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Internet Analysis - Performance Models -. G.U. Hwang Next Generation Communication Networks Lab. Division of Applied Mathematics KAIST. 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 - PowerPoint PPT Presentation
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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.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST3
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.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST4
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST5
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST6
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST7
Let
The autocovariance function of {bn}
The autocorrelation function of {bn}
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST8
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. Division of Applied Mathematics, KAIST9
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.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST10
The expectation of the Pareto distribution
The distribution of the forward recurrence time Yr of the Pareto distribution
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST11
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. Division of Applied Mathematics, KAIST12
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. Division of Applied Mathematics, KAIST13
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST14
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST15
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. Division of Applied Mathematics, KAIST16
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.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST17
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.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST18
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. Division of Applied Mathematics, KAIST19
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. Division of Applied Mathematics, KAIST20
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:
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST21
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. Division of Applied Mathematics, KAIST22
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.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST23
A lower bound for the queue length Since Yt has stationary increments, we get
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST24
Hence, from the fact that Yt » N(0,t2H) we get
where (x) denotes the distribution function of a standard normal R.V.
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST25
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST26
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. Division of Applied Mathematics, KAIST27
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. Division of Applied Mathematics, KAIST28
The complementary distribution function of the Packet interarrival times
exponential distribution
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST29
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST30
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST31
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
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST32
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. Division of Applied Mathematics, KAIST33
Traffic appears LRD at large time scales
original ACF
ACF using moving averages
Next Generation Communication Networks Lab. Division of Applied Mathematics, KAIST34
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.