Transcript traffic

Traffic Modeling
Nelson L. S. da Fonseca
Institute of Computing
State University of Campinas
Campinas, Brazil
Network Workload: “Traffic”
• Telecom networks built to transport
traffic
• Flow of bytes or packets or messages
REDE
Multimedia Networks
Quality of Services
• “Perception of the quality of the
transfer of information expressed by
quantitative metrics”
Aplicações
Aplicações
QoS
QoS
REDE
QoS
Aplicações
QoS
Aplicações
Quality of Service
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Why Traffic Models Matter?
Queuing in Multimedia Networks
Telephone
Computer
Computer
Why is Traffic Modeling
Important?
• Need for performance evaluation and
capacity planning
• Understand the nature of traffic
generated by applications and their
transformation along the network
• Conception of effective traffic control
mechanisms: congestion control, flow
control, routing …
Why is Traffic Modeling
Important?
• Accurate performance prediction
requires realistic traffic models
• Can use in analysis or testing (“synthetic
traffic”)
• Synthetic traffic matches real traces,
is more efficient to use
An Example
• Traffic: 5 synchronous streams of
299,040 cells/sec
• Buffer size B, line rate 312,500
cells/sec
• An ATM cell 53 bytes
An Example
• G/D/1/B or nD/D/1
r = 0.957
– E(W) = 2.7 msec
– W < Wmax = 4 x 3.2 = 12.8 msec
– No losses for B=4 or bigger
• If Poisson: M/D/1, already big difference
– E(W) = r m / 2(1- r) = r 3.2 /2(1- r) = 35.6 msec!
– Need B = 249 for cell loss 10-9!
• Conclusion: Mean is not enough; smoothness or
burstiness affects a lot
Our Goal
• To understand relevance of traffic
characteristics with respect to network
performance and QoS
• Identify and familiarize with common
traffic models
Mean and Variance
E[ X ] 

 XP{X }
X  
VAR[ X ]  E[( X  E[ X ]) ]
2
VAR[ X ] 

 ( X  E[ X ]) P{X }
2
X  
Mean and Variance
Mean amount of work arrived at each slot
=4
Variance =
4.0 x 0.5  4.0 x 0.5  16.0
2
2
Mean and Variance
Mean amount of work arrived= 4.0
Variance of the amount of work arrived
=
4.0 x 0.5  4.0 x 0.5  16.0
2
2
Mean and Variance
• Both processes have the same mean
and variance. But what is their
impact on queuing?
Mean and Variance
• Consider a queue with a server, service
rate of two cells per time unit and
buffer space of two cells.
2/1
Mean and Variance
When the first stream
Feeds the queue
Service = 2/1
No loss!
Mean and Variance
But when the second flow
feeds the queue
2/1
then 1/5 pkts are lost!
Correlation
Low Correlation
High Correlation
Autocovariance
Variance
v=Autocovariance Integral
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
LRD versus SRD
autocorrelation
LRD
SRD
lag
Origins of Correlations…
• Superposition of weakly correlated
sources
• Size of files
• User behavior
• Protocol dynamics
• etc ….
Video
• MPEG
Video
• 500x500 pixels por frame = 250.000
pixels por frame, generating 2
Mbits/frame for an 8-bit gray scale at
a rate of 30 frames/sec
60
Mbits/seg. If three colors are used,
the transmission rate is 180 Mbits/sec
Compression
Variable Bit
Rate
Video
probalility (1E-3)
20
videophone
bit rate
videoconference
scene changes
100-150 frames
15
tv scene
10
5
Frame sequence
5
10
15
bit rate Mbit/s
20
25
30
Variable Bit Rate
BT
SCR
PCR
Burstiness
• Variable with time scales
Traffic Bursts and Scales
• Hierarchical view
– Call
– Burst
– Packet
• How to characterize:
Mean? Peak? Variance?
Autocorrelation?
Burstiness
• Most traffic types in high-speed
networks are bursty
• Burstiness is mainly due to
autocorrelation
• Renewal models assume autocorrelation
away for tractability
• Performance prediction non-realistic
without burstiness
Taxonomy of Models
• Renewal and IID: no dependence
• Short Range Dependent (SRD)
–
–
–
–
–
Markov modulated, Interrupted Poisson Process etc.
Semi Markov
MAP
Fluid: useful for analysis and also for simulation
Regression models – classical statistics
• Long range Dependent (LRD)
– Monofractal (self-similar)
– Multifractal
SRD Models
Voice Source
• Human voice bursts of 0.4 a 1.2 sec
durations and periods of silence of 0.6 a
1.8 sec (exponentially distributed);
• Sampling at 125 mseg intervals with 8
bits coding 170 cells (53 bytes)
ATM/sec
ON
OFF
Superposition of Voice Sources
Superposition of Voice Sources
N   
 i    

 i     
i
  


   
01...i 1
i 
0
1 2 ...i
N i
Superposition of Voice Sources
 N 0  1  0
N 0  [  ( N 1) ]1  2 2  0
...
N  (i 1) i 1  [i  ( N 1) ] i  (i  1) i1  0
...
 N 1  N N  0
Superposition of Voice Sources
  [ 0 , 1... n ]
M  0
 Nλ Nλ 0 ...


α  [α  (N  1)λ (N  1)λ ...


M   0 2α  (N  2)λ  2α ... 


0 0 3α ...


... ... ... ...


Fluid-Flow Equations
• x – buffer occupancy;
• Each voice source generates V cells/sec
during talk spurt with duration of 1/ sec;
• A channel with capacity VC cells/sec has C
units of information per second;
• N sources;
• Fi(t,x) – system state, at time t there are i
sources in state on and the buffer occupancy
is x;
Fluid-Flow Equations
dF0 ( x)
 C
  NF0 ( x)  F1 ( x)
dx
dF1 ( x)
(1  C )
 NF0 ( x)  [(N  1)   ]F1 ( x)  2F2 ( x)
dx
dF2 ( x)
(2  C )
 ( N  1)F1 ( x)  [(N  2)  2 ]F2 ( x)  3F3 ( x)
dx

dFN ( x)
( N  C )
 FN 1 ( x)  NFN ( x)
dx
Fluid-Flow Equations
• Fj(x) = Prob. [j sources on, buffer
occupancy≤x]
N
  i  ij e
i 0
zi x
0 j  N
Markov Modulated Process
• Arrival rate depends on understanding
Markov chain
1
2
n
1
2
n
Superposition of Voice Sources
1
2
1
2
Superposition of Voice Sources
Superposition of Voice Sources
Superposition of Voice Source
2-states MMPP
1 0 
_/\ _  

0

2

 q11 q12 
 

q
q
22 
 21
Markov Modulated Process
• M/G/1 – type
• Efficient algorithms
 B0
B
 1
0

0

P
0

 .
 .

 .
  xP
A0
A1
A2
0
0
.
.
.
0
A0
A1
A2
0
.
.
.
0
0
A0
A1
A2
.
.
.











M/M/1

0
0
1  
  1   

0

 0

1   


P 0
0

1   
 .
.
.
.

.
.
.
 .
 .
.
.
.











Markov Modulated Process
x  xP
x0 

x
v
v 0
Bv

xk   xk  v 1 Av
v 0

x
k 0
k
T
e
1
Markov Modulated Process
single
Arrivals
batch
discrete
Time
continuous
Markov Modulated Process
• Markov Modulated Poisson Process –
MMPP (continuous time, single arrival)
• Batch Markovian arrival process – BMAP
(continuous time, batch arrival)
Markov Modulated Process
• Discrete Time Batch Markovian Arrival
Process – D-BMAP (discrete time, batch
arrivals)
• Discrete Time Markovian Arrival
Process (D-MAP)
Regression Models
• Autoregressive models
• TES: Transform-ExpandSample
– Uses modulo-1 operations
– Correlated numbers with
desired PDF
– Marginal by inverse
transform
– Numerical fitting of
correlation
• DAR(p), for example p=1:
LRD Models
LRD Models
• Monoscaling – Monofractal – self-similar
• Multiscaling - Multifractal
LRD versus SRD
autocorrelation
LRD
SRD
lag
The Hurst Parameter
Long Range Dependence
• Hyperbolic decay of autocorrelation
r (k )  H (2 H  1)k 2 H  2

 r (k )  
k  
if
k 
1 H 1
2
The Parameter H
• The Hurst parameter, H, quanitifes the
degree of self-similarity in a time-series
• 0.5 < H < 1 :
• H = 0.5 :

• 0 < H < 0.5 :
non-summable

k  r (k )  

r (k )   summable
k  


k  
not common
r (k )  0
LRD
SRD
Tail distribution and LRD
Pr{Z   }
c
1 e
Pr{Z   }

Pr{Z   }
 
 1  
 1
Pr{Z   }
  1
t 
Examples
Video
Application
Hust Parameter Peak/Mean
(GOP)
Student siting at
workstation
Simpsons
0.53
0.18
0.88
0.43
Terminator 2
0.80
0.35
Mr. Bean
0.95
4.1
Impact of Long-Range Dependence
• Implications to network design and control may be
crucial
• Self similar models: burstiness across time-scales
• loss probability:
New Era: Self Similarity and LRD
• Observations in early nineties:
– Scaling in Ethernet traffic
– Persistence of video traffic
– Similarly for WAN
• Traffic did not scale or become
smoother as expected
• Poisson assumption questioned
• Birth of “self similar” era
Scaling
• The absence of a
characteristic scale
in a time series.
• The whole and its
parts cannot be
statistically
distinguished from
each other
Self-Similar Traffic (Ethernet)
1 Second Intervals
10 Second Intervals
60 Second Intervals
100 Second Intervals
Half Hour Intervals
1 Hour Intervals
Poisson Process
Interval = 10
Interval = 1
700
650
600
550
500
450
400
350
300
70
65
60
55
50
45
40
35
30
0
20
40
60
80
100
0
20
Interval = 100
40
60
80
100
80
100
Interval = 1000
7000
6500
6000
5500
5000
4500
4000
3500
3000
70000
65000
60000
55000
50000
45000
40000
35000
30000
0
20
40
60
80
100
0
20
40
60
Self-Similar Process
• The sample path of a process X(t) and
those of a re-scaled version cH X(t/c)
by dilatting the time axis by a factor of
c>0 and the amplitude axis by a factor
of H cannot be statistically
distinguished from each other
Self-Similarity
• By analyzing the number of packets or number of
bytes, we observe an invariance of the traffic
behavior in differente time-scales, from
milliseconds to minutes to even hours:
Self-Similar
• Aggregate Process
mi
1
( m)
X (i ) 
 X (t )
m t m (i 1)1
Self-Similar
• X(t) is exactly second-order self-similar with
parameter H (1/2 < H < 1)
2
2H
2H
2H
 (k )  ((k  1)  2k  (k  1) )
2
• X(t) is asymptotically second-order self-similar if

2H
2H
2H
lim  (k )  ((k  1)  2k  (k  1) )
m 
2
2
(m)
Self-Similar
• Self-similar with stationary increments
 (k ) 

2
2
(t
2H
ts
2H
s
• The increment process satisfies
1 H
X d m
var( X
(m)
X
( m)
)  m
2

2H
)
Self-Similar
• The Fractal Brownian Motion is a self
similar with Gaussian stationary
increments and the increment process is
called Fractal Gaussian Noise
Self-Similar
• An aggregate of a large number of onoff sources with Pareto distribution
produces an FBM
The Logscale Diagram
 = 2H + 1
• Confidence intervals
• Linear behavior
The Logscale Diagram
Wavelet Transform
Wavelets from a length-8 Daubechies filterbank. From top to
bottom: y0,0(t), y1,3(t), y3,22(t)
• Projecting a signal onto locally oscillating
waveforms
Scaling (dilating and
compressing) and shifting
y j ,k (t )  2 j / 2y (2 j t  k )
Wavelet Transforms
X (t )   CX ( j0 , k ) j0 ,k   d X ( j, k )y j ,k (t )
k
j  j0 k
CX(j0,k) – coarse-resolution
dX(j,k) – wavelet coefficients – finer scales
Logscale Diagram
• Wavelet coefficients and variance of data
related
•
 1 nj
2
Y j  log( j )  log  d X ( j , k ) 
 n k 1

• Logscale diagram  log(Yj) x Yj
Multifractal
• At small time scales burstiness does not
follow the same pattern at higher scales
• h(t), Hoelder exponents, highly rearying as
a function of t
Multifractal
• When Gaussian, local self-similar
• Non-stationary increments
Multiplicative Cascades
• Initial mass is
iteratively divided
into parts which size
is defined by
• Multipliers 
random variable 
define the weight of
each part
Binomial Cascade
•
X ((2k  1) / 2
n 1
)  X ( 2k / 2
n 1
i
M
 ki ( X (1)  X (0))
i 1
•
M 2nk1  M 2nk11  1
n 1
d
)
Multiscaling Trees
• Binomial cascades – each level of the
tree corresponds to a different level of
aggregation
Multifractal Brownian Motion
• Generalization of Fractal Brownian
Motion
• Gaussian process
• In the neighborhood of t an mBm can be
approximated by an fBm
fBm
mBm
Multiscaling Diagram
j
(q)
 1 / n j k d X ( j , k ) , q  R
• Fractal
q
E X  (t )  Cq 
• Multifractal
q
qH
E X  (t )  Cq 
q
(q)
Multiscale Diagram
• A lack of alignment in the diagram
strongly suggests multifractality
Linear Multiscale Diagram
hq   q
1
q
2
• Fractal – horizontal line
Is it possible to observe both
mono and multi scaling in a trace?
• Yes, but at different time scale due to
aggregation
• Large scale – monoscaling
• Small scale – multiscaling
• Boundary between scaling patterns –
cutoff scaling
Scaling – time domain
Detecting the cut-off time scale
Congestion Window of TCP
Simple Model for the
Throughput of TCP
throughput =
w * MSS
Bytes/sec
RTT
The Inverse-square Root Law
The Inverse-square Root Law
• At a loss of a packet which occurs with
probability p the window size is reduced
from W to W/2
• Number of packets transmitted between
two losses – area of trapezoidal
• T – duration of inter loss period
1 T W

 W 
2 RTT  2

The Inverse-square Root Law
• Number of packet
transmitted = 1/p
• Rate of increase of
window size – one
packet per round
trip time (RTT)
• T = RTT x W/2
W
4
W
 1
W   

 2
 p
W 
8
3p
The Inverse-square Root Law
• Average sending rate
1
p
X ( p) 
RTT  W
1
RTT

2
3
2p
A TCP Model
• Window grows 1/W at every ack reception
• “round” period in which there is a back-toback transmission of W packets. It ends at
the reception of the first ack for this w
packets
• Duration of a round – one RTT
• An ack is sent at every b packets received,
W/b acks received in a round => next round
W’ = W +1/b
A TCP Model
• Packet loss is round independent
• If a packet is lost all subsequent
packets in the round are lost
• Triple duplicate (TD), timeout (TO) and
receiver window limitation
A TCP Model
Triple Duplicate
W
W2
W1
W3
A1
A2
A3
TDP2
TDP1
t
TDP3
LEGENDA
Pacotes enviados
Pacote reconhecido
Pacote perdido
Wi1
2
i
Ocorre 3 ACK's idênticos
TD termina
 i -1
3
2
5
1
4
1
2
b
i
3
Xi
4
b
b
TDPi
No de RTT
Último RTT
A TCP Model
• B(p) – throughput as a function of the
loss rate
• TDP – period between two TD losses
E[Y ]
B 
E[ A]
A TCP Model
• After the first loss, i, occurs Wi – 1 packets
are sent in the round. Thus Yi = i+Wi – 1
packets are sent
E[Y ]  E[ ]  E[W ]  1

E[ ]   (1  p)
k 1
k 1
1
p 
p
k
A TCP Model
1 p
E[Y ] 
 E[W ]
p
E[ A]  ( E[ X ]  1) RTT
• X – round in which loss occurred
A TCP Model
Wi 1
Xi
Wi 

2
b
2
E[W ] 
E[ X ]
b
A TCP Model
Yi 
X i / b 1

k 0
 Wi 1

 k b   i 

 2

X iWi 1 X i  X i


 1   i 

2
2  b

X i  Wi 1


W

1

  i
i
2  2

A TCP Model
1 p
E[ X ]  E[W ]

 E[W ] 
 E[W ]  1  E[  ]

p
2  2

2b
8(1  p)  2  b 
E[W ] 



3b
3bp
 3b 
E[W ] 
 1
8
  
3bp

E[W ] 
8
3bp


p
2
A TCP Model
2b
2b(1  p)  2  b 
E[ X ] 



6
3p
 6 
 2b
E[ A]  RTT 

 6

2

2b(1  p)  2  b 

 1

3p
 6 

2
A TCP Model
1
B p  
RTT
 1 
3

 o
 p
2bp


A TCP Model
Timeout
W
A TCP Model
Timeout
TD
TO
2TO
4TO
t
A TCP Model
Timeout
E[ M ]
B
E[ S ]
Si  Zi
•
•
•
•
TD
 Zi
TO
Mi – number of packets sent during Si
Si – interval including TDi and TOi periods
ZiTO – duration of sequence of time-outs
ZiTD – time interval between two consecutive
time-out sequences
A TCP Model
Timeout
ni
E[ M ]  E[ Yij ]  E[ Ri ]
j 1
E[ S ]  E[
ni
 A ]  E[Z
j 1
ij
TO
]
• Yij – the number of packets sent in the jth TD period
in the interval ZiTD
• Aij – the duration of each TD period
• Ri – number of packets sent during timeout period
• ni – number of TD periods in ZiTD
A TCP Model
Timeout
E[Y ]  Q  E[ R]
B
E[ A]  Q  E[ Z TO ]
Q  min(1, 3 w)  E[W ]
1
E[ R] 
p
2
3
4
5
6
1

p

2
p

4
p

8
p

16
p

32
p
E[Z TO ]  T0
1 p
A TCP Model
Timeout
• Q – probability that a loss ending a TDP
is a TO
• w – number of packets in a round
• Number of TOs in a TO sequence –
geometric distribution
• Timeout increases duration according to
2i-1 T0 up to i =6
A TCP Model
Time Out
B p  
1

2bp
3bp 
2


RTT
 T0 m in 1, 3
p 1  32 p

3
8 



W
A TCP Model
Window Limitation
TD
TO
2TO
4TO
t
A TCP Model
Window Limitation
Wmax
Wmax U i


2
b
U i Wmax
Yi 
(
 Wmax )  ViWmax
2
2
• Window grows for U periods and stay
with its maximum value during V periods
A TCP Model


 Wmax
1
B p   min
,
 RTT RTT 2bp  T min1, 3 3bp  p 1  32 p 2
0



3
8












A TCP Rate Expression with
General Loss Distribution
1
X
RTT p

1 R(0)   v k R(k )
2
k 1
• X – sending rate
• R – correlation between intervals with (multiple)
packet loss
• v– multiplicative increase factor of TCP window
TCP + RED

W t  
1
W t  W t  Rt 


 pt  Rt 
Rt 
2
Rt  Rt 
N t 


C

W t ,
q 0


Rt 

qt   


m ax 0 ,C  N t  W t , q  0
Rt 




R t  
q t 
 Tp
C
Link Dimensioning with TCP
PS - bottleneck
IS – TCP connections
Envelope Processes
• Needed for admission control, link
dimensioning etc
• Exact solution typically impossible
• Classical queuing: approximate fit and
numerical solutions
• More modern approaches:
– Effective or equivalent bandwidth
– Envelope processes
Envelope Processes
Upper bounds to the amount of
traffic arrived
Envelope Process
• Deterministic EP  absolute upper
bound  deterministic guarantees
• Probabilistic EP  allows violation
defined by violation probability 
statistical guarantees
QoS Guarantees
• Deterministic guarantees
Pr{delay  d}  1
• Statistical guarantees
Pr{delay  d }  
Deterministic Envelope Process
A(t1, t2 )  Aˆ (t2  t1 )t1  t2
• If EP is increasing and sub-additive , the long
term rate average rate exists (the envelope rate)
•
Aˆ (t )
ˆ  lim
t 
t
EP is not unique - Minimum Envelope Process
(MEP) – the tigthest bounding function
• subadditive ( A(t1  t2 )  A(t1 )  A(t2 ))
Deterministic Envelope Process
Cruz’s Envelope Processes (,)
A(0, t )    t
A(0, t )
lim

t 
t
unregulate
d traffic
token
buff
er
*≤
 tokens
 tokens/sec
Example of
a leaky
bucket
regulator
VBR Video
The (ˆ , ˆ ) EP
• Concave, piecewise linear function
• n (, r) pairs
• In a time interval t, the traffic is
restricted to
Aˆ (t )  min {   t}
1i n
i
i
• Enforcement by a cascade of Leaky
buckets
• T-Spec in Intserv adopts a dual leaky
bucket
The (ˆ , ˆ ) EP
The D-BIND EP
• Multi rate-interval process {(Rk, Ik), k = 1,2..K}
piecewise linear function
Rk I k  Rk  1I k 1
ˆ
A(t ) 
(t  I k )  Rk I k , I k 1  t  I k
I k  I k 1
How many EP’s?
Queuing with EP’s
• The Lindley’s equation
q(t  1)  max{0, q(t )  a(t )  c}
Queuing with EP’s
Queuing with EP’s
Queuing with EP’s
Convolution
min-plus Calculus
• Replaces integral with infimum
operation and the multiplication
with summation
A  B  inf { A( )  B (t   )}
 R
Convolution
min-plus Calculus
Convolution
min-plus Calculus
Service Curve
• Minimum Service Curve
B(t )  A  S (t )
Service Curve
Network Calculus
Tandem Network
Probabilistic EP’s
• Allow a certain amount of violating
traffic to the EP bound
• Leads to higher utilization of resources
Bounded Burstiness Processes
• The cumulative traffic is allowed to exceed its
EP but a probability decaying with the degree
the EP is exceeded
• Stochastically bounded burstiness with upper
rate  and bounding function f ():


n

f
(
u
)(
du
)
 


ntimes
Pr{A(s, t )   (t  s)   }  f ( )
Bounded Burstiness Processes
• Superposition Ai(t) = {i, fi()} i=1,2
A1(t) + A2(t) = {1 +2, f1(p)+f2((1-p))},
where 0<p<1, is also SBB
• For working conserving service and A(t)
SBB, the output process is SBB with {,
f()+(1/c –) } and the queue is bounded
• Exponential Bounded Bourstiness
f() = e- where  and  are source
specific (also sum of exponential)
Chang’s EP
• Deterministic bound to the
moment generating function of
A(t), i.e., the Fourier transform of
A(t)
1

log Ee
A( t1 ,t 2 )
 Aˆ ( , t 2  t1 ), t1  t 2
Kurose’s Bound
• The traffic flow is characterized by a family
of random variables {B(t1),B(t2),…} that
stochastically bounds the source over
respective intervals with length tk
• In working conserving server the output is
bounded by the same bounding family of the
input
• Delay bounded
Rate Variance EP
• Rather than bounding the cumulative
traffic, the rate variance EP bounds the
variance over intervals of length t
 A( s, s  tk ) 
RV (tk )  VAR

tk


• The accumulated process
 A (s, s  t )
i
i
k
can be bounded by a Gaussian process
t
i k
2
RVi (t k )
Fractal Envelope Process
W (t  u )  W (t ) 
lim 
 BH (t ) (u )uR 

H (t )
 0

uR 

Â(t )  at  t
   2 log
H
Multifractal Envelope Process
fBm
Multifractal Envelope Process
t
Â(t )   a  κσH(x)x
H(x)1
0
Â(t )  at  t
H
dx
Multifractal Envelope Process
Multifractal Envelope Process
Multifractal LB
A Comparison
Queue fed by an EP
Queue fed by an EP
Statistical Multiplexing
Envelope Process of an
Aggregate of Multifractal Flows
0
1
2
N
 N 2


2 H i ( x ) 1
2 2 Hi ( x) 
ai     i H i ( x) x
  i x
 dx

i 1
 i 1
 i 1

T N
Â(t )  

Envelope Process of an
Aggregate of Multifractal Flows
Equivalent Bandwidth
The Bandwidth needed to provide
the required Quality of Service
to a Flow
2
(t)
1
0
0.5
1
1.5
2
2.5
3
1.5
2
2.5
3
1.5
2
2.5
3
2.5
3
-1
2
ABW(t) = C - (t)
1
0
0.5
1
-1
C
2
1
0
1 t
 ( s ) ds
t 0
0.5
1
-1
ABW (t   , t ) 
1
C   ( s)  ds


 
t
t
2
1
0
-1
1 t
 C   ( s)  ds

0
t
0.5
1
1.5
2
Effective (Equivalent) Bandwidth
1

log Ee
A( t1 ,t 2 )
 Aˆ ( , t 2  t1 ), t1  t 2
Effective (Equivalent) Bandwidth
• Deterministic bound to the
moment generating function of
A(t), i.e., the Fourier transform of
A(t)
• Replace z in the z-transform by

e
Effective (Equivalent) Bandwidth
Ak (t1 ,t2 )
E[e
Ak (t1 ,t2 )
ln E[e
Aˆk ( ,t2 t1 )
]e
]  Aˆk ( , t2  t1 )
Aˆk ( , t2  t1 )  aˆk ( )(t2  t1 )  ˆ k ( )
Effective (Equivalent) Bandwidth
q(t  1)  max{0, q(t )  a(t )  c}
E[e
q ( t )
t
]   E[e
 ( A( t  s ,t )  sC )
]
s
K
Aˆ ( , t )  aˆ ( )t  ˆ ( )   aˆk ( )t  ˆ k ( )
k 1
K
ˆ ( )  ˆ k ( )
k 1
K
aˆ ( )   aˆ k ( )
k 1
Effective (Equivalent) Bandwidth
E[e
q ( t )
ˆ ( )
]e
t
e
s[ aˆ ( ) C ]
s 0
t 
aˆ ( )  C
Effective (Equivalent) Bandwidth
• Chernoff Bound
x
q
P[q  x]  e E[e ]
• Space Parameter
1
1
  ln( )
x PL
Effective (Equivalent) Bandwidth
11
Ak (t )
lim
ln[ Ee
]
t  t 
Effective (Equivalent) Bandwidth
Poisson Source
Ak ( t )
E[e
]e
Ck  k
PL  105
9
PL  10
 k t (1e )

e 1

  11.5 / x
  20.7 / x
Equivalent Bandwidth of
Multifractal Flows
N
 N 2


2 H i ( t ) 1
2 2 H i (t ) 
   i H i (t )t
  i t

 i 1
 i 1


1
2
N
 C   ai
i 1
2
(t)
1
0
0.5
1
1.5
2
2.5
3
1.5
2
2.5
3
1.5
2
2.5
3
2.5
3
-1
2
ABW(t) = C - (t)
1
0
0.5
1
-1
C
2
1
0
1 t
 ( s ) ds
t 0
0.5
1
-1
ABW (t   , t ) 
1
C   ( s)  ds


 
t
t
2
1
0
-1
1 t
 C   ( s)  ds

0
t
0.5
1
1.5
2