1. On Modeling Feedback Congestion Control Mechanism of TCP using Fluid Flow Approximation and Queuing Theory　
Hisamatu Hiroyuki
Department of Infomatics and Mathematical Science,
Graduate School of Engineering Science,
Osaka University, Japan
Thank you for your Introduction, Mr. Chairman.
And now, I’m going to speak out my research.
The title is “On Modeling Feedback Congestion Control Mechanism of TCP
using Fluid Flow Approximation and Queuing Theory”

2. Background TCP (Transmission Control Protocol)
Transport layer protocol
Congestion control mechanism
Analysis of the TCP until today
Assuming a constant packet loss probability
Statistical behavior
Real Network
Packet loss probability has changed according to packet　transmission rate
At first, I’m going to introduce you Research Background.
TCP, the objective of this research, is transport layer protocol in internet and has
congestion control mechanism.
In the literature, there have been a great number of analytical studies on TCP.
Most of those studies have focused on the statistical behavior of TCP
by assuming a constant packet loss probability in the network.
However, the packet loss probability, in reality, has changes
according to packet transmission rate from TCP connections

3. Objective
Model the interaction between two systems as a feedback system
Network seen by TCP
M/M/1/m queuing system
Congestion control mechanism of TCP
Fluid flow approximation
Background Traffic are also taken account of
For modeling the changes of packet loss probability in network
according to packet transmission rate from TCP connections,
we model the interaction between Network and the congestion control mechanism of TCP
as a feedback system;
that is, both the congestion
Control mechanism of TCP running on a source host and the network seen by TCP are modeled
by dynamic systems
In addition, our analytic model, both TCP traffic and background traffic are taken account of

4. Analytic Model TCP connections Take account of background traffic
source hosts
destination hosts
router
TCP traffic
background traffic
This figure displays our analytic model.
That is, TCP connections share one bottleneck link
and Background traffic except TCP Exists on the bottleneck link

5. Modeling total TCP and Network
Modeling Network as a feedback system
Network seen by TCP
Congestion control mechanism of TCP seen by network
Congestion control
Mechanism of TCP
window size
Network seen
by TCP
Next, we model the total network as feedback system
That is, we model the interaction between the congestion control mechanism of TCP
The congestion control mechanism of TCP is a window based flow control mechanism,
And it dynamically changes the window size according to occurrence of packet losses in the network.
So we model congestion control mechanism of TCP, where it input is
the packet loss probability in the network
and its output is the window size of TCP
On the other hand, the network seen by the TCP behaves,
When the number of packets entering the network increases,
some packets are waited at the buffer of the router.
This sometimes causes buffer overflow,
resulting in a packet loss.
So the packet loss probability becomes large
when the number of packet entering the network increases.
Thus, we model the network seen by TCP,
Where its input is window size and its output is packet loss probability
packet loss
probability

6. Modeling Network using Queuing Theory
Assume bottleneck router is a Drop-Tail router
Model by M/M/1/m queue
Incoming traffic
TCP traffic
background traffic
Arrival rate of the background traffic lB p
Packet loss
probability ・ w1 wN
TCP
Window size
M/M/1/m m
We assume there exists only a single bottleneck link in the network.
And we also assume the bottleneck router adopts a Drop tail router,
If the network is stationary, bottleneck router can be modeled by a single queue
Let N be the number of TCP connections,
w sub i is window size and r sub i is the round trip time of the ith TCP connections,
the transmission rate from ithe TCP connections can be approximated by w sub i divided r sub i,
taking account of the background traffic,
the average packet arrival rate at the bottleneck router, lambda is given like this.
And then we can obtain the packet loss probability from the queuing theory l

7. Changes of TCP Window Size
Congestion avoidance phase
Increase window size at every receipt of ACK packet
Decrease window size at every detection of packet loss
Detect from receipt of duplicate ACKs
Detect from time-out mechanism
The congestion control mechanism of TCP is quite complicated,
In this research,we model only the main part of the congestion control mechanism of TCP,
We model the essential behavior of TCP in its Congestion avoidance phase,
the window based flow control mechanism and the loss recovery mechanism
Including the fast retrancemit

8. Modeling TCP using Different Approaches
4 analytic models
Model A1：
Assume a constant packet loss probability
Derive window size of TCP connection in steady state
Model A2 ：
Approximate A1 when packet loss probability is
very small
Model B：
Window size change at every receipt of ACK packet
and detection of packet loss
Model C：
Evolution of window size between two succeeding packet loss
Next, we introduce 4 analytic models called A1, A2, B, C.
They derived from different modeling approaches.
Model A1:
by assuming a constant packet loss probability in the network,
describing the window size of a TCP connections in steady state.
steady state is the state after enough time passed
Model A2:
when the packet loss probability is very small,
approximated the expression of the window size of TCP connection in Model A1
Model B:
Window size changes at every ACK receiving and every detecting packet loss
Model C:
this analytic approach uses a discrete-time model, where a time slot
Corresponds to the duration between two succeeding packet losses

9. Simulation Model :arrival rate Bottleneck Router UDP m:buffer size
Destination
：link capacity
:propagation delay
:5+0.5i [packet/ms]
:5 [packet/ms]
:5+i [ms]
:5 [ms]
:2 [packet/ms] N
:10 m
:50 [packet]

10. Network Model Relation between offered traffic load and packet
loss probability
M/M/1/m queuing system
Simulation result
0.5 1
1.5 2
2.5 3
0.01
0.1
Offered Traffic Load
Packet Loss Probability
Simulation
M/M/1
M/M/1/m
This figure shows the relation between the offered traffic load and the packet loss probability.
These values are measured at the bottleneck router for every 10[ms]
In the figure, the packet loss probabilities obtained from well-known results of M/M/1/m
And M/M/1 are also plotted.
This figure shows the dynamics of the network at a relatively small time scale can be
well modeled by the M/M/1/m model
M/M/1/m models dynamics of network correctly

11. TCP Model Relation between packet loss probability and window size
Congestion control mechanism of TCP
Simulation result
Simulation
Model A1
Model A2
Model B
Model C 5 10 15 20
0.001
0.01
0.1 1
Packet Loss Probability
Average Window Size [packet]
By comparing with simulation results,
we discuss how accurate four analytic models of
TCP capture the relation between the window size and packet loss probability.
This figure shows the relation between the packet loss probability
and the window size obtained using model A1, A2,B,C,respectively.
We also plot the simulation results which are measured at the bottleneck router for every 1[s]
This figure shows when the packet loss probability is less than 0.02,
analytic models A1, A2, and B show good agreement with simulation results.
On the other hand, when the packet loss probability is more then 0.03,
analytic models B and C show Good agreement
A1, A2, B show good agreement
B and C show good agreement

12. Transient Behavior Transient Behavior
Dynamics of the window size form its initial value
to its equilibrium value
Use Model B for Congestion control model of TCP
Using the analytic model presented before, we analyze the transient behavior of
TCP in the congestion avoidance phase.
By the word transient behavior,
we mean the dynamics of the window size from its initial values to its equilibrium value.
This time we adopt the model B from our proposed model as the congestion control model of TCP
time
Window size

13. Transient Behavior Analysis
Modeling the network as a discrete-time model
Time slot:duration between succeeding ACK
packets received
Network state
w(k): window size of TCP connections
P(k): packet loss probability
For given initial values, the evolution of the network
state can be obtained
we model both the congestion control mechanism of TCP and the network
as Interconnected discrete-time system,
where the states of these models change at every time slot.
The state of the network at slot k is then fully described
by the window size and the packet loss probability
For given Initial values of the window size and the packet loss probability,
the evolution of the window size and the packet loss probability can be numerically obtained.
From model B and well known results of the M/M/1/m queuing system,
The state transition equations are obtained.
Rho of k is offered traffic load at the bottleneck router at slot k

14. Numerical Example:Case of Different Propagation Delays
0.5 1
1.5 2 5 10 15 20 25 30 35
Average Window Size [packet]
Time [s]
t = 10 [ms]
t = 30 [ms]
t = 50 [ms]
We next present several numerical examples.
In following numerical examples, unless explicitly noted,
the initial window size is 1[packet],
the initial packet loss probability is 0,
the number of TCP connections are 10,
the capacity of the bottleneck link is 5[packet/ms],
and its propagation delay is 15[ms]
and the buffer size of the bottleneck router m is 50 [packet]
the evolution of the window size in the congestion avoidance phase
for the propagation delay tau of 10, 30, 50 [ms],
one can find that the window size becomes large as the propagation delay increases.
This can be intuitively understood from the increased bandwidth delay product.
In addition, as the propagation delay becomes small,
Ramp-up time of the window size becomes short
and convergence speed of the Window size becomes slow.
This is because, from a control theoretical viewpoint,
the feedback gain in the congestion avoidance phase of TCP is changed
according to the round trip time.
thus increasing the propagation delay implies decreasing the feedback gain
When propagation delay is small
Ramp-up time of the window size becomes short
The window size oscillates for long

15. Numerical Example:Case of Different Amount of Background Traffic
lB = 0
lB = 2
lB = 4.5 5 10 15 20 25 30 35
Average Window Size [packet]
0.5 1
1.5 2
Time [s]
This figure shows the evolution of the window size in the congestion avoidance phase
for The amount of background traffic of 0, 2 and 4.5[packet/ms]
From this figure, one can find that the window size in steady state becomes small
as the amount of background traffic increases,
this indicating that TCP suffers less throughput when the amount of background traffic increases
One can also find the increase rate of the window size is independent of the amount of background traffic.
This is because, in the congestion avoidance phase,
TCP increases the window size by one packet per a round trip time.
When the amount of the background traffic is large
The window size of steady state is small
The increase rate of the window size is independent of the amount of the background traffic

16. Conclusion and Future Work
Model the dynamics of TCP
Feedback system consisting of two systems
Transient behavior Analysis of TCP
Propagation delay
The amount of the background traffic
Future work
Rigorous analyses of stability and transient behavior of TCP
In this paper we have modeled both the congestion control mechanism of TCP and the network
as a feedback system,
and have analyzed the transient behavior.
Heavily dependent on the propagation delay of the bottleneck link,
but is almost Independent of the amount of background traffic.
We have also shown that the operation of TCP in the congestion avoidance phase is less stable
when the amount of background traffic is small
or the propagation delay of the bottleneck link is short.
In this paper, we have analyzed the transient behavior of TCP by iteratively calculating state transitions.
However, rigorous analyses of the stability and the transient behavior of TCP are possible .
we are currently working on such rigorous analyse