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

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. 53 rd IETF, Minneapolis, March 18 2002

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 on a particular vector • Three datasets: # 1, # 3 and # 4 2

Delay Measurement Measurement Record (300 seconds) • M packets with fixed length of 576 bytes dispatched periodically • GPS synchronized send ( t s ) and receive ( t r ) times of each packet measured/recorded • One-way delay d = t r − t s • The total number of sent ( M ) and received ( ≤ M ) packets recorded • Duplicated and dropped packets recorded 3

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 × 10 6 bps, packet length = 576 bytes, utilization ρ = 1 . 0. 4

Delay Measurement Statistics (300 second measurement period) 1 � M r • Mean delay: < d > = i =1 d i M r • Standard deviation: s 2 = � M r 1 i =1 ( d i − < d > ) 2 M r − 1 • Minimum and maximum delay d min , d max • Probability density (pdf). Autocorrelations. Power spectra. 5

25 160 150 20 140 delay d (ms) delay d (ms) 15 130 120 10 110 5 100 0 100 200 300 0 100 200 300 time from beginning of record (seconds) time from beginning of record (seconds) Typical time series of delay over a 300 second measurement record. Left; dataset # 1. Right; dataset # 4.

3 10 600 average delay average delay min. delay min. delay 500 max. delay max. delay 400 300 2 10 ms ms 200 1 10 100 -16 -12 -8 -4 0 4 8 12 16 20 24 28 32 0 8 16 24 32 40 48 56 64 72 80 hour from midnight, Monday 07/02/2001 hour from midnight, Thursday 09/13/2001 Average, minimum and maximum delay over consecutive 300 second measurement records. Left; dataset # 1. Right; dataset # 4.

3 10 average delay 2 10 standard deviation 2 10 average delay standard deviation ms ms 1 10 1 10 0 10 0 10 -1 10 -16 -12 -8 -4 0 4 8 12 16 20 24 28 32 0 8 16 24 32 40 48 56 64 72 80 hour from midnight, Monday 07/02/2001 hour from midnight, Thursday 09/13/2001 Average delay & standard deviation over consecutive 300 second measurement records. Left; dataset # 1. Right; dataset # 4.

Measurement, Record 10090, dataset 1 Measurement, Record 35581, dataset4 Shifted Gamma distribution Shifted Gamma distribution 0.5 11 10 0.45 9 0.4 8 0.35 7 0.3 6 pdf pdf 0.25 5 0.2 4 0.15 3 0.1 2 0.05 1 0 0 100 105 110 115 120 7.5 7.6 7.7 7.8 7.9 8 delay (ms) delay (ms) Delay pdfs over typical 300 second measurement record. Left; dataset # 1. Right; dataset # 4. Approximated by shifted Gamma distribution [Mukherjee (1992)].

0 0 10 10 -1 10 -1 10 -2 10 -2 10 -3 10 pdf pdf -3 -4 10 10 -5 10 -4 10 -6 10 -5 10 -7 10 -6 -8 10 10 1 2 3 10 10 10 100 200 300 400 500 delay (ms) delay (ms) Delay pdfs over whole dataset. Left; dataset # 1. Right; dataset # 4.

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1 0.9 0.8 0.7 Fraction of packets 0.6 0.5 0.4 0.3 dataset 4 0.2 dataset 1 0.1 -1 0 1 2 10 10 10 10 P (% minimum delay window) Average fraction of packets in measurement record with delay within P % minimum delay window for datasets #1 and # 4

1 1 0.8 0.8 0.6 0.6 C(T) C(T) 0.4 0.4 0.2 0.2 0 0 0 100 200 300 0 100 200 300 T (secs.) T (secs.) Autocorrelation function over one typical 300 second measure- ment record. Left; dataset # 1. Right; dataset # 3.

0 10 Dataset1 Dataset3 Dataset4 -1 10 pdf -2 10 -3 10 0 1 2 10 10 10 Correlation time (seconds) Pdf of delay autocorrelation time over typical 300 second measurement records. Three datasets.

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 N/ 2 − 1 , ω k = 2 π k d k e iω k t , d − k = d ∗ ˆ � d ( t ) = , k , T k = − N/ 2 d k ( ω ) | 2 • Power spectrum = | ˆ 14

0 -1 10 10 -2 10 -3 10 power spectrum power spectrum -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 -10 10 -5 -4 -3 -2 -1 0 1 -5 -4 -3 -2 -1 0 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 -1 ) -1 ) frequency (sec frequency (sec Power spectrum of measured delay time series. Left, dataset # 1. Right, dataset # 4

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