1
Analytical Models of IP Networks with TCP
Traffic
Towards an Analytical Model of the Internet
Presentazione per lesame finale di Dottorato di

• Michele Garetto

Tutore prof. Marco Ajmone Marsan
2
Outline

• Problem statement
• Modeling approach
• Analytical models for greedy flows
• Analytical models for finite flows
• Network solution
• Analysis of TCP performance
• Conclusions

3
The problem
IP core

4
Problem Statement
Input variables

• IP network link speed, propagation delays,
  buffer sizes, AQM parameters (or drop-tail),
  routing
• TCP flows number of connections (greedy flows),
  connection establishment rates and flow length
  distributions (finite flows), segment size,
  maximum window size,

Output variables

• IP network link utilizations, queuing delays,
  packet loss probabilities,
• TCP flows average window size and throughput
  (greedy flows), completion times (finite flows)

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Outline

• Problem statement
• Modeling approach
• Analytical models for greedy flows
• Analytical models for finite flows
• Network solution
• Analysis of TCP performance
• Conclusions

6
Modeling approach
TCP sub-models one for each end-to-end path
(ingress router-egress router)

Queues one for each (unidirectional) link
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Protocol-network models decoupling
Modeling approach
TCP sub-models
Network model

• Iterate with a Fixed Point Solution

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TCP sub-model the queuing network approach

• Start from the Finite State Machine (FSM) of the
  protocol (any version)
• Associate an /G/? queue to every state
• Number of customers in the queue is the number of
  connections in that state
• Customers move from one queue to another based on
  probabilistic transitions, following the protocol
  dynamics
• in the case of short-lived flows, classes are
  introduced to represent the number of segments
  still to be transferred

The queuing network can be viewed as a
Stochastic Finite State Machine (SFSM)
describing the protocol behavior
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Outline

• Problem statement
• Modeling approach
• Analytical models for greedy flows
• Analytical models for finite flows
• Network solution
• Analysis of TCP performance
• Conclusions

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Analytical models for greedy flows

• TCP sub-models
• The queuing network approach
• Closed queueing network, single class
• Network model
• Open queueing network of M/M/1/B queues
• Solution of each queue in isolation

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TCP model (greedy flows)
Closed queuing network model of TCP Tahoe
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Results single bottleneck topology
Congested buffer (Drop Tail)
S1
R1
S2
R2
R3
S3
45 Mb/s


Sn
Rn

• Unidirectional data traffic
• One congested queue (drop-tail)
• Different propagation delays (wide-area)

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Results single bottleneck
0.1
Packet loss probability
model - B512
B128
0.01
ns simulations - B512
B128
0.001
0
100
200
300
400
Number of connections
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Window size distributions
0.20
model - 25 connections
ns sim - 25 connections
0.15
model - 100 connections
Probability density function
ns sim - 100 connections
0.10
0.05
0.00
0
10
20
30
40
50
60
Window size
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Results multi-bottleneck
S2
S3
S4
R4
80 Mb/s
80 Mb/s
20 Mb/s
R2
20 Mb/s
80 Mb/s
20 Mb/s
S1
R1
45 Mb/s
R3
S5
R5
Setup
Pair 1
Pair 2
Pair 3
Pair 4
Pair 5
1
100
25
25
50
50
2
25
100
50
10
50
3
10
40
20
10
30
4
100
25
10
20
200
5
20
200
100
50
10
6
20
10
200
100
400
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Results multi-bottleneck
End-to-end packet loss probability
Throughput (b/s)
0.1
1e6
sim
sim
y x
y x
0.01
pair 1
pair 1
pair 2
pair 2
pair 3
pair 3
pair 4
pair 4
1e5
pair 5
pair 5
1e5
1e6
0.01
0.1
mod
mod
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Outline

• Problem statement
• Modeling approach
• Analytical models for greedy flows
• Analytical models for finite flows
• Network solution
• Analysis of TCP performance
• Conclusions

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Finite flows definition of g
The normalized goodput g (or nominal link
utilization) on a given link is defined as
follows
(The actual traffic intensity ? includes also
retransmissions arriving at the link, and
acknowledgement belonging to other flows in the
network)
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Analytical models for finite flows

• The overall solution is obtained in two steps

• Step 1 Analysis of the IP network
• Computation of queuing delays and
• packet loss probabilities in each queue
• Fixed Point Approximation
• Simplified TCP model
• Step 2 Analysis of TCP performance
• To be done for a given e2e pair and a given flow
  length
• Using packet loss probabilities and RTT obtained
  by Step 1
• Detailed TCP model

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Outline

• Problem statement
• Modeling approach
• Analytical models for greedy flows
• Analytical models for finite flows
• Network solution
• Analysis of TCP performance
• Conclusions

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Finite flows network solution

• The IP network is modeled as an open queueing
  network (a queue for each link)
• TCP traffic is not Poisson !
• Traditional approaches based on M / M / 1 queues
  do no apply to IP networks carrying TCP traffic
• Which queue model to use ?

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Queue models with finite flows
1
ns sim
g 0.8 Flow dist geom (20)
0.1
B 8
0.01
Probability
0.001
0.0001
1e-05
0
50
100
150
200
250
Number of packets in the queue
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Queue models with finite flows

• A queue with batch arrivals is needed to capture
  the burstiness of TCP traffic
• A simple M X / M / 1 model, where batches
  correspond to the clusters of packets transmitted
  every RTT by the TCP sources, provides an
  accurate (conservative) prediction of the queue
  behavior
• We build a simple Stochastic Finite State
  Machine to compute the batches produced by TCP
  sources for the case in which the flow length
  distribution is geometrically distributed

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TCP model in case of a geometric flow length
distribution
? An Open Queuing network, with a single class of
customers
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Extension to long-tail flow length distributions

• A generic flow length distribution (in
  particular long-tail) can be approximated by a
  mixture of geometric distributions

• Each geometric component is analyzed separately
  by a SFSM
• Arrival rates of batches of the same size are
  then added up to obtain the actual batch size
  distribution

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Realistic flow length distribution Access link
PoliTo
Fitted mixture of 7 geometric distributions
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Example of long-tail distribution
Mixture of 3 geometric components
overall distribution
? 0.89 - mean 10
? 0.1 - mean 100
? 0.01 - mean 1000
overall mean 28.9
Probability
10000
Flow length (packets)
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Geometric decomposition
Flow length pdf
log
log
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Queue models with finite flows

• The solution provided by the M X / M / 1
  model can be refined considering that packets
  within a batch do not enter the queue
  simultaneously, but arrive one after the other
• Define
• An approximate solution of the queue with
  spread batch arrivals is provided by an MMPP /
  M / 1 model, where the modulating chain is the M
  / M / 8 that represents the number of batches
  arriving at the queue

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Finite flows step 1 (network solution)

• A network of M X / M / 1 / B queues does not
  allow a product form solution
• The product form solution provides a conservative
  (pessimistic) prediction of the behavior of the
  queues
• Some approximations can be adopted to reduce the
  errors caused by solving each queue in isolation

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Identification of Uncongested Queues
C1
Q
C

CM

• If , queue Q is can be marked
  as uncongested

In most cases a negligible error is introduced by
simply assuming zero waiting queue and no loss

• Uncongested queues are not solved using the M
  X /M/1/B model

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Limitation of the model impact of correlated
batch sizes
M X / M / 1 model
0.1
0.1
0.01
0.01
Probability
0.001
0.001
0.0001
0.0001
g 0.8 Flow dist 3 geom
1e-05
1e-05
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
Number of packets in the queue
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Results of step 1 single bottleneck
1
B 128
sim flow dist geom (20)
mod flow dist geom (20)
sim flow dist geom (60)
0.1
mod flow dist geom (60)
Packet Loss Probability
0.01
0.001
Existence of an admissible solution
0.0001
0.8
0.85
0.9
0.95
1
Normalized Goodput (g)
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Results of step 1 meshed topology
155 Mb/s
2
45 Mb/s
1
3
B
A
C
D
E
4
5
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Results of step 1 meshed topology
Average number of packets in the queues
Average packet loss probabilities in the queues
80
0.1
y x
y x
70
60
0.01
50
sim
sim
0.001
40
30
0.0001
20
10
1e-05
0
1e-05
0.0001
0.001
0.01
0.1
0
10
20
30
40
50
60
70
80
mod
mod
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Results of step 1 meshed topology
Examples of queuing delay distributions
1
0.1
Link D-4 (94 )
sim
mod
0.01
probability
0.001
0.0001
Link 1-A (71 )
Link C-D (34 )
sim
sim
mod
mod
1e-05
0
20
40
60
80
100
120
Queue length (packets)
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Outline

• Problem statement
• Modeling approach
• Analytical models for greedy flows
• Analytical models for finite flows
• Network solution
• Analysis of TCP performance
• Conclusions

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Finite flows Analysis of TCP performance

• The network solution (step 1) provides for each
  end-to-end path
• Average round trip time
• Average packet loss probability
• Loss event probability for each window size
• The behavior of TCP flows of a given size over a
  given path can now be explored in details using
  an Open Multiclass Queueing Network (OMQN)
• From the solution of the OMQN it is possible to
  compute the average completion time of the flows

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TCP sub-model (finite flows)
?e
Open multiclass queuing network of TCP Tahoe
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Results single bottleneck
10
10 packets
20 packets
5.0
100 packets
2.0
Average completion time (s)
1.0
0.5
0.2
0.75
0.8
0.85
0.9
0.95
1
g
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Example of multi-bottleneck topology Tandem
network
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Tandem network completion times
158
y x
10
1
model
0.1
(sec)
0.1
1
10
simulation
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Completion times distributions

• An approximate path analysis technique applied
  to the open multiclass queueing network is used
  to compute also distribution and quantiles of
  completion times

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Distribution for the number of active connections

• An M / M / m queue proves to be a good model
  to predict the distribution for the number of
  active connections. The number of servers is
  computed by matching the average value obtained
  from the OMQN

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Conclusions

• Analytical models based on Queuing Networks have
  been used to describe the behavior of both IP
  networks and TCP sources
• Traditional Markovian analysis can be applied
  successfully to study complex TCP/IP networks
• We obtain not only average values, but also
  distributions very close to simulation results
• The proposed methodology can be used for the
  analysis and design of high-speed networks where
  simulations are not feasible

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Finite connections convergence of the FPA
(unique solution)
1
TCP - Link utilization (g) 0.95
0.1
.
Packet loss probability ( p )
0.01
M/M/1/B (B 32)
0.001
0.0001
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Load ( ? )
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Finite connections convergence of the FPA
(double solution)
Unstable point
Stable point
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Comparison greedy flows / finite flows
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Comparison greedy flows / finite flows
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Queue models with finite flows
1
M X / M / 1 / B - ? 0
MMPP / M / 1 / B - ? 1
MMPP / M / 1 / B - ? 10
0.1
MMPP / M / 1 / B - ? 100
M / M / 1 / B
0.01
Average Packet Loss Probability
X geom (10)
B 128
0.001
0.0001
1e-05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Traffic intensity
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Graphical User Interface
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Model / simulation / measurementsAccess link
PoliTo
(86 link utilization)