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

1. References for M/G/1 Input Process
Krunz and Makowski, Modeling Video Traffic Using M/G/1 Input Process, IEEE JSAC, vol. 16, , 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, , 2000
M. Zukerman, T.D. Neame and R.G. Addie, Internet traffic modeling and future technology implications, INFOCOM 2003,
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2. 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 burst generates packets during its holding time.
We assume the system is in the steady state.
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3. The {bn} process
arrivals
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4. 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
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5. 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 1 2 3 4 5 6 7 8 9
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6. The autocovariance function of {bn}
Let
The autocovariance function of {bn}
The autocorrelation function of {bn}
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7. The M/G/1 arrival model is long range dependent if E[X2] = 1.
Then
The M/G/1 arrival model is
long range dependent if E[X2] = 1.
short range dependent if E[X2] < 1.
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8. 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.
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9. The expectation of the Pareto distribution
The distribution of the forward recurrence time Yr of the Pareto distribution
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10. 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.
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11. 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
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12. A sample path of the M/Pareto arrivals
 = 0.4,  = 1.18 , =
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13. The autocorrelaton function
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14. 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, , 1998
M/G/1 input process is used to model video traffic encoded by DCT.
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15. 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.
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16. 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.
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17. 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)).
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18. 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
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19. Suppose that a stochastic process Yt
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:
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20. 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.
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21. q(t) = sups· t [A(t) - A(s) - C(t-s)]
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.
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22. A lower bound for the queue length
Since Yt has stationary increments, we get
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23. Hence, from the fact that Yt » N(0,t2H) we get
where F(x) denotes the distribution function of a standard normal R.V.
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24. Next Generation Communication Networks Lab.

25. 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.
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26. Traffic model in the backbone
T. Karagiannis et. al, A nonstationary Poisson view of internet traffic, INFOCOM 2004,
Traffic appears Poisson at sub-second time scale
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27. The complementary distribution function of the Packet interarrival times
exponential distribution
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28. 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
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29. 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
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30. 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
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31. 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
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32. Traffic appears LRD at large time scales
original ACF
ACF using moving averages
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33. 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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