Evidence for long-tailed distributions in the Internet Allen B. - PowerPoint PPT Presentation

Evidence for long-tailed distributions in the Internet Allen B. Downey Wellesley College – p.1

� � � Self-Similarity No shortage of explanations... ON/OFF model M/G/ model Protocol models – p.2

� ccdf test Samples ccdf test (n=10,000) from 1 Pareto and lognormal 0.1 Prob {X > x} distributions with similar tail 0.01 behavior. 0.001 lognormal sample pareto sample 1e-4 0 5 10 15 20 log2 (x) – p.3

� � � File sizes on a Web server Sizes of 15,160 File Sizes from Calgary dataset 1 files at the University of 1/4 Calgary. Prob {file size > x} 1/16 By conventional 1/64 goodness of fit, 1/256 Pareto wins. 1/1024 lognormal model Tail behavior is 1/4096 pareto model not long-tailed. actual ccdf 1/16384 1KB 32KB 1MB 32MB File size (bytes) – p.4

� ✁ � � Numerical differentiation Numerical Estimated derivative of ccdf derivatives are 0 noisy. -0.5 Testing for Inverse slope trends is robust. -1 Tail curvature = 0.141, p-value -1.5 0.001. -2 1/16 1/64 1/256 1/1024 1/4096 P (X > x) – p.5

� � � � Files sizes on another Web server Files sizes from File Sizes from Saskatchewan dataset 1 University of Saskatchewan. 1/4 Prob {file size > x} 1/16 Pareto model fits well. 1/64 1/256 Two-mode lognormal model 1/1024 fits well. lognormal model 1/4096 actual ccdf Tail curvature 1/16384 test is no help. 1KB 32KB 1MB 32MB File size (bytes) – p.6

� � � � Interarrival times, TCP packets 4 million TCP packet interarrival times 1 interarrivals from LBL and DEC 0.1 datasets. Prob {time > x} 0.01 Very consistent 0.001 between 10^-4 datasets. 10^-5 Pareto model Some signs of lognormal model 10^-6 straightness. actual cdf Extreme tail hard .001 .01 0.1 1 10 100 1000 10^4 x (seconds) to characterize. – p.7

� � � Interarrival times, TCP connections 782,000 TCP connection interarrival times connections in 1 LBL CONN-7. 0.1 Feldmann Prob {time > x} 0.01 reports that 0.001 Weibull fits the bulk. Pareto model 10^-4 lognormal model Fits the tail well, Weibull model 10^-5 actual cdf too. 10^-6 0.1 1 10 100 1000 x (seconds) – p.8

� � Interarrival times, web requests 135,000 Web request interarrival times requests from 1 instrumented 0.1 browsers at BU. Prob {time > x} 0.01 Hard to characterize tail 0.001 behavior. 10^-4 Pareto model lognormal model 10^-5 actual cdf 10^-6 .001 .01 0.1 1s 10 100 10^4 10^6 x (seconds) – p.9

� � http transfer times 135,000 Web request transfer times transfers. 1 Lognormal 0.1 model fits the Prob {time > x} 0.01 extreme tail. 0.001 10^-4 Pareto model lognormal model actual cdf 10^-5 0.1 1 10 100 1000 x (seconds) – p.10

� � Throughput For each Throughputs, BU dataset transfer, divide 1.0 size by transfer lognormal model Prob {throughput > x} 0.8 time. actual ccdf Across paths 0.6 and time, 0.4 throughput is roughly 0.2 lognormal. 0.0 10 100 1000 10^4 10^5 x (bytes/second) – p.11

� � � ftp transfer times 105,000 FTP transfer times transfers in LBL 1 CONN-7. 0.1 Not so clear that Prob {time > x} this is lognormal. 0.01 Paxson used 0.001 two-stage Pareto 10^-4 model. Pareto model lognormal model actual ccdf 10^-5 0.1 1 10 100 1000 10^4 10^5 10^6 x (seconds) – p.12

� � ftp throughput Again, roughly Throughputs, LBL dataset lognormal. 1.0 Top end lognormal model Prob {throughput > x} 0.8 actual ccdf compressed by hw limitations. 0.6 0.4 0.2 0.0 10 100 1000 10^4 10^5 x (bytes/second) – p.13

� � � ✁ ftp burst sizes Two or more ftp burst sizes 1 transfers with 4s between. 1/4 1/16 56,000 bursts. Prob {size > x} 1/64 Fairly convincing 1/256 straight line. 1/1024 1/4096 lognormal model Pareto model 1/16384 actual cdf 1 32 1KB 32KB 1MB 32MB x (bytes) – p.14

� � � ftp burst lengths How many ftp burst lengths 1 transfers in a burst? 1/4 1/16 Prob {length > x} 85% are 1/64 singletons. 1/256 Lognormal? 2^-10 Pareto? 2^-12 lognormal model Pareto model 2^-14 actual cdf 1 10 100 1000 x (# transfers) – p.15

� � � http burst lengths 456,000 http http burst length, Trace A 1 connections from Charzinski lognormal model Pareto model trace. 1/32 Prob {length > x} actual cdf 70% of 2^-10 connections make a single request. 2^-15 Tail behavior 2^-20 hard to 1 10 100 1000 characterize. x (# transfers) – p.16

� � More http burst lengths 739,000 http burst length, Trace B 1 connections recorded by lognormal model Pareto model Charzinski. 1/32 Prob {length > x} actual cdf Pretty clearly 2^-10 lognormal. 2^-15 2^-20 1 10 100 1000 x (# transfers) – p.17