The Impact of Long-Range-Dependent
Traffic on Network Performance
George Lin
Ph.D. Defense Presentation
Aug. 18, 2000
Outline
Introduction
– long-range-dependent traffic
– motivation
– overview and contributions
Performance analysis with LRD input traffic
– off-line performance analysis
finite buffer queueing analysis
reassembly and multiplexing queueing analysis
– on-line sensitivity queueing analysis
– summary
Concluding remarks and future directions
Aug. 18, 2000 George Lin - Defense Presentation 2
Introduction
Long-Range-Dependent Traffic
Network traffic exhibits long-range-dependent (LRD)
property
– LAN
– WAN
– Internet (WWW)
– VBR video
Implication
– autocorrelation function decays slower than exponential
– bursty at a wide range of time scales
Aug. 18, 2000 George Lin - Defense Presentation 4
Long-Range-Dependent Traffic
Ethernet trace Poisson arrivals / i.i.d. packet sizes
1.5 104 1.5 104
Bytes per sample
Bytes per sample
1 104 1 104
5000 5000
0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
sample count (Sample interval = 0.01 sec.) sample count (Sample interval = 0.01 sec.)
1 105 1 105
Bytes per sample
Bytes per sample
5 104 5 104
0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
sample count (Sample interval = 0.1 sec.) sample count (Sample interval = 0.1 sec.)
4 106 4 106
Bytes per sample
Bytes per sample
2 106 2 106
0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
sample count (Sample interval = 10 sec.) sample count (Sample interval = 10 sec.)
Aug. 18, 2000 George Lin - Defense Presentation 5
Long-Range-Dependent Traffic
LRD Traffic Models
– Fractional Gaussian noise (FGN)
– Fractional autoregressive integrated moving average (FARIMA)
– Wavelet
Heavy-tailed on/off sources
compelling physical evidence
file sizes and packet trains are heavy tailed
M/G/
– video, FTP, TELNET; source behavior
– model LRD traffic with G having a heavy tailed distribution
– model SRD traffic with G having an exponential distribution
Aug. 18, 2000 George Lin - Defense Presentation 6
Motivation
It’s important to study the impact of LRD traffic
Off-line performance analysis
– existing methods
assume infinite buffer queues driven by LRD input traffic
study the asymptotic tail behavior of the buffer overflow probability
– buffer overflow probability is the probability that the buffer occupancy exceeds a given threshold
value in the steady state
– potential limitations of existing methods
asymptotic tail behavior often capture only the most slowly decreasing term of
the buffer overflow probability
apply to a limited range of parameter values
crucially rely on the infinite buffer assumption
– derive the buffer overflow probability rather than the loss probability
– loss probability is the fraction of lost work in the steady state
Aug. 18, 2000 George Lin - Defense Presentation 7
Motivation
On-line sensitivity queueing analysis
– existing methods
determine performance sensitivities with respect to network parameters
– determine how performance measures vary with the changes in network parameters
determine the performance sensitivities by observing a sample path
– potential limitations of existing methods
only determine performance sensitivities with respect to continuous
parameters
crucially rely on the Markov structure of the systems
Aug. 18, 2000 George Lin - Defense Presentation 8
Overview and Contributions
Study the impact of LRD traffic
– Finite buffer qeueing analysis
develop an off-line method which assumes a buffer with finite capacity and
are based on non-asymptotic method
determine the significance of finite buffer assumption and non-asymptotic
analysis
– loss probability
– we show that existing analysis assuming infinite buffer significantly underestimates the network
performance
– Reassembly and multiplexing queueing analysis
develop off-line methods which are based on non-asymptotic analysis
determine practical impact of LRD traffic and non-asymptotic analysis
– buffer overflow probability
– frame loss probability
– for reassembly queueing, we show that LRD traffic has no significant impact
– for multiplexing queueing, we show that existing asymptotic analysis significantly
underestimates the impact of LRD traffic when the buffer size is small
Aug. 18, 2000 George Lin - Defense Presentation 9
Overview and Contributions
Study the impact of LRD traffic
– On-line sensitivity queueing analysis
develop an on-line method which utilizes the proportional relationship
determine performance sensitivity with respect to discrete parameters for a
queueing system with LRD traffic
– loss probability
– mean queue length
– mean delay
– we show our method is useful for systems which are not amenable to existing on-line methods
Aug. 18, 2000 George Lin - Defense Presentation 10
Performance Analysis with LRD
Input Traffic
Off-Line Performance analysis
Finite buffer queueing analysis
Reassembly and multiplexing queueing analysis
On-line sensitivity queueing analysis
Summary
Finite Buffer Queueing Analysis
Goal: Investigate the significance of the finite buffer
assumption
– analyze the performance of a network multiplexer with finite
buffer capacity
network multiplexers are fundamental building block for sharing network
resources such as bandwidth and buffer space
– obtain loss probability based on non-asymptotic method
rather than buffer overflow probability
determine the implication of using buffer overflow probability as the
performance measure (or to approximate loss probability)
– determine the impact of LRD traffic
Aug. 18, 2000 George Lin - Defense Presentation 12
Queueing Model
General fluid input process
– include a large class of LRD
and SRD processes
M/G/
Gaussian process
– the instantaneous input rate Lu
Lu takes on integer values
P[Lu=k], E[Lu] t
total work flow in [0,t], L u du.
if (0 < X u B)
0
Buffer with finite capacity B
dX u L u c or ( X u , 0, L u c) Single server with constant
output rate c
du or ( X u B, L u c)
0
otherwise buffer occupancy Xu
Aug. 18, 2000 George Lin - Defense Presentation 13
Loss Probability
~
Let X t denote the stationary buffer occupancy in the steady state,
~
X t lim X s
s
~
Let L t denote the stationary input rate in the steady state,
~
L t lim L s
s
amount of lost work in [0,s]
Loss Probability lim
s amount of arriving work in [0,s]
E[ work lost rate]
E[work arriving rate]
1
~ ( k c) P[ X t B, L t k ]
~ ~
E [ L t ] k c
Aug. 18, 2000 George Lin - Defense Presentation 14
Loss Probability
The joint probability in the loss probability is given as follows.
P[ X t B, L t k ] (c n)
~ ~ ~ ~
P[Ws w, L t s n, L t k ]ds ,
0 n0
w w B cs
t
~
where Ws L u du.
t s
Proposition If the following condition holds:
i) the input process in the steady state is stationary and ergodic, and
ii) the average input rate is less than the constant output rate,
then buffer full probability is given as follows.
Buffer Full Probability = P[ X t B] (c n)
~ ~
P[Ws w, L t s n]ds ,
0 n0
w w B cs
t
~
where Ws L udu.
t s
Aug. 18, 2000 George Lin - Defense Presentation 15
Loss Probability
High level proof for the Proposition:
By extending Benes analysis, we show that
P[ X t x ] 1 (c n)
~ ~ ~
P[Ws w, L t s n, X t s {0, B}]ds
0 n
w w x cs
1 ( c n)
~
P[Ws w, L t s n]ds
0 n
w w x cs
~ ~
( c n) P[Ws w, L t s n,0 X t s B]ds
0 n
w w x cs
~ ~
P[ X t B] 1 (c n) P[Ws w, L t s n]ds
0 n
w w x cs
~ ~ ~
P[ X t B] 1 P[ X t B] (c n) P[Ws w, L t s n]ds
0 n
w w x cs
Aug. 18, 2000 George Lin - Defense Presentation 16
Finite Buffer Queueing Analysis
Example
~
L t is characterized by a M / G / process
M/P/ , H1=0.55,
M/P/ , H3=0.9,
M/M/
~
~ ( E[ L t ]) k E [ Lt ]
~
P[ L t k ] e
k!
~
System load = E[ L t ] / c
c = 1.55 Mbps
Aug. 18, 2000 George Lin - Defense Presentation 17
Finite Buffer Queueing Analysis
Results
M/P/ , H1=0.55
This figure compares loss probability
System load = 0.77, 0.58, 0.38 with buffer overflow probability
– buffer overflow probability is the
probability that buffer occupancy
exceeds a given threshold value,
where the threshold value equals to
the buffer capacity of the
corresponding finite buffer system
– simulation results agree with our
analysis.
– buffer overflow probability
significantly overestimate the loss
probability
Aug. 18, 2000 George Lin - Defense Presentation 18
Finite Buffer Queueing Analysis
Results
This figure shows the impact of
LRD traffic when buffer size is
small
– LRD traffic has significant
impact on network performance
even when the buffer size is
small
Aug. 18, 2000 George Lin - Defense Presentation 19
Summary of Finite Buffer Queueing
Analysis
Investigate the significance of finite buffer assumption
– Buffer overflow probability (existing analysis) significantly
overestimates the loss probability, and designing networks using
buffer overflow probability as the performance measure will cause
inefficient network utilization
– Existing analysis underestimate the impact of LRD traffic when
the buffer capacity is small
Aug. 18, 2000 George Lin - Defense Presentation 20
Performance Analysis with LRD
Input Traffic
Off-Line Performance analysis
Finite buffer queueing analysis
Reassembly and multiplexing queueing analysis
On-line sensitivity queueing analysis
Summary
Reassembly and Multiplexing
Queueing Analysis
Goal: study the buffer requirements of reassembly and
multiplexing operations in networks and determine
practical impact of LRD traffic based on non-asymptotic
mehod
– intermediate network elements
routers
connectionless servers
– interworking units
network gateways
application gateways (e.g., transcoders)
Aug. 18, 2000 George Lin - Defense Presentation 22
IP over ATM
Exit Router
Packets ATM
Switch
Higher Higher
Layer Layer
IP IP IP
AAL AAL AAL
ATM ATM ATM ATM ATM
PHY PHY PHY PHY PHY
Aug. 18, 2000 George Lin - Defense Presentation 23
Queueing Model
Frame
ON/OFF Work Batches Reassembly Aggregated LRD Input Process
Sources E[A] Queue – ON/OFF Sources
R Aggreg.
1 LRD X1 Frame – Work-Batches; frames (MTU), cells
Multiplexing
R Input + Queue – M/G/; , E[A],
2 Process X2
– LRD, SRD
MTU
R +
Nt XNt Frame Reassembly Queue
– accumulate and reassemble
– infinite buffer with a given threshold value
ON/OFF Source Frame Multiplexing Queue
Work-Batch (A)
– re-segment and transmit
ON OFF
State R 0 State – infinite buffer with a given threshold value
Cell T3 Frame (MTU)
T2 T1
Aug. 18, 2000 George Lin - Defense Presentation 24
Differences in Queueing Models
Finite buffer queueing analysis Reassembly and multiplexing
– assume general fluid input queueing anaylsis
process, and use M/G/ as an – assume M/G/ process with
example the notion of frame (because
– multiplexing queue with finite we obtain frame loss
buffer capacity probability)
– reassembly queue with infinite
buffer
– multiplexing queue with
infinite buffer
Aug. 18, 2000 George Lin - Defense Presentation 25
Analysis of the Both Queues
Performance measures
– buffer overflow probability
the probability that buffer occupancy exceeds a certain threshold value
in an infinite buffer system
provides an upper bound to loss probability of the corresponding
finite buffer system
– frame loss probability
the ratio between the number of lost frames and the number of
total frames in the steady state
a frame with cells arriving when the buffer occupancy exceeds
the threshold value is lost
Aug. 18, 2000 George Lin - Defense Presentation 26
Frame Reassembly Queue Results:
Impact of LRD Traffic
• This figure indicates that
LRD traffic and Markov
traffic yield similar queueing
behavior
Aug. 18, 2000 George Lin - Defense Presentation 27
Frame Multiplexing Queue Results:
Impact of LRD Traffic
• The Figure indicates that
LRD traffic and Markov
traffic yield similar behavior
when the buffer size is small,
but yield diverse behavior
when the buffer size is large.
Aug. 18, 2000 George Lin - Defense Presentation 28
Summary of the Reassembly and
Multiplexing Queueing Analysis
Frame reassembly operation
– LRD does NOT have a significant impact
finite MTU size reduces the negative effects of LRD
– MTU size has a significant impact
Frame multiplexing operation
– LRD has a significant impact
especially when target loss probability is small
– MTU size is not a factor
Aug. 18, 2000 George Lin - Defense Presentation 29
Performance Analysis with LRD
Input Traffic
Off-Line Performance analysis
Finite buffer queueing analysis
Reassembly and multiplexing queueing analysis
On-line sensitivity queueing analysis
Summary
On-Line Sensitivity Queueing Analysis
Goal: develop a new on-line performance sensitivity
estimation method for systems with discrete parameters
and LRD traffic
– examine in real-time how performance measures would vary with
the changes in system parameters
example application: simulate the system with a given set of parameters, then
obtain the entire performance measure vs. system parameter(s) curve with the
simulation data
– discrete parameters + LRD traffic raise difficulties
exiting methods rely on the partial or complete knowledge of the Markov
structure of the system
Aug. 18, 2000 George Lin - Defense Presentation 31
Overview of Our Method
Proportional Relationship Method Obtain the steady state probabilities
of the nominal system from the
observed sample path
Obtain the performance measure of
the nominal system
Obtain the steady state probabilities
of the perturbed system by utilizing
the proportional relationship
Obtain the performance measure of
the perturbed system
Calculate the differences between
the performance measures of the
two systems
Aug. 18, 2000 George Lin - Defense Presentation 32
Queueing Model
Controlled stream
– customers arrive in batches
– batch size is probabilistically determined
by the queue length
Uncontrolled stream
– customer arrive in batches
– batch size is governed by an underlying
Markov chain
a (j ,N ,c) P[ An1 ,c) k |Yn( N ) j ]
i
k
(N i i infinite or finite number of states
structure of the Markov chain is unknown
ai(,N,k,u ) P[ An1 ,u ) k , Pn1 j| Pn i ]
j
i (N i
Buffer and single server
P[ A ( N i ,u )
n 1 k | Pn 1 j ] P[ Pn 1 j| Pn i ] – nominal system with capacity N1
– perturbed system with capacity N0
Yn(Ni ) min((Yn( Ni ) 1) An 1i ) , N i )
1
(N
Zn 1i ) ((Yn( Ni ) 1) An 1i ) N i )
(N (N
Aug. 18, 2000 George Lin - Defense Presentation 33
Example 1
Markov Uncontrolled Stream
Nominal buffer capacity, N1=50
Perturbed buffer capacity, N0=40
Aug. 18, 2000 George Lin - Defense Presentation 34
Example 2
LRD Uncontrolled Stream
Nominal buffer capacity, N1=200
Perturbed buffer capacity, N0=150
Aug. 18, 2000 George Lin - Defense Presentation 35
Summary of
On-Line Sensitivity Queueing Analysis
Develop the proportional relationship for on-line
sensitivity queueing analysis
– apply the proportional relationship method and perform sensitivity
analysis of the feedback controlled queueing system with respect
to buffer capacity
– we show the proportional relationship method successfully
perform sensitivity analysis with respect to discrete parameters for
a system with LRD traffic
– we show that our method is comparable with simulation methods
Aug. 18, 2000 George Lin - Defense Presentation 36
Performance Analysis with LRD
Input Traffic
Off-Line Performance analysis
Finite buffer queueing analysis
Reassembly and multiplexing queueing analysis
On-line sensitivity queueing analysis
Summary
Summary
Performance analysis with LRD traffic
– Finite buffer queueing analysis
loss probability
we show that existing analysis assuming infinite buffer significantly
underestimates the network performance
Existing analysis underestimate the impact of LRD traffic when the buffer
capacity is small
– Reassembly and multiplexing queueing analysis
buffer overflow probability and frame loss probability
we show that LRD has no impact on reassembly operations
we show that existing asymptotic analysis significantly underestimates the
impact of LRD traffic when the buffer size is small
Aug. 18, 2000 George Lin - Defense Presentation 38
Summary
Performance analysis with LRD traffic
– On-line sensitivity queueing analysis
utilize proportional relationship
determine performance sensitivity with respect to discrete parameters for a
queueing system with LRD traffic
– loss probability
– mean queue length
– mean delay
– we show our method is useful for systems which are not amenable to existing on-line methods
Aug. 18, 2000 George Lin - Defense Presentation 39
Concluding Remarks
and
Future Directions
Concluding Remarks
Contributions
Provide new methods for performance analysis and
performance optimization with LRD traffic
Aug. 18, 2000 George Lin - Defense Presentation 41
Future Directions
Off-line performance analysis extension
– application to designing static network components or configuring
quasi-static network parameters
– Finite buffer multiplexing queueing analysis extension
Obtain asymptotic loss probability in close form
apply to admission control based on a priori characterizations
On-line sensitivity analysis extension
– application to configuring dynamic network parameters
apply the proportional relationship method to more realistic models
apply the proportional relationship method to dynamic buffer allocation
apply to admission control based on measurement data
– improve estimation efficiency for LRD traffic
Aug. 18, 2000 George Lin - Defense Presentation 42