Trace II Measurements (continued)

Piecewise Minimum Algorithm

- Effect of the congestion in the network delay is removed

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Simulation

- Purpose of simulation is to examine the average performance of skew estimation algorithms.
- Assume exponential distribution of end-to-end delay with varying mean.

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Simulation (continued)

Linear Programming Algorithm

Paxson’s Algorithm

- Linear Programming algorithm has less variance as compared to

Paxson’s algorithm.

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Conclusion

- The linear regression and piecewise minimum algorithms demonstrated a poor performance over traces of Internet delay measurements.
- As compared to Paxson’s, Linear Programming Algorithm is unbiased and has less variance.

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References:

- Estimation and removal of clock skew from Network Delay Measurements- Sue B. Moon, Paul Skelly ,Don Towsley
- On Calibrating Measurements of Packet Transit Time – Vern Paxson

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