Quality of Service (QoS) Analysis of an Internet

Traffic Trace Over Gigabit Ethernet

Yu Chen and Izzat Darwazeh

Department of Electronic and Electrical Engineering, University College London (UCL), London, the U.K.

Email: {y.chen, i.darwazeh}@ee.ucl.ac.uk

Abstract—An Internet traffic trace over gigabit Ethernet con-tains information on arrival time and size of every packet overthe collection period. An important problem is to model thelatency and packet losses within the trace. The Lindley equationis a general form of describing the evolution of queueing delayprocesses and queue length processes, both of which are tightlyassociated with the latency and packet losses. We report on ouruse of the Lindley equation to analyse publicly-available link-layer data from a gigabit Ethernet gateway and discuss theperformance of packet queueing delay and queue length basedon the data.

I. INTRODUCTION

Quality of Service (QoS) is crucial to multimedia services

in wireless and wired-line communication networks. Two im-

portant QoS metrics are latency and packet loss. The question

of how to model these two metrics in real systems is appealing

but remains open.

The modelling of these two metrics in IP networks depends

on the statistical nature of the packet inter-arrival and packet

size distributions, which have been extensively studied. Self-

similar [1], long-range dependent [2], heavy-tail distributed

[3] or Poisson processes [4] are commonly assumed to char-

acterise traffic statistics.

On the other hand, the use of the Lindley equation to analyse

Internet packet traces was first reported by Park et al. in

2005 [5]. Fischer and Bevilacqua used the equation to analyse

Internet packet traces [6]. Kim et al. used the equation to

evaluate the data/voice services in CDMA 2000 systems [7].

The Lindley equation has two related versions; one describes

the evolution of packet queueing delay processes and the other

is for the evolution of queue length processes. However, the

work mentioned above only considered the packet queueing

delay processes and their performances.

In this paper we use both versions of the Lindley equation

to analyse the Internet traffic over gigabit Ethernet. A trace

of the Internet traffic is from the University of Massachusetts

Amherst (UMASS). To the authors best knowledge, this is

the first work that uses the Lindley equation to analyse a

realisation of a queue length process from real measured traffic

data.

The remainder of this paper is organised as follows: In

Section II, we model a gigabit Ethernet gateway as a queueing

model. Section III presents and discusses two versions of the

Lindley equation that relate to packet queueing delay and

queue length. The Internet traffic trace used in this paper is

described in Section IV. In Section V, we present analyses

by applying the Lindley equation to an Internet trace. These

analyses include a numerical evaluation of packet size, packet

delay and queue length distributions. Section VI concludes this

paper.

II. SYSTEM MODEL

A gigabit Ethernet system can be modelled as a

G/G/1/∞/FIFO queueing model, as is shown in Fig. 1. Packets

are randomly generated from higher layers. Let Vn denote

the random variable of the nth packet arrival instant and Tn

denote the random variable of the nth inter-arrival time, which

is the time difference between the nth packet arrival instant

and (n+1)st packet arrival instant. Their relation is shown in

Tn = Vn+1 − Vn (1)

Let Ln denote the random variable of the size of the nth

packet. The probabilistic distributions of Tn and Ln are

considered to be general. Fig. 1 also shows an example of

a sequence of packets arriving at the queueing system. The

arrivals are a sequence of unequal-height impulses, which are

randomly scattered in a time line; the heights of impulses

indicate their packet sizes.

Incoming packets are first stored in a buffer and later served

by one server. The capacity of the server is fixed to a constant

value c. In a gigabit Ethernet system, such a value equals 1

Gbps. The buffer size is assumed to be infinite. Packets in the

buffer are served in a fair manner according to the First-In-

First-Out (FIFO) discipline.

We further denote the service time of the nth packet by Sn.

Sn is assume to be equal to its packet size divided by the

capacity c and is expressed by

Sn =Ln

c(2)

It is clear to see that Sn is a generally distributed random

variable because of the statistical nature of Ln.

III. LINDLEY EQUATION

From the perspective of QoS, the concepts of packet queue-

ing delay and queue length are tightly associated with the

latency experienced by packets and packet losses, respectively.

The use of the first version of the Lindley equation that

describes the evolution of packet queueing delay processes

in Ethernet is introduced in Section III-A and the use of the

second version of Lindley’s equation is introduced in Section

III-B.

ServerFIFO

Buffer

capacity, c

0 1 2 3 4 5 6 7 8 9

Arrival Packets

ll

ll

l

t t t t1

12

2

3

3

4

4

5

Fig. 1: System model of a gigabit Ethernet system

A. Lindley Equation for Packet Queueing Delay Processes

Denote by Dn the queueing delay value of the nth packet.

Dn is a random variable. For a G/G/1/∞/FIFO queueing

model, Dn is given by the first version of the Lindley equation:

Dn = max{0, Dn−1 + Sn−1 − Tn−1} (3)

Eq. (3) is a recursive equation, stating the relation between the

queueing delay of the nth and (n − 1)st packets with effect

of the service time of the (n− 1)st packet and the (n− 1)st

inter-arrival time. Furthermore, (3) is regarded as a general

form for calculating packet queueing delay processes [8].

In an Ethernet system, by replacing Sn−1 with Ln−1/cbased on (2), (3) can be rewritten as

Dn = max{0, Dn−1 +Ln−1

c− Tn−1} (4)

The packet delay distribution considered in this paper is

the complementary cumulative density function (CCDF) of

packet queueing delay Pr{Dn > Dmax}, where Dmax is

called the delay bound. For a given delay bound Dmax1,

the probability Pr{Dn > Dmax1} is called the delay bound

violation probability (DBVP).

B. Lindley Equation for Queue Length Processes

Suppose that the system is observed every To seconds. We

denote A[n] as the number of bits that arrives at the system

during the time interval [(n− 1)To, nTo), i.e.,

A[n] =

j+m∑

i=j+1

Li,

Vj+1 ∈ [(n− 1)To, nTo), · · ·Vj+m ∈ [(n− 1)To, nTo)

and C[n] as the number of bits that the serve is capable to

serve during the time interval [(n−1)To, nTo). A[n] and C[n]are both random variables and are called the bit arrival and

bit service at the nth observation epoch, respectively. In an

Ethernet system, C[n] can be simply expressed as

C[n] = cTo (5)

Denote by Q[n] the queue length at the nth observation

epoch. Q[n] is a random variable. For a G/G/1/∞ queueing

model, Q[n] is given by the second version of the Lindley

equation:

Q[n] = max{0, Q[n− 1] +A[n]− C[n]}

= max{0, Q[n− 1] +A[n]− cTo} (6)

Similar to the attributes of (3), (6) is also a recursive equation

that is used for calculating queue length processes. Moreover,

it is worth noting that in contrast to the first version of the

Lindley equation, (6) does not even need the FIFO assumption.

The queue length distribution considered in this paper is

CCDF of queue length Pr{Q[n] > Qmax}, where Qmax is

called the maximum queue-length bound. For a given queue-

length bound Qmax1, the probability Pr{Q[n] > Qmax1} is

called the queue-length bound violation probability (QBVP).

IV. TRACE OF THE INTERNET TRAFFIC OVER GIGABIT

ETHERNET

In our analysis, we downloaded a trace of the Internet Traffic

over gigabit Ethernet from UMASS trace repository [9]. The

trace is from a gigabit Ethernet connection entering UMASS

on 14th, November, 2004. It was a gigabit Ethernet system

operating in a Fibre link at a capacity of 1 Gbps. Further

information on how they collected the data can be found on

the same website.

The total size of the file is 382 MB. We used Wireshark to

filter out the information of arrival instants and packet sizes out

from the file and then sorted packets based on their arrivals.

After processing the file, 11,976,472 packet arrival instants

and sizes were obtained over a period of 81.63 seconds.

V. DATA ANALYSIS

This section presents several analyses carried out with the

trace. We used R1 software for statistical analysis. Fig. 2

shows traffic loads per second over the collection period (81.63

seconds). It is found that the traffic loads are fairly constant

during this period. For the entire duration, the mean inter-

arrival time and the standard deviation of the inter-arrival time

are 6.90 µs and 13.85 µs, respectively, which indicates that it

is not Poisson traffic.

Since the traffic loads per second are shown to be stationary

in Fig. 2, we use the information of packet arrival instants and

packet sizes of the first 1,000,000 packets (approximately 6.9

seconds) from the trace, i.e.,

v = {v1, v2, v3, · · · v1000000} (7)

l = {l1, l2, l3, · · · l1000000} (8)

Eqs. (7) and (8) are used to analyse the packet size distri-

bution, packet queueing delay distribution and queue length

distribution in this Section.

1www.r-project.org

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90

Tra

ffic

Lo

ad (

Gb

ps)

Time (second)

Fig. 2: Traffic load per second

A. Packet Size Distribution

On the basis of (2), analysing packet sizes is equivalent to

analysing service times. Using (8), we plot the packet size

distribution in Fig. 3. The histogram of packet sizes is shown

in Fig. 3a. The majority of the packets have either extremely

small or large packet sizes (Thompson et al. reported and

discussed similar observations in [10]). We further show the

empirical probability mass functions (PMF) for the packets

less than 100 bytes in Fig. 3b and for the packets greater than

1480 bytes in Fig. 3c. From these two figures, the minimum

packet size is 64 bytes and the maximum packet size is 1518

bytes, which conform to the standard of Ethernet II. Moreover,

these two types of packets (64-byte and 1518-byte packets) are

mostly common in the trace.

B. Packet Queueing Delay Distribution

For analysing the packet queueing delay distribution, the

first step is to obtain a sequence of packet queueing delay

samples from (7) and (8). According to (1), the inter-arrival

time of the nth and (n+ 1)st packets is obtained by

tn = vn+1 − vn (9)

Applying (9) to (7), we have a sequence of inter-arrival times

t = {t1, t2, t3, · · · t999999} (10)

Assume that the packet queueing delay of the first packet d1is zero. From (4), the queueing delay of the nth packet is

calculated by

dn = max{0, dn−1 +ln−1

c− tn−1} (11)

where the values of ln−1 and tn−1 are from (8) and (10).

By recursively calculating (11), we have a sequence of packet

queueing delay

d = {d1, d2, d3, · · · d1000000} (12)

Fig. 4 shows the empirical and estimated CCDFs of packet

queueing delay. The x-axis is delay bound (the unit is mi-

crosecond), and the y-axis is DBVP in log scale. The empirical

0

0.001

0.002

0.003

0.004

0.005

0.006

0 200 400 600 800 1000 1200 1400 1600

Den

sity

Packet Size (Byte)

(a) Histogram of packet size

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

60 65 70 75 80 85 90 95 100

Rel

ativ

e F

req

uen

cy

Packet Size (Byte)

(b) Empirical PMF of packets less than 100 bytes

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1480 1485 1490 1495 1500 1505 1510 1515 1520

Rel

ativ

e F

req

uen

cy

Packet Size (Byte)

(c) Empirical PMF of packets greater than 1480 bytes

Fig. 3: Packet size distribution

result is shown in a red solid line. The estimated CCDF

is calculated by the curve fitting technique based on the

assumption that the CCDF is exponentially distributed. This

function is plotted in a black line with circle marks. From the

figure, the empirical and estimated CCDFs are well matched

especially when a delay bound is greater than 20 µs and less

than 50 µs.

10-6

10-5

10-4

10-3

10-2

10-1

100

0 10 20 30 40 50 60 70 80

Del

ay B

ou

nd

Vio

lati

on

Pro

bab

ilit

y

Delay Bound (µs)

Empirical CCDF

Estimated CCDF

Fig. 4: CCDF of packet queueing delay

C. Queue Length Distribution

Following the procedure of Section III-B, we first specify

that the value of To equals 5 µs. Since v and l cover

the observation duration of 6.9 seconds, there are 1379006

observation samples of arrived bits, i.e.,

a = {a[1], a[2], a[3], · · · a[1379006]} (13)

Using (6) recursively, we have a sequence of queue length

samples, i.e.,

q = {q[1], q[2], q[3], · · · q[1379006]} (14)

Fig. 5 shows the empirical and estimated CCDFs of queue

length. The x-axis is queue length bound, which is measured

by the number of bits, and the y-axis is QBVP in log scale. The

empirical result is shown in a red solid line and the estimated

CCDF is plotted in a dark line with dots. We use the curve

fitting technique with exponential distribution assumption to

estimated CCDF as well. Similar to the result shown in Fig.

4, the empirical and estimated CCDFs of queue length are in

good agreement when a queue length bound is between 10000

bits and 60000 bits.

Finally, it is shown in Figs. 4 and 5 that the tail distributions

of packet queueing delay and queue length are exponentially

bounded. Such findings may be explained by the Effective

Bandwidth theory [11] [12] (the Effective Bandwidth theory

approximates packet queueing delay and queue length distri-

butions to be exponentially distributed).

VI. CONCLUSION

Packet queueing delay distributions and queue length distri-

butions are particularly important to Quality of Service (QoS)

provisioning. Previous research used the Lindley equation to

analyse packet queueing delay distributions. In theory, the

Lindley equation contains two recursive equations that easily

generates packet queueing delay processes and queue length

processes. Therefore, this paper applied these two equations to

analyse a publicly available Internet traffic trace over gigabit

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 10000 20000 30000 40000 50000 60000 70000 80000

Qu

eue-

Len

gth

Bo

un

d V

iola

tio

n P

rob

abil

ity

Queue-Length Bound (bit)

Empirical CCDF

Estimated CCDF

Fig. 5: CCDF of queue length

Ethernet. It was found that the tails of the empirical com-

plementary cumulative density functions (CCDFs) of packet

queueing delay and queue length are exponential bounded.

These findings may be explained by the Effective Bandwidth

theory.

ACKNOWLEDGEMENT

This research has been supported by the Cooperating Ob-

jects Network of Excellence (CONET), funded by the Euro-

pean Commission under FP7 with contract number FP7-2007-

2-224053.

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