Internet Analysis - Performance Models -

1. Internet Analysis- Performance Models - G.U. Hwang Next Generation Communication Networks Lab. Division of Applied Mathematics KAIST

2. References for M/G/1 Input Process • Krunz and Makowski, Modeling Video Traffic Using M/G/1 Input Process, IEEE JSAC, vol. 16, 733-748, 1998 • Self-similar Network Traffic and Performance Evaluation, Eds. K. Park and W. Willinger, John Wiley & Sons, 2000. • B. Tsybakov, N.D. Georganas, Overflow and losses in a network queue with a self-similar input, Queueing Systems, vol. 35, 201-235, 2000 • M. Zukerman, T.D. Neame and R.G. Addie, Internet traffic modeling and future technology implications, INFOCOM 2003, 587-596. Next Generation Communication Networks Lab.

3. The M/G/1 arrival model • Consider a discrete time system with an infinite number of servers. • During time slot [n,n+1), we have Poisson arrivals with rate l and each arrival requires service time X according to a p.m.f. sn, n¸ 1 where E[X]<1. c.f. a customer arriving at the M/G/1 system can be considered as a burst. • When there are bn busy servers in the beginning of slot [n,n+1), the number of packets generated is bn. c.f. Each burstgenerates packets during its holding time. • We assume the system is in the steady state. Next Generation Communication Networks Lab.

4. The {bn} process arrivals Next Generation Communication Networks Lab.

5. The process {bn} of the M/G/1 arrivals • Let Yk denote a Poisson random variable with parameter lP{X¸ k}, which denotes the number of bursts arriving at [n-k,n-k+1) and being still in the system at time [n,n+1). bn = åk=11 Yk = Poisson R.V. with parameter lE[X]. n-5 n-4 n-3 n-2 n-1 n n+1 n+2 n+3 n+4 Next Generation Communication Networks Lab.

6. the stationary version of {bn}n¸ 0 • b0 : the initial number of bursts • a Poisson r.v. with parameter E[X] • the length of each initial burst is according to the forward recurrence time Xr of X the forward recurrence time X 0 1 2 3 4 6 7 8 9 5 Next Generation Communication Networks Lab.

7. Let • The autocovariance function of {bn} • The autocorrelation function of {bn} Next Generation Communication Networks Lab.

8. Then • The M/G/1 arrival model is • long range dependent if E[X2] = 1. • short range dependent if E[X2] < 1. Next Generation Communication Networks Lab.

9. A Pareto distribution • A random variable Y is called to have a Pareto distribution if its distribution function is given by where 0 < g < 2 is the shape parameter and d (> 0) is called the location parameter. • Remarks: • If 0 <  < 2, then Y has infinite variance. • If 0 < · 1, then Y has infinite mean. Next Generation Communication Networks Lab.

10. The expectation of the Pareto distribution • The distribution of the forward recurrence time Yr of the Pareto distribution Next Generation Communication Networks Lab.

11. The M/Pareto arrival process • When the service times are Pareto distributed given above, we have M/Pateto input process (or Poisson Pareto Burst input process). • Now let A(t) be the total amount of work arriving in the period (0,t]. • We assume that each burst in the system generate r bits per slot. Next Generation Communication Networks Lab.

12. The mean and variance of A(t) • If we define H = (3-)/2 and 1<<2, then the M/Pareto input process is asymptotically self-similar with Hurst parameter H. • c.f. Var[Yt] = t2H Var[Y1] for a self-similar process Yt Next Generation Communication Networks Lab.

13. A sample path of the M/Pareto arrivals = 0.4,  = 1.18 , = 0.9153 Next Generation Communication Networks Lab.

14. The autocorrelaton function Next Generation Communication Networks Lab.

15. c.f. M/G/1 for S.R.D. • Krunz and Makowski, Modeling Video Traffic Using M/G/1 Input Process, IEEE JSAC, vol. 16, 733-748, 1998 • M/G/1 input process is used to model video traffic encoded by DCT. Next Generation Communication Networks Lab.

16. Fractal Brownian Motion • Consider a self-similar process Yt and wide sense stationary increments Xn. Recall that • For 0 < H · 1, we can show that the function r(t,s) is nonnegative definite, i.e., for any real numbers t1, , tn and u1,,un, i=1nj=1n r(ti,tj) ui uj¸ 0. Next Generation Communication Networks Lab.

17. Definition of a joint normal distribution The vector X = (X1,,Xk), is said to have a joint normal distribution N(0,) if the joint characteristic function is given by where E[Xi] = 0 for all 1· i · m and =(mn) is the covariance matrix defined by mn = E[XmXn] for 1· m,n · k. Next Generation Communication Networks Lab.

18. Definition of a Gaussian process A stochastic process Yt is Gaussian if every finite set {Yt1,Yt2,,Ytn } has a joint normal distribution for all n. • From classical probability theory, there exists a Gaussian process whose finite dimensional distributions are joint normal distributions N(0,) where  = (r(t,s)). Next Generation Communication Networks Lab.

19. A self-similar Gaussian process Yt with stationary increments Xn having 0 < H < 1 is called a fractional Brownian Motion (fBm). • If E[Yt] = 0 and E[Yt2] = 2 |t|2H for some  > 0 for a Gaussian process, then we get Next Generation Communication Networks Lab.

20. Theorem Suppose that a stochastic process Yt • is a Gaussian process with zero mean, Y0 = 0, • E[Yt2] = 2 |t|2H for some  > 0 and 0 < H < 1, • has stationary increments; then {Yt} is called a fractional Brownian motion. c.f. The self-similarity comes from the following: Next Generation Communication Networks Lab.

21. c.f. The fractional Gaussian Noise The increment process of the fractional Brownian motion with Hurst parameter H is called the fractional Gaussian Noise (fGN) with Hurst parameter H. Next Generation Communication Networks Lab.

22. Consider a queueing system with input process At = t + Yt where Yt is a normalized fBM,i.e., E[Yt2] = 1. • Then the queue content process q(t) is given by q(t) = sups· t [A(t) - A(s) - C(t-s)] where C is the output link capacity. • Assume that q = limt!1 q(t) exists. Next Generation Communication Networks Lab.

23. A lower bound for the queue length • Since Yt has stationary increments, we get Next Generation Communication Networks Lab.

24. Hence, from the fact that Yt»N(0,t2H) we get where F(x) denotes the distribution function of a standard normal R.V. Next Generation Communication Networks Lab.

25. Next Generation Communication Networks Lab.

26. The superposition of ON/OFF sources • Consider an ON/OFF source with the following properties • The ON periods are according to a heavy tail distribution • The OFF periods are either heavy tailed or light tailed with finite variance. • The superposition of N ON/OFF sources is shown to behave like the fractional Brownian Motion when N is sufficiently large. Next Generation Communication Networks Lab.

27. Traffic model in the backbone • T. Karagiannis et. al, A nonstationary Poisson view of internet traffic, INFOCOM 2004, 1558-1569. • Traffic appears Poisson at sub-second time scale Next Generation Communication Networks Lab.

28. The complementary distribution function of the Packet interarrival times exponential distribution Next Generation Communication Networks Lab.

29. Traffic follows a non-stationary Poisson process at multi-second time scale points of rate changes relative magnitude of the change in the slope change of free region Next Generation Communication Networks Lab.

30. The change of Hurst parameters • Hurst parameters of time intervals of length 20 sec • the reasons for change: • self-similarity of the original traffic • the change in routing • the change in the number of active sources Next Generation Communication Networks Lab.

31. Autocorrelation for the magnitude of rate changes (i.e., the height of the spikes in Fig. 7) a negative correlation at lag 1 95 % C.I. for 0 Next Generation Communication Networks Lab.

32. The complementary distribution function for the lengths of the change of free intervals (the stationary intervals) exponential distribution A Markovian random walk model would be a good candidate Next Generation Communication Networks Lab.

33. Traffic appears LRD at large time scales original ACF ACF using moving averages Next Generation Communication Networks Lab.

34. Summary • Due to the high variability of the internet traffic • it is very difficult to give good mathematical models and additionally estimate the traffic parameters. • continuous traffic measurements should be done to reflect the changes of the internet traffic characteristics on performance models. Next Generation Communication Networks Lab.