On Modeling Round-Trip Time Dynamics
of the Internet using System Identification

Hiroyuki Ohsaki

1

, Mitsushige Morita

2

, and Masayuki Murata

1
1

Cybermedia Center, Osaka University
1-30 Machikaneyama, Toyonaka, Osaka 567-0043, Japan

foosaki, muratag@cmc.osaka-u.ac.jp

2

Graduate School of Engineering Science, Osaka University
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

m-morita@ics.es.osaka-u.ac.jp
Abstract. Understanding the end-to-end packet delay dynamics of the
Internet is of crucial importance since it directly affects the QoS (Quality
of Services) of realtime services, and it enables us to design an efficient
congestion control mechanism. In this paper, we measure the round-trip
time, and build a mathematical model representing its dynamics using
system identification. We first measure, as the input and output data for
system identification, the packet inter-departure time from a source host
and the corresponding round-trip time measured by the source host.
ICMP (Internet Control Message Protocol) is utilized to measure the
round-trip time for each packet. We next model the network, seen by a
specific source host, as a dynamic SISO (Single-Input and Single-Output)
system. Using measurement results obtained from three different network
configurations, we investigate how accurately the round-trip time
dynamics of the Internet can be modeled with the system identification.
1 Introduction

Understanding the end-to-end packet delay dynamics of the Internet is of crucial
importance since (1) it directly affects the QoS (Quality of Services) of realtime
applications, and (2) it enables us to design an efficient congestion control mechanism
for both realtime and non-realtime applications. For non-realtime applications,
a delay-based approach for congestion control mechanisms, rather than a
loss-based approach as used in TCP (Transmission Control Protocol), has been
proposed (e.g., [1, 2]). The main advantage of such a delay-based approach is, if
it is properly designed, packet losses can be prevented by anticipating impending
congestion from increasing packet delays.
For a long time, queueing theory has been extensively used as a powerful tool
to analyze packet-switched networks. In general, the queueing theory assumes
stationarity of the network, and allows us to obtain several performance measures
such as the average packet delay and the average packet loss probability.
However, the stringent limitation of the queuing theory is its difficulty to analyze
the dynamic behavior of the network. Several measurement-based studies suggest
that the end-to-end packet behavior in the Internet is quite dynamic [3--5].
Another approach, being different from the queueing theory, should therefore be
taken to investigate the packet delay dynamics of the Internet.

2 Hiroyuki Ohsaki et al.

In [6], the authors have proposed a novel approach to model the end-to-end
packet delay dynamics of the Internet. The main idea of the approach is treating
the network, seen by a specific source and destination pair, as a black-box, and
modeling the end-to-end packet delay dynamics using system identification [7].
The end-to-end packet delay dynamics are modeled as a SISO (Single-Input and
Single-Output) system based on the ARX (Auto-Regressive eXogenous) model.
The input to the system is a packet inter-departure time from the source host,
and the output is a (one-way) end-to-end packet delay variation measured by
the destination host.
This paper is a direct extension of [6], and primarily focuses on an applicability
of our approach to real networks. However, there is a major difference in the
modeling approach. In [6], the network seen by a specific source and destination
pair is modeled as a black-box. On the contrary, in this paper, the network seen
by a specific source host is modeled; that is, the output is the round-trip time
variation instead of the end-to-end packet delay variation. Although modeling
the round-trip time dynamics suffers from more measurement noise than modeling
the end-to-end packet delay, the modeling approach taken in this paper is
easier to implement, so that desirable for practical purposes.
After discussing advantages and disadvantages of several measurement methods
for the round-trip time, we present a measurement method using ICMP
(Internet Control Message Protocol) to collect the input and output data for determining
the coefficients of the ARX model. Since almost all hosts and routers
respond to ICMP packets, this method can be used in various network environments.
We then collect the input and output data from real networks. Three
network configurations are used including both wired and wireless LANs. Using
the input and output data obtained, coefficients of the ARX model are determined
using the least-square method. We investigate how accurately the ARX
model can represent the round-trip time dynamics of the Internet.
This paper is organized as follows. In Section 2, we summarizes related works
in recent publications. In Section 3, a black-box approach for modeling the
round-trip time dynamics of the Internet using the ARX model is explained.
In Section 4, we discuss several measurement methods of the round-trip time,
in particular, for collecting the input and output data for system identification.
Section 5 shows several measurement and modeling results, and discuss how accurately
the ARX model can capture the round-trip time dynamics. In Section 6,
we discuss several possible applications of our approach, followed by conclusion
of this paper.
2 Related Works

In the literature, there have been several measurement-based studies regarding
the end-to-end packet delay [3, 4, 8, 9] and the end-to-end path characteristics [5,
10]. In [3], the authors have examined the end-to-end packet delay and loss
behavior in the Internet using small UDP probe packets. In [4], the authors have
examined the correlation between packet delay and packet loss experienced by
a continuous-media traffic source, based on measurements of per-packet delays
and packet loss. In [8], a large number of TCP measurements have been used to
discuss two estimation problems: estimation of the retransmission timer (RTO)
for a TCP connection, and estimation of the available bandwidth. In [9], the

On Modeling Round-Trip Time Dynamics of the Internet 3
source host destination host
physical layer
network layer

packet
interdeparture

time
ICMP Echo
Request
ICMP Echo
Reply
physical layer
network layer
Fig. 1. Modeling round-trip time dynamics as SISO system

authors have presented an approach to characterize loss and delay characteristics
of a transmission link based on end-to-end multicast measurements. In [5], the
packet dynamics of the Internet have been analyzed based on measurements of
about 20,000 TCP data transfers. In [10], the routing behavior of the Internet
has been analyzed based on measurements of about 40,000 end-to-end traceroute
results. However, those studies are limited to a statistical behavior of the endto
-end packet delays and/or path characteristics. In other words, the dynamics
of the packet delay of the Internet, which is the main concern of this paper, has
not been investigated.
Aside from analyses of the end-to-end packet delay, another area of measurementbased
studies is regarding a black-box modeling of the network traffic [11--15].
In [11], the authors have proposed a traffic model for wide-area TCP traffic by
characterizing several distributions of, for example, the packet inter-arrival time
and the number of bytes transferred. In [12], the authors have proposed a fast
algorithm to construct a CMRP (Circulant Modulated Rate Process) for traffic
modeling. In [13], CMRP and ARMA (Auto-Regressive Moving Average) have
been discussed as a traffic model. In [14, 15], a measurement-based tool for traffic
modeling and queueing analysis has been developed, which uses CMPP (Circulant
Modulated Poisson Process) for a traffic model. Those studies are closely
related to our black-box modeling approach, but there is a significant difference.
Those studies have focused on traffic modeling based only on outputs (i.e.,
observed amount of traffic). On the contrary, this paper focuses on modeling
the round-trip time dynamics based on both inputs (i.e., packet inter-departure
time) and outputs (i.e., round-trip time variation). In other words, this paper
focuses on how the round-trip time of a packet sent from a source host is affected
by its past packet transmission process.

3 Black-Box Modeling and System Identification

As depicted in Fig. 1, the network seen by a specific source host, including underlying
protocol layers (e..g, physical, data-link, and network layers), is considered
as a black-box. Our goal of this paper is to model a SISO system describing the
round-trip time dynamics: i.e., the relation between a packet sending process
from the source host and its resulting round-trip time observed at the source
host. Effects of other traffic (i.e., packets coming from other hosts) are modeled
as noise. As the input to the system, we use a packet inter-departure time from

4 Hiroyuki Ohsaki et al.
ARX model

u(k) y(k)

input
(packet inter-departure time)
output
(round-trip time variation)

e(t)

noise
(other traffic)
Fig. 2. ARX model for modeling roundtrip
time dynamics

AR model
or
ARMA model
y(k)
e(t) noise
output
(packet arrival rate)
Fig. 3. AR model or ARMA model for
modeling network traffic

the source host: i.e., the time interval between two consecutive packet transmissions
from the source host. Use of the packet inter-departure time is straightforward
since it directly affects the end-to-end packet delay. As the output from
the system, we use a round-trip time variation measured by the source host:
i.e., the difference in two consecutive round-trip times. We choose the roundtrip
time variation, instead of the round-trip time itself, as the output from the
system. This choice is for reducing unstationarity of noise (i.e., effect of other
traffic) on the measured round-trip time since the aggregated network traffic at
a packet-level time scale is not stationary [16].
In this paper, the ARX model is used and its coefficients are determined using
system identification [7]. Figure 2 illustrates a fundamental concept of using the
ARX model for modeling the packet delay dynamics. The input to the ARX
model is a packet inter-departure time from the source host, and the output
from the ARX model is a round-trip time variation measured at the source host.
Effects of other traffic (i.e., packets coming from other hosts) are modeled as
the noise to the ARX model. Letting u(k) and y(k) be the input and the output
data at slot k, the ARX model is defend as

A(q) y(k) = B(q) u(k 0 n d ) + e(k) (1)
where A(q) and B(q) are given by

A(q) = 1 + a 1 q

01

+ : : : + a na q

0na

B(q) = b 1 + b 2 q

01

+ : : : + b n b q

0n b +1
In the above equations, e(k) is unmeasurable disturbance (i.e., noise), and q

01

is the delay operator; i.e., q

01

u(k) j u(k 0 1). The numbers n a and n b are the
orders of polynomials. The number n d corresponds to delays from the input to
the output. For compact notation, i is introduced as

i = [n a ; n b ; n d ] (2)
In our case, u(k) and y(k) correspond to k-th packet inter-departure time and

k-th round-trip time variation. All coefficients of the polynomials, an and b n , are
parameters of the ARX model, and are to be determined from input and output
data using system identification. Refer to [7] for the detail of the ARX model.
Our approach of a black-box modeling using the ARX model is distinctive
from other black-box approaches, which model network traffic using the

On Modeling Round-Trip Time Dynamics of the Internet 5

AR (Auto-Regressive) model or the ARMA (Auto-Regressive Moving Average)
model [13, 17, 18]. Figure 3 illustrates a typical usage of the AR model or the
ARMA model for modeling network traffic. Comparing Figs. 2 and 3, the ARX
model has the input whereas either the AR model or the ARMA model does
not. In other words, only the ARX model can represent the dynamics, i.e., how
the past input data affects the future output data.
Note that the ARX model has a drawback in modeling the round-trip time
dynamics; i.e., the ARX model is a linear time-invariant model, so it cannot rigorously
capture non-linearity of the round-trip time dynamics. But it should be
noted that the ARX model is applicable in various control engineering problems.
This is because non-linear dynamical systems operating around the stable point
can be well approximated by a linear system [7]. In Section 5, we will investigate
how accurately the round-trip time dynamics can be described by the ARX
model.
The system identification problem for the ARX model is formulated as a
minimization problem, where the cost function is given by a loss function [7].
Because of space limitation, only the outline is shown in this paper, and interested
readers should refer to [7] for more detail.
Let ` be a vector of all coefficients and /(k) be a vector of all past n a outputs
and n b inputs, respectively.

` = [a 1 ; : : : ; ana ; b 1 ; : : : b n b ]

T

(3)

/(k) = [0y(k 0 1); : : : ; 0y(k 0 n a );

u(k 0 n d 0 1); : : : ; u(k 0 n d 0 n b )]

T

(4)
Using Eq. (1), the output from the ARX model  y(kj`) is given by
 y(kj`) = /

T

(k) ` (5)
The loss function VN (`; Z

N

) is defined as the sum of all squared prediction errors
for N input and output data.
VN (`; Z

N

) =
1

N

N

X

k=1

(y(k) 0  y(kj`))

2

(6)
where Z

n

is the past input and output data defined as

Z

N

= fu(1); y(1); : : : ; u(N); y(N)g (7)
The solution
 ` N that minimizes the above loss function is easily obtained by the
least squares method:
 ` N =

"

N

X

k=1

/(k)/

T

(k)

#01 N

X

k=1

/(k)y(k) (8)
4 Measurement Methods

For collecting the input and output data from real networks, it is necessary to
send a series of probe packets into the network, and to measure their resulting
round-trip times. For sending probe packets, one of the following protocols can
be used.

6 Hiroyuki Ohsaki et al.

-- TCP (Transmission Control Protocol)

-- UDP (User Datagram Protocol)

-- ICMP (Internet Control Message Protocol)
In what follows, we discuss advantages and disadvantages of these protocols for
sending probe packets to collect the input and output data, in particular, for
system identification.
TCP has a feedback-based congestion control mechanism, which controls the
packet sending process from a source host according to the congestion status of
the network. Since it is an ACK-based protocol, it is easy for the source host
to measure the round-trip time for each packet. However, TCP is not suitable
for sending a probe packet because of the following reasons. First, for system
identification purposes, the input data (i.e., the packet inter-departure time)
should contain diverse frequencies. So the white noise, which equally contains
all frequencies, is the ideal input data for system identification [7]. However, the
packet inter-departure process of TCP would have limited frequencies. Second,
most of system identification techniques assume an independence between the
input and output data. However, because of a feedback-based nature of TCP,
the packet inter-departure time is dependent on the past round-trip times, so
the independence assumption cannot be satisfied with TCP.
On the contrary, UDP has no feedback-based control. The packet interdeparture
time of UDP can be freely controlled. However, UDP is a one-way
protocol. The destination host must perform some procedure to measure the
round-trip time for each packet at the sender side. One possible way is to use

ICMP Destination Unreachable message as in the traceroute program [19]. When
the host receives a UDP packet to an unreachable port, it returns ICMP Destination
Unreachable message to the source host. The source host can therefore
measure the round-trip time by observing the elapsed time between the UDP
packet transmission and the receipt of the corresponding ICMP packet. However,
as specified in [20], generation of ICMP Destination Unreachable messages is limited
to a low rate. Use of ICMP Destination Unreachable message is therefore
not desirable to collect the input and output data for system identification.
ICMP is a protocol to exchange control messages such as routing information
and node failures [21]. Since ICMP has no feedback-based control, the interdeparture
time of ICMP packets can be freely controlled. Also it is easy to
measure the round-trip time at the source host by using ICMP Echo Request and

ICMP Echo Reply messages, as in the ping program. Although a part of network
devices have a rate limitation for transmitting ICMP Echo messages [22], many
network devices respond to ICMP Echo messages. So this method can be used
in various network environments.
In this paper, we therefore choose ICMP Echo message as a probe packet.
More specifically, the source host sends a series of ICMP Echo Request messages
to the destination host, and the destination host returns ICMP Echo Reply
messages. We have modified the ping program to dynamically change the packet
inter-departure time (originally fixed at one second).
The detailed algorithm is described below.
Sender Algorithm (Source Host):

S1) Send ICMP Echo Request message of 1,500 bytes including IP and ICMP
headers. The payload of the ICMP packet holds the timestamp of the packet
transmission.

On Modeling Round-Trip Time Dynamics of the Internet 7

S2) Randomly choose the packet inter-departure time from the exponential distribution
in order to schedule the next ICMP packet transmission.

S3) Go to S1.
Receiver Algorithm (Source Host):

R1) Wait for the receipt of ICMP Echo Reply message.

R2) Extract the timestamp from the payload.

R3) Calculate the round-trip time from the current time.

R4) Calculate the round-trip time variation from the previous round-trip time.

R5) Go to R1.
The destination host copies the payload of the received ICMP Echo Request
message to the returning ICMP Echo Reply message. Thus, the ICMP Echo Reply
packet contains the timestamp placed by the source host at its transmission
time. This enables precise measurement of the round-trip time at the source
host.

5 Modeling from Measured Data

Network Configurations

We have measured three sets of input and output data from the following three
network configurations.

N1) 100 Mbps LAN without background traffic

N2) 100 Mbps LAN with background traffic

N3) 11 Mbps wireless LAN and 100 Mbps LAN
In the network configuration N1, both the source and destination hosts are
directly connected to a single 100 Mbps switch. Because of a direct connection,
there exists no background traffic, and the output data (i.e., the round-trip
time variation) suffers from little observation noise. This network configuration
enables us to investigate how accurately the round-trip time dynamics can be
modeled in a high-speed and non-congested network.
The network configuration N2 is a 100 Mbps LAN, which consists of five
100 Mbps switches connected in serial. The network configuration N2 is a private
LAN in our laboratory, where about 50 client computers and 10 server
computers are connected. There are five switches between the source and destination
hosts. Since intermediate switches process traffic from other computers,
the round-trip time measured at the source host might be affected by existence
of the background traffic. This network configuration is for investigating how
the background traffic deteriorates the accuracy of the ARX model.
In the network configuration N3, the destination host is equipped with a
11 Mbps wireless LAN interface. The base station is connected to the network
configuration N2. There are five 100 Mbps switches between the source host
and the base station. In this case, the wireless LAN, which is much slower than
100 Mbps LAN, is the bottleneck. The round-trip time is expected to be significantly
larger than other network configurations.
In each network configuration, we have collected both the packet interdeparture
time u(k) and the round-trip time variation y(k) using the approach

8 Hiroyuki Ohsaki et al.
120 140 160 180 200
0
0.5
1
1.5
2
sequence number
(a) Packet inter-departure time

120 140 160 180 200
0
0.5
1
1.5
2
2.5
3
sequence number

round-trip
time
[ms]
(b) Measured round-trip time
120 140 160 180 200
-0.5
0
0.5
sequence number

round-trip
time
variation
[ms]
(c) Measured round-trip time
variation

105 110 115 120 125 130
-0.5
0
0.5
sequence number

measured data
model output
(d) Measured data y(k) and
model output y

3

(k)

Fig. 4. Results in network configuration N1 (100 Mbps LAN w/o Background Traffic)

described in Section 4. The source host sent 10,000 probe packets with an exponentially
distributed inter-departure time. Note that lost packets are not included
in the measured input and output data. Of all input and output data
collected, we use the input and output data of 100 packet samples for coefficients
determination and model validation of the ARX model. In what follows,
we discuss how accurately the round-trip time dynamics can be modeled by the
ARX model.

Network Configuration N1

In the network configuration N1, the mean packet inter-departure time has been
set to 0.2 ms, resulting in the average packet transmission rate of 43.2 Mbps
and the average round-trip time of 0.8 ms. Shown in Fig. 4 are results in the
network configuration N1 for i = [8; 8; 1]. This figure shows: (a) the packet
inter-departure time u(k), (b) the measured round-trip time, (c) the measured
round-trip time variation y(k), and (d) comparison between the measured output
data and the model output. More specifically, the "measured output data" is
the measured round-trip time variation y(k), and the "model output" is the
simulated output from the ARX model, which is defined as

y

3

(kj`) = /

3T

(kj`) ` (9)
where

/

3

(kj`) = [0y

3

(k 0 1j`); : : : ; 0y

3

(k 0 n a j`);

u(k 0 n d 0 1); : : : ; u(k 0 n d 0 n b )] (10)

On Modeling Round-Trip Time Dynamics of the Internet 9
120 140 160 180 200
0
0.5
1
1.5
2
sequence number
(a) Packet inter-departure time

120 140 160 180 200
0
0.5
1
1.5
2
2.5
3
sequence number

round-trip
time
[ms]
(b) Measured round-trip time
120 140 160 180 200
-0.5
0
0.5
sequence number

round-trip
time
variation
[ms]
(c) Measured round-trip time
variation

105 110 115 120 125 130
-0.5
0
0.5
sequence number

measured data
model output
(d) Measured data y(k) and
model output y

3

(k)

Fig. 5. Results in network configuration N2 (100 Mbps LAN w/ Background Traffic)

Note the difference between  y(kj`) and y

3

(kj`); i.e.,  y(k) is a 1-step ahead prediction
from the measured inputs and outputs, whereas y

3

(kj`) is a simulated
output only from the measured inputs assuming zero noise. There are several
techniques for checking accuracy of the ARX model obtained by system identification
[7]. Comparing the measured output data and the model output is one
of the most intuitive approaches.
Figure 4(c) shows that the amplitude of the round-trip time variation is very
small, whereas the packet inter-departure time dynamically changes. This is
because there is no background traffic between the source and destination hosts.
A slight change in the round-trip time would be caused by the processing delay
variation at the host and/or by a timer granularity of the operating system, since
the network is not a bottleneck in the network configuration N1. Figure 4(d)
indicates that the ARX model cannot capture the round-trip time dynamics
in the network configuration N1. Namely, the model output y

3

(k) is almost
unchanged, although the measured round-trip time changes. This is caused by
the weak correlation between the packet inter-departure time and the measured
round-trip time; that is, in the network configuration N1, the round-trip time
is almost independent of the packet inter-departure time.

Network Configuration N2

Figure 5 shows results in the network configuration N2 for i = [8; 8; 1]. In this
case, the mean packet inter-departure time has been set to 0.6 ms, resulting the
average packet transmission rate of 18.0 Mbps and the average round-trip time
of 1.8 ms. Figure 5(c) shows that the amplitude of the round-trip time variation

10 Hiroyuki Ohsaki et al.
120 140 160 180 200
0
10
20
30
40
50
60
70
80
sequence number
(a) Packet inter-departure time

120 140 160 180 200
0
5
10
15
20
25
30
35
40
sequence number

round-trip
time
[ms]
(b) Measured round-trip time
120 140 160 180 200
-15
-10
-5
0
5
10
15
sequence number
(c) Measured round-trip time
variation

105 110 115 120 125 130
-15
-10
-5
0
5
10
15
sequence number

round-trip
time
variation
[ms]

measured data
model output
(d) Measured data y(k) and
model output y

3

(k)

Fig. 6. Results in network configuration N3 (11 Mbps Wireless LAN + 100 Mbps LAN)

is larger than that of the network configuration N1. The main reason for such a
large amplitude would be the effect of the background traffic; that is, the roundtrip
time tends to become large when the network is congested. It can be found
that the model output y

3

(kj`) and the measured output y(k) roughly coincide
but slightly differ. This is because the measured round-trip time variation is
disturbed by other traffic, which is unknown so that not included in the model
output y

3

(k).
Network Configuration N3

Results in the network configuration N3 for i = [8; 8; 1] are shown in Fig. 6.
In this case, the mean packet inter-departure time has been set to 12.0 ms,
resulting the average packet transmission rate of 967 Kbps and the average
round-trip time of 16.7 ms. Figure 6(c) shows that the amplitude of the roundtrip
time variation is much larger (about 10 ms) than the previous cases, N1

and N2. Figure 6(d) indicates that the round-trip time dynamics is not correctly
modeled by the ARX model. It is probably because the transmission delay at
the wireless link is significantly changed, resulting in a large measurement noise.
From these observations, we conclude that the round-trip time dynamics can be
modeled by the ARX model when the network is moderately congested.

Choice of Model Orders and Number of Samples

In the above results, the orders and the delay of the ARX model is fixed at

i = [8; 8; 1]. In general, the accuracy of the ARX model is dependent on the

On Modeling Round-Trip Time Dynamics of the Internet 11
40 50 60 70 80 90 100
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
number of samples
Fig. 7. Relation between loss function
and the number of samples

0 5 10
15 20 0
10
20
0.02
0.04
0.06
0.08
n

b

n a

loss
function
Fig. 8. Relation between loss function
and the orders of the ARX model

choice of the orders and the delay of the ARX model, and the number of samples
used for system identification. It is therefore desirable to carefully choose i and
the number of samples to minimize the loss function VN (`; Z

N

) (i.e., the sum of
all squared prediction errors).
Figure 7 shows the relation between the loss function JN (`) and the number
of samples used from the input and output data in the network configuration

N2. In this figure, the orders and the delay of the ARX model is fixed at i =
[8; 8; 1], while the number of samples is changed from 40 to 100. This figure
shows a tendency that, as the number of samples increases, the loss function first
decreases and then gradually increases. The similar tendencies are observed in
other network configurations N1 and N3, although the results are not included
here.
We next show the relation between the orders of the ARX model and the
loss function VN (`; Z

N

) in Fig. 8. This figure uses 100 samples from the input
and output data in the network configuration N2, and the orders of the ARX
model, n a and n b , are changed from 1 to 20, respectively. This figure indicates
that the loss function increases as the n b increases. On the contrary, the choice
of n a has little effect on the loss function.
Another important factor in determining the orders of the ARX model is the
highest frequency that should be captured by the ARX model. Namely, the ARX
model is able to capture higher frequency of the output data (i.e., the round-trip
time variation) with larger n a and n b . Moreover, the ARX model requires more
computational burden and becomes less stable as the orders increase [7]. So the
orders of the ARX model should be determined by taking account of a trade-off
among accuracy, complexity, and stability.
6 Discussion and Conclusion

We discuss several possible applications of our approach --- modeling the roundtrip
time dynamics of the Internet using the ARX model. Details of these topics
will be discussed in the forthcoming paper, but it is worthwhile to discuss how
our approach can be applied to various problems. The first and straightforward
application would be to use our approach to understand the round-trip time
dynamics of the Internet. We can analyze the round-trip time dynamics through

12 Hiroyuki Ohsaki et al.

the ARX model. Because the ARX model is one of LTI (Linear Time Invariant)
models, various analysis techniques for LTI models in time- and frequencydomain
can be utilized. The second application would be to predict the future
round-trip time from the ARX model. As have shown in Section 5, the roundtrip
time of a packet is considerably disturbed by background traffic. Hence, it
is difficult to predict the far future round-trip time. However, the ARX model
can predict the near future round-trip time, which would be useful to, for example,
QoS controlling mechanisms. As noted in Section 1, the third and possibly
most important application would be to design a delay-based congestion control
mechanism. Once the ARX model capturing the round-trip time is obtained,
it would be possible to apply the optimal control theory to design an efficient
delay-based congestion control mechanism. Congestion control of the Internet
is a difficult problem because of its complexity such as heterogeneity of various
network elements and non-negligible propagation delays. However, we believe
that combination of the ARX model and the optimal control theory would help
us to design a more efficient congestion control mechanism. We are currently
working on designing a delay-based congestion control mechanism for stream
video applications.
In this paper, we have proposed a novel approach to model the round-trip
time dynamics of the Internet using system identification. The main idea is to
model the network, seen by a specific source host, as a linear time-invariant ARX
model. The input to the ARX model is the packet inter-departure time from
the source host, and the output is the round-trip time variation measured at the
source host. With the ICMP-based measurement method, we have collected three
sets of the input and output data from real networks. Using the measurement
results, we have determined coefficients of the ARX model, and have investigated
how accurately the ARX model captures the round-trip time dynamics. We have
found that the ARX model can capture the round-trip time dynamics when the
network is moderately congested. We have also found that, when the network is
not congested or the measured round-trip time is noisy, the ARX model fails to
capture the round-trip time dynamics.
As a future work, it is important to validate effectiveness of our modeling
approach for a through set of input and output data obtained from various
network configurations. We are currentry measuring the input and output data
in working LAN and WAN environments [23].
Acknowledgement

This work was supported in part by Research for the Future Program of Japan
Society for the Promotion of Science under the Project "Integrated Network Architecture
for Advanced Multimedia Application Systems"(JSPSRFTF97R16301)
.
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