internet analysis - performance models -

Internet Analysis- Performance Models -

G.U. HwangNext Generation Communication Networks Lab.

Division of Applied MathematicsKAIST

Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST2

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.


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 and each arrival requires service time X according to a p.m.f. n, 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 burst generates packets during its holding time.

We assume the system is in the steady state.


The {bn} process

21b22 b

43 b 44 b 35 b26 b 37 b

28 b39 b 310 b

011 b 012 b 113 b

arrivals


The process {bn} of the M/G/1 arrivals

Let Yk denote a Poisson random variable with parameter P{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 = k=11 Yk

= Poisson R.V. with parameter E[X].

n n+1 n+2 n+3 n+4n-2n-3 n-1n-5 n-4


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

5 6 7 8 932 40 1

X

the forward recurrence time


Let

The autocovariance function of {bn}

The autocorrelation function of {bn}


Then

The M/G/1 arrival model is long range dependent if E[X2] = 1. short range dependent if E[X2] < 1.


A Pareto distribution A random variable Y is called to have a Pareto

distribution if its distribution function is given by

where 0 < < 2 is the shape parameter and (> 0) is called the location parameter.

Remarks: If 0 < < 2, then Y has infinite variance. If 0 < · 1, then Y has infinite mean.


The expectation of the Pareto distribution

The distribution of the forward recurrence time Yr of the Pareto distribution


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.


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


A sample path of the M/Pareto arrivals

A trace of X(t)

0

2

4

6

8

10

12

14

1 319 637 955 1273 1591 1909 2227 2545 2863 3181 3499 3817 4135 4453 4771

time slot

nu

mb

er

of

bu

sy

se

rve

rs

= 0.4, = 1.18 , = 0.9153


The autocorrelaton functioncomparison of the autocorrelation

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1 840 1679 2518 3357 4196 5035 5874 6713 7552 8391 9230

k (lag)

auto

corr

elat

ion

synthetic trace value

theoretical value


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.


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=1nj=1

n r(ti,tj) ui uj ¸ 0.


Definition of a joint normal distributionThe 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.


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)).


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


TheoremSuppose 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:


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.


Consider a queueing system with input processAt = t + Yt

where Yt is a normalized fBM,i.e., E[Yt2] = 1.

Then the queue content process q(t) is given byq(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.


A lower bound for the queue length Since Yt has stationary increments, we get


Hence, from the fact that Yt » N(0,t2H) we get

where (x) denotes the distribution function of a standard normal R.V.


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.


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


The complementary distribution function of the Packet interarrival times

exponential distribution


Traffic follows a non-stationary Poisson process at multi-second time scale

points of rate changes

change of free region

relative magnitude ofthe change in the slope


The change of Hurst parameters

Hurst parameters of time intervals of length 20 secthe reasons for change:• self-similarity of the original traffic• the change in routing• the change in the number of active sources


Autocorrelation for the magnitude of rate changes(i.e., the height of the spikes in Fig. 7)

95 % C.I. for 0a negative correlation at lag 1


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


Traffic appears LRD at large time scales

original ACF

ACF using moving averages


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.

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