Modelling internet packet traffic congestion in networks using

Modelling internet packet traffic congestion in networks using

MODELLING INTERNET PACKET TRAFFIC CONGESTION
IN NETWORKS (USING CHAOTIC MAPS)
DAVID ARROWSMITH
SCHOOL OF MATHEMATICAL SCIENCES
QUEEN MARY, UNIVERSITY OF LONDON
LONDON E1 4NS, UK
www.maths.qmul.ac.uk/~arrow/sharjah
university of sharjah 8/5/2003
COMMUNICATIONS - PACKET TRAFFIC
empty packet
non-empty packet
012
time
n
1994 - "On the self-similar nature of Ethernet traffic", Leland et al, Comp Comm Rev
"bursty activity" of packet rates over many time-scales
conventional "Poisson" models for over optimistic on queue performance
"Poisson-like" view of aggregated sources suggests smoother traffic
increasing bandwidth/capacity is not always appropriate
SRD and LRD OUTPUT
N= BATCH SIZE of AVERAGED DATA
1
1
0.8
0.8
0.6
0.6
N=100
0.4
0.2
0.4
0.2
200
400
600
800
1000
1
1
0.8
0.8
0.6
0.6
0.4
N=10000 0.4
0.2
0.2
200
400
600
800
Short range dependent
1000
200
400
600
800
1000
200
400
600
800
1000
Long range dependent
AUTOCORRELATION
BINARY TIME SERIES:
Xt ∈ {0, 1},
t = 1, 2, 3, . . .
MEAN:
2
µ = E(Xt ) = E (Xt )
NEW TIME SERIES- averages of batch sizes N:
VARIANCE:
Var(Xt ) =
E(Xt2 )
2
( )
Yt N
E(Xt )
1
=
N
(t+1)N
X
Xt
j=tN +1
AUTOCORRELATION FORMULA :
E(Xt Xt+k ) E(Xt )E(Xt+k )
E(Xt Xt+k )
µ
p
c(k) =
=
µ (1 µ )
( Var(Xt )Var(Xt+k ) )
AUTOCORRELATION FOR LRD TRAFFIC :
POWER
LAW DECAY
c(k)
k
−β
, β ∈ (0, 1)
2
AUTOCORRELATION FOR SRD TRAFFIC :
EXP
DECAY
c(k)
α− k , α > 1
ITERATION OF MAPS
ORBIT WEBS
1
initial point
x0
iteration
xn = f ( xn-1 )
orbit
x0 , x1 , x 2 , x 3 ...
sequence of orbital points
is geometrically
shown by the WEB of lines
0
0
x0 d
x1
1
CHAOTIC MAPS
and DIGITAL OUTPUT

x n +1
orbit
8
2
x0 , x1 , x2 , x3 , . . .
3 6
5
1
4 7
0

OFF

ON
0
xn 

output from orbit of map
time n
CODING OF ORBITS
ON
OFF
OFF
OFF
DIGITAL
OUTPUT
0 0 0 0 1 1
0 0 1 1 0 0
0 1 0 1 0 1
ON
1
1
0
ON
OFF
ON
map produces all possible
OFF/ON sequences
1
1
1
0 
time n
INTERMITTENCY EFFECT
obstruction
slow change long range
dependence
fast change short range
dependence
Intermittency allows small incremental changes in the orbit
INTERMITTENCY and LINEAR MAPS
d
intermittency map
f (x) = x+ax
m
tangency with y=x at x=0
implies intermittency at x=0
and small iterative changes
in the values of x
PROBABILITY of ESCAPE
to the region x > d in more than n
iterations of the map f
is given by
INTERMITTENCY(POWER LAW DECAY)
Prob (n) ~ n
-(2-m)/(m-1)
LINEAR(EXPONENTIAL DECAY)
Prob (n) ~ 2 -n
0
0
x0
d
OUTPUT COMPARISON OF MAPS
1
0.8
A
0.6
0.4
0.2
50
100
150
200
50
100
150
200
250
300
BURST
0.8
B
0.6
0.4
0.2
OFF
BURST
250
300
STATISTICAL OUTPUT
( scales 10 100 1000 )
Poisson process
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
200
400
600
800
1000
200
400
600
800
200
1000
400
600
800
1000
Piece-wise linear map
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
200
400
600
800
1000
200
400
600
800
1000
200
400
600
800
200
400
600
800
1000
Piece-wise intermittency map
1
0.6
0.4
0.8
0.5
0.6
0.4
0.3
0.3
0.2
0.4
0.2
0.1
0.2
0.1
200
400
600
800
1000
200
400
600
800
1000
1000
DYNAMICS - STATISTICS INTERFACE
NEW AREA OF RESEARCH
INTERMITTENCY IN DYNAMICS
- CORRELATION IN STATISTICS
- LONG RANGE DEPENDENCE IN APPLICATIONS
CONJECTURES (Erramilli, Leland and others, 1995)
AUTOCORRELATION
SINGLE INTERMITTENCY MAPS (SCHUSTER)
DOUBLE INTERMITTENCY MAPS (BARENCO and A.)
fi (x) =
{
x + axm1 ,
x - b(1 - x)m2
0<x<d
d<x<1
c (k)
k
−γ
γ =(2− m )/(m−1),
Max m i
OUTPUT LOAD
invariant measure of map
enables calculation of LOAD = number of packets/unit time
ratio of 0's to 1's
invariant
measure
map
d
1-d
d
invariant
measure
map
d
1-d
B
A
1-d
= area A : area B
load = 1-d
load = area B
A
d
B
1-d
INTERMITTENCY and AUTOCORRELATION
The map segments (Erramilli et al. ) are
m
m
1
f (x) = x + ax
, x < d , and, f (x) = x - b(1-x) 2 , x > d
- there are tangencies with y = x at x = 0 and x=1 respectively.
Usual description of packet rate behaviour is the HURST parameter
H=1- β/2 ε [0.5,1] where β ε [0,1] is the correlation decay coefficient.
Erramilli (cj. 1995) H= (3m - 4)/ (2m - 2) where m = max (m1 , m )
2
Hurst parameter
H=0.5 for m=1.5
H=1.0 for m=2.0
SRD
LRD
RECTANGULAR GRID NETWORK MODEL
hosts can source,
transfer and receive
packets
random host destination
routers can transfer
packets
every node has a buffer
for queueing packets
packets at head of
queue move one step
closer to destination for
each time step
ONSET of CONGESTION
`Phase transition' for Poisson-like and LRD traffic
LRD Traffic Source
Poisson-Like Traffic Source
10
4
0.4
0.35
10
10
3
throughput
average lifetime
0.3
2
0.25
0.2
0.15
0.1
0.05
=0.16
10
=0.16
1
0
0.2
0.4
0.6
0.8
1
load
0
0
0.2
0.4
0.6
0.8
1
load
SRD
LRD
using intermittency
ρ
=0.16
LRD Traffic Source
Poisson-Like Traffic Source
10
4
0.4
m1=m2=1.95
10
10
0.3
3
throughput
average lifetime
m1=m2=1.95
2
0.2
0.1
10
1
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
load
10
4
0.6
0.8
1
0.6
0.8
1
0.3
3
2
0.2
0.1
10
1
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
load
10
load
4
0.4
m1=m2=1.5
m1=m2=1.5
10
10
0.3
3
throughput
average lifetime
1
m1=m2=1.8
throughput
average lifetime
10
0.8
0.4
m1=m2=1.8
10
0.6
load
2
0.2
0.1
10
1
0
0.2
0.4
0.6
load
0.8
1
0
0
0.2
0.4
load
10
10
10
10
4
10
average router queue length - Poisson
average host queue lehgth - Poisson
COMPARATIVE
QUEUE
LENGTHS
10
3
2
1
0
10
10
10
10
4
λ = 0.156
λ = 0.294
λ = 0.303
λ = 0.389
3
2
1
0
HOST/ROUTER - LRD/POISSON
0
2
4
6
8
10
x 10
pre-critical load
λ = 0.294
post-critical load
λ = 0.303
heavy congestion
λ = 0.389
10
10
10
10
10
4
10
3
2
1
0
0
2
4
6
8
2
4
6
8
10
x 10
4
10
x 10
average router queue length - LRD
λ = 0.156
average host queue length - LRD
below critical load
0
4
10
10
10
10
4
4
3
2
1
0
0
2
4
6
8
10
x 10
4
NETWORK MODELS
different types of graph:
REGULAR:
(a) triangular
(b) hexagonal,
and
(a)
(b)
(c) SMALL-WORLD
random connections added
to increase connectivity
(d) SCALE-FREE
−γ
Prob(vertex valency =n) = n
with exponent γ [2,3].
∋
(c)
(d)
DEPLETED RECTANGULAR LATTICE
probability of link deletion = 0.25
DEPLETED LATTICE QUEUES
Transmission control algorithm-congestion
Fukada et al (2000)
Success
No
Idle
(n=0)
Collision
Send
Yes
Waiting time
n
1 <= k < 2
Yes
n < backoff limit
Increment
backoff
counter
(n++)
No
n = backoff limit
n: collision counter
k: waiting time
TRANSMISSION CONTROL PROTOCOL (TCP)
TCP gives CLOSED LOOP MODELS
Packets transmitted only if earlier packet arrival is confirmed
There is a current `window' size for the number of packets that can be sent.
The packet production at node
fi (x) =
i at time
x + axm1 ,
x - b(1 - x)m2
Startup mechanism for TCP at node
If window size is
t=n
0<x<d
d<x<1
dout(x) =
0,
1
0<x<d
d<x<1
i
wi (n), the source will send pi (n) = min(wi (n), si (n))
Residual file size =
si (n) no of iterates of f to the next transition of "ON-OFF''
TCP DYNAMICS AT START-UP
xi ( n) < d “OFF”
wi (n + 1) = 0
xi (n + 1) = f (xi (n))
xi (n) > d “ON”
wi (n + 1) =
1,
if xi (n - 1) < d “OFF”
min(2wi (n), wmax ), otherwise
xi (n + 1) = f pi (n) (xi (n))
CRITICAL LOADS - various networks
Packet lifetime τ
dependence on
the scaled packet
load in the network
λ − packet rate at hosts
d av − average length
of route
There are two clusters the average lifetimes
of packets for scale-free
(SF) graphs go critical
much earlier than those
for small world (SW) and
regular grid (R) networks
450
SW/R
SF
τ
400
350
300
packet lifetime
ρ − density of hosts
500
250
200
150
100
50
0
0
0.2
0.4
0.6
0.8
1
ρ λ d av
- criticality
Dt = total distance of packets to
destination at time t LxL grid
2
D
= D + ρλ L dav
t+1
t
-L
2
Local/Global Critical points ( λ c' / λc )
Mean field theory
1.4
Values of λ c' measured from simulation
Values of λ c measured from simulation
1.2
1
0.8
λc = 1/(ρ dav )
0.6
0.4
0.2
λ'c = 1/(ρ dav- ρ +1)
0
0
0.2
pre-congestion <=> non-increasing D
t
0.4
0.6
0.8
1
ρ
Host density
0.8
λc
=
1/ ρ d
av
0.7
0.6
0.5
λc = 1/(ρ dav )
0.4
0.3
0.2
0.1
λc' = 1/(ρ dav- ρ +1)
0
0
0.2
0.4
0.6
Host density
0.8
1
ρ
MODELLING CONGESTION
LRD can arise from various sources data streams, network structures, aggregation
modelling techniques have been developed to
introduce LRD, with varying load, into networks
erratic onset of congestion is a robust feature of
LRD modelling in various networks
ultimate aim is to provide effective control protocols
which respond efficiently to bursty congestion
onset behaviour