[PPT] - Statistics of One-Way Internet Packet Delays Andrew Corlett CQOS PowerPoint Presentation

SLIDE 1

Statistics of One-Way Internet Packet Delays

Andrew Corlett CQOS Inc., Irvine, CA

D. I. Pullin

California Institute of Technology Stephen Sargood Nortel Networks UK Ltd. 53rd IETF, Minneapolis, March 18 2002

SLIDE 2

Delay Measurement

The Data

Packet transmission between two CQOS C-nodes (a vector).
Internet transmission (not a dedicated link)
Packets transmitted periodically over 300 second periods
Each 300 second period → ‘Measurement Record’
A dataset consists of many sequential Measurement Records
n a particular vector
Three datasets: # 1, # 3 and # 4

2

SLIDE 3

Delay Measurement

Measurement Record (300 seconds)

M packets with fixed length of 576 bytes dispatched

periodically

GPS synchronized send (ts) and receive (tr) times of each

packet measured/recorded

One-way delay d = tr − ts
The total number of sent (M) and received (≤ M) packets

recorded

Duplicated and dropped packets recorded

3

SLIDE 4

Delay Measurement

Dataset parameters

Dataset Number of Measurement Pkts/300 Occupancy Vector records period (days) secs (M). fraction ν (Hop Count) # 1 621 2.2 611 0.0063 Local (11) # 3 2033 7.1 9556 0.092 Local (11) # 4 1017 3.5 730 0.0073 London (22) Parameters defining datasets # 1, # 3 and # 4. Bandwidith = 1.5 × 106 bps, packet length = 576 bytes, utilization ρ = 1.0.

4

SLIDE 5

Delay Measurement

Statistics (300 second measurement period)

Mean delay: < d >=

1 Mr

Mr

i=1 di

Standard deviation: s2 =

1 Mr−1

Mr

i=1(di− < d >)2

Minimum and maximum delay dmin, dmax
Probability density (pdf). Autocorrelations. Power spectra.

5

SLIDE 6

time from beginning of record (seconds) delay d (ms)

100 200 300 5 10 15 20 25

time from beginning of record (seconds) delay d (ms)

100 200 300 100 110 120 130 140 150 160

Typical time series of delay over a 300 second measurement

record. Left; dataset # 1. Right; dataset # 4.

SLIDE 7

hour from midnight, Monday 07/02/2001 ms

16 -12
8
4

4 8 12 16 20 24 28 32 10

1

10

2

10

3

average delay

min. delay
max. delay

hour from midnight, Thursday 09/13/2001 ms

8 16 24 32 40 48 56 64 72 80 100 200 300 400 500 600 average delay

min. delay
max. delay

Average, minimum and maximum delay over consecutive 300 second measurement records. Left; dataset # 1. Right; dataset # 4.

SLIDE 8

hour from midnight, Monday 07/02/2001 ms

16 -12
8
4

4 8 12 16 20 24 28 32 10

1

10 10

1

10

2

10

3

average delay standard deviation

hour from midnight, Thursday 09/13/2001 ms

8 16 24 32 40 48 56 64 72 80 10 10

1

10

2

average delay standard deviation

Average delay & standard deviation over consecutive 300 second measurement records. Left; dataset # 1. Right; dataset # 4.

SLIDE 9

delay (ms) pdf

7.5 7.6 7.7 7.8 7.9 8 1 2 3 4 5 6 7 8 9 10 11 Measurement, Record 10090, dataset 1 Shifted Gamma distribution

delay (ms) pdf

100 105 110 115 120 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Measurement, Record 35581, dataset4 Shifted Gamma distribution

Delay pdfs over typical 300 second measurement record. Left; dataset # 1. Right; dataset # 4. Approximated by shifted Gamma distribution [Mukherjee (1992)].

SLIDE 10

delay (ms) pdf

10

1

10

2

10

3

10

6

10

5

10

4

10

3

10

2

10

1

10

delay (ms) pdf

100 200 300 400 500 10

8

10

7

10

6

10

5

10

4

10

3

10

2

10

1

10

Delay pdfs over whole dataset. Left; dataset # 1. Right; dataset # 4.

SLIDE 11

————————————————————————-

SLIDE 12

P (% minimum delay window) Fraction of packets

10

1

10 10

1

10

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dataset 4 dataset 1

Average fraction of packets in measurement record with delay within P% minimum delay window for datasets #1 and # 4

SLIDE 13

T (secs.) C(T)

100 200 300 0.2 0.4 0.6 0.8 1

T (secs.) C(T)

100 200 300 0.2 0.4 0.6 0.8 1

Autocorrelation function over one typical 300 second measure- ment record. Left; dataset # 1. Right; dataset # 3.

SLIDE 14

Correlation time (seconds) pdf

10 10

1

10

2

10

3

10

2

10

1

10 Dataset1 Dataset3 Dataset4

Pdf of delay autocorrelation time over typical 300 second measurement records. Three datasets.

SLIDE 15

Delay Measurement

Power spectrum of delay

Delay time series consists of concatenated records
Dataset # 1; 380,030 packets.

Dataset # 4; 729,844 packets

Fourier series for delay time series

d(t) =

N/2−1

k=−N/2

ˆ dkeiωk t, , ωk = 2 π k T , d−k = d∗

k,

Power spectrum = |ˆ

dk(ω)|2

14

SLIDE 16

frequency (sec

1)

power spectrum

10

5

10

4

10

3

10

2

10

1

10 10

1

10

10

10

8

10

6

10

4

10

2

10

frequency (sec

1)

power spectrum

10

5

10

4

10

3

10

2

10

1

10 10

1

10

9

10

7

10

5

10

3

10

1

Power spectrum of measured delay time series. Left, dataset # 1. Right, dataset # 4

SLIDE 17

Delay Measurement

Summary; statistics of one-way delay

Pdf of one-way delay well approximated by shifted-Gamma

distributions.

Data subject to varying degrees of non-stationarity
Mean delay time series show:

⊲ strong temporal variability in local datasets ⊲ lesser temporal variability in international dataset

Autocorrelation time for delay series ∼ 2 − 10 seconds
Power spectra show only expected dominant frequencies

16