5 Application of Network Tomography

8/14/2019 5 Application of Network Tomography

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5 Application of Network Tomographyin LANs

5.1 The Applicability of the Delay Model

The notions of the delay model provide useful instruments to conduct a link-level delay

distribution inference in a network. The measurements can be obtained by adopting one of

the techniques described in the Section 4 such as Ping, Traceroute or Pathchar. The choiceof the delay model and the reliability of the measurements are to be decided ointly. !nly in

this case it is possible to obtain optimal result from the estimations. "ctually, it is necessary

to know how high is the gap, if it e#ist, between the theoretical instruments provided by the

delay model and their applicability. The aim of this section is to reduce this gap and allow

the simplest solutions to obtain reliable results.

The present section describes in details the procedure to calculate the estimate of the link

delay distribution. $t provides, in particular, a general process to compute the iterative

analyses of the steady probability distribution described in the Section %.

The reliability of the results can be tested only by implementing the inference model to a

network. "n application in a &"' is the easiest way to proof the theoretical forecast, and to

test a large scale theory such as $nternet Tomography to a small network. Successively it is

mentioned the choice of Ping to obtain the measurements. "lthough, some changes of

Ping(s program, are also described to provide reliable measurement to the inference model.

The applicability of a model covers a delicate aspect of an inference model. $t is not always

possible to test an inference model with simple solutions. )ased on the analysis of link

delay estimation by &o Presti in a &"', this can be possible and the aim of this *hapter is

to provide its process of work.

5.2 Modeling Delay

&et T+,& be a logical tree . represents the set of vertices, with the source /, the

receivers 0 and the branch point of phys. The packets pair is sent down along the tree


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from the root. &et us denote for each node ka random variable kD taking values in the

positive real line { }R+ . kD is the random delay that a packet could spend to traverse

the link , f k k L . /D represents the delay on the root and by convention kD +/. kD +

indicates a dropped packet along the link. The key hypothesis is that the independence of

the measured delays along each link can be assumed. 2or this reason kD are assumed to be

independent. $t is possible to define the cumulative delay e#perienced along a path from

the root to the node k, like the sum of each random variable kD along the path,

jkj k

X D

= .

"ccording to the discreti3ation by &o Presti described in the Section %, each link delay is

discreti3ed to a set +5/,q,6q,7,iq,)q,8, where q is the width bin, )9: is the number of

possible bin, and represents the event of packet lost. The probability distribution of kD is

denoted by k

e#pressed in the 1quation %.4 in the Section %, and

is the set ofdistribution for each link k e#pressed in the 1quation %.; in the Section %.

" measurement is the one way delay from the root to the end receivers represented by the

set 0k. The set of children of a generic node is denoted by d. &et us define

R k k RX X = as the one way delay occurred on the way from the source to the receivers

belonging to the set 0k. RX is also discreti3ed to the set . 2igure %.< in the Section %,

depicts the space of RX , in which R +#. The 1quation %.:= in the Section % shows

the way to obtain > k d . ?sing the 1@ algorithm to ma#imi3e the likelihood function

, A L X D in the 1quation %.:B, the ma#imum likelihood estimate of>

k d , with k is

> >

k k

kk

d Q

n d n d d

n d n

== ;.:

&et us focus now on the computation of > kn d . Cnowing > kn d for each d, the

probability distribution along the k-th link k d can be obtained. $n the 1quation %.6< in

the Section %, let us call the measurement ijx depending on i, simply Rx . This is the case,

in which 0k is composed of two end receivers. $f there are more subsets of two end

receivers of 0k, it is necessary to introduce a summitry with i, inde#es.

The computation of > kn d is quite complicated. The second step of 1@ algorithmDB,6=,6EF

shows how > kn d can be computed.

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; "pplication of 'etwork Tomography in &"'s

>:> D G F

nm

R Rk l km

n d P D d X x=

= = =

> D G F

l

R RR R Rk

xn x P D d X x

= = =

>

>

D G F>

D F

l

lR R

R R k lR k

x R R

P X x D dn x d

P X x

= ==

=

;.6

Having the link k fi#ed, the count > kn id for each iq can be calculated in the 1quation

;.6. The iterative algorithm is e#pressed by the distribution k computed at the step l. &et

us focus the attention on each of the components of the 1quation ;.6.

Computation of 0n#

Rx is a 6 # nvector. n represents the number of packets pairs sent to the end receivers of

the set 0. 1ach line of Rx belongs to a finite space defined by R +#. Rn x representsthe number of times the same discreti3ed measurement is present in the vector Rx .

Computation of >D Fl R RP X x = and

>

D G Fl R R kP X x D d = =

This is the most comple# aspect of the computation. &et 0k the set of receivers

descendent from the node k. $f k0, there will be assumed 0k+5k8.

&et us define jR k j R kY Y = as the delay observed from receivers descendent from

node k. &et R k R k f kZ Y Y= be the delay measured from node fk down to the

receivers of k. &et us denote A D Fk R k R k R k z P Z z = = , k,

num R k

R kz .$f ,

for e#ample , the tree is composed of three vertices, 2igure ;.: shows the

corresponding R kz for a node k.

;:


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The A k R kz respects the following recursion

A k R k k R k z z = k0 ;.%

A A jk R k k R j

d Q j d kz d z d

= k0 ;.4

$n the 1quation ;.4 R k R j j d kz z = .

f(k)

k

R(k)

0

f(k)

k

R(k)

0

2igure ;.:I 2or each k it is possible to calculate R kz . 2rom the total R kY delayfrom the root till the set 0k the , f kY is subtracted.

There are the necessary instruments to calculate >D FR RlP X x

= . $n fact, it is sufficient to

observe that :R RY Z= , where : denotes the child node of the root node /.

The 1quation ;.; shows that the delays observed at the receivers are equal to the delay

e#perienced from the root down to the receivers of :, which represent the set 0.

/: : : : RR R f RZ Y Y Y Y Y= = = ;.;

$t is possible then to define the computation of >D FR RlP X x

= as

;6


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: ::>> >D F A A

ll l

R R RRP X x z y

= = =

;. D G FR Rl kP X x D d = = can be obtained in a similar way, even if it is

more complicated than >D FR RlP X x

= , because the probability in this case is calculated by

conditioning to each possible value of d and for each measurement Rx . The approach to

compute this probability is the following I

,> >

FD G F Dl lk d

R R R RkP X x D d P X x

= = = =

;.B

where the distribution , >k dl

is obtained from > l by setting > J :l

d = when d+d( and

> J /l

d = otherwise. This process will be e#plained better in the ne#t section where it is

described the practical application and the computation of these probabilities. $t is possible

to define the computation >D G FR Rl kP X x D d

= = asI

,

: ,> >

F >D G F D A l lk d

lR R R R Rk k d

P X x D d P X x y

== = = =

;.=

'ow all the probabilities in the 1quation ;.6 are computed. $t is vital to understand that

the 1quation ;.6 works, when a branch node k is chosen. Then > kn d must be calculatedfor each d. The computation of the 1quation ;.B is rather difficult, because it depends not

only on the measurement but also on the current d. The implicit difficulty is that in order to

calculate each probability, it is necessary to know the current distribution of the other link,

which belongs to the path. $t must be calculated for each link, and this increases the

computational burden of the inference algorithm. The 1quation ;.6 shows the recursion of

the equation, where > l

k d represents the probability delay distribution of link k,

calculated in d and in the previous step. The choice of the initial distribution is vital if a

fast convergence is required. $t is described in D


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The pseudo code to compute the link delay distribution will be described in the present

section. $t is composed of the above mentioned 1@ algorithm.

The variable l represents the current step. The variable threshold represents the

parameter which allows the algorithm to know if the ma#imum is reached. $n fact, by

comparing the current inferred estimate with the previous one, the difference betweenthem, can be defined. This value is e#pressed by the variable threshold . $f the threshold is

low, more iterations are necessary. The threshold defines the surface of the

multidimensional ma#imum point. The ma#imum will be a point with a null surface, only

if there is infinite equality of each component of the current estimate to each component of

the previous one. $n this case the threshold is represented by the value 3ero. $n the practical

application it is advisable to use a large threshold and then to decrease it until the

computational burden will become too heavy and the number of the iterations be too high.

?sually, a threshold of one percent is sufficient. $f the tree has few nodes, it is possible to

decrement the threshold. The problem can be contained in the appro#imation processes of

the compiling program. The pseudo code implemented by 2rancesco &o Presti D6EF is

depicted in the 2igure ;.6 .

procedure main 5

l+/ A choose > l A do 5 K computation of the e#pected counts

for each k

for each d

> kn d + / A

for each Rx R

> kn d 9 +

,

,

>L :, ,>

>L :,

lk d l

R kl

compute yRn x d

compute yR

end

end

end

Kcomputation of the current estimates

for each k

for each d

:>

> Al kk

n dd

n + =

end

end

l 99A

;4


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8 while :> >G Gl l M threshold A

> > l = A 8

double function compute_k! z! "5

if k 0

return >k z A

else

return

L , , Ak R j

d Q j d kd compute j z d

8

2igure ;.6I The pseudo code.

5.$ Application to the %o"ter Lab LAN

The goal of the present work is to apply 'etwork Tomography on a &"'. 'etwork

Tomography is usually referred to $nternet Tomography because it focuses on a large scale

network. The aim of this work is to test a delay probability distribution inference in a small

scale network. "pplying the mentioned algorithm to a &"', the theory to an accessible and

bordered network can be used. "ll the e#periments have be conducted in the 0outer &ab of

the Nepartment of Nistributed System of the ?niversity of Ouer3burg ermany. Thephysical network, to which the analysis of probability distribution delay is applied, is

depicted in the 2igure ;.%

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Ethernet V.21 eria! !ink (2MBits/s)

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Fast Ethernet IEEE 802.3u (100MBits/s)

Ethernet V.21 eria! !ink (2MBits/s)

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2igure ;.%I The physical tree is composed by the root ?ll and the end receivers

&atona and enus.

The host ?ll represents the root or the source. &atona and enus are the end receivers. npacket pairs are sent from ?ll to &atona and enus. 2igure ;.% shows the physical tree.

"pplying the @odel Tree described in the Section %.6 , it is possible to make a graph of the

logical tree shown in the 2igure ;.4. 1ach link is labeled with a number. The branch node 6

is k, and the node : is its parents fk. 0k, the set of end receivers that will receive the

packet pairs sent from the root, are nodes % and 4.

2igure ;.; shows the tree, to which the &o Presti algorithm will be applied. The algorithm

works on the link separated by a branch node. $n this case the branch node is the number 6.

The goal is to estimate the delay distribution for the links 6, % and 4.

0 1 2

4

3

Ull Hera Zeus

Venus

Latona

R""t f(k) k

R(k)

2igure ;.4. &ogical tree. Nenote the node k, its parent fk and the set of

receivers 0k.

" packet pair is sent from root to host % and 4. The first member of the packet pair will

reach the host 4 and the second member the host %. 2igure ;.< shows how the two members

of the packet pair travel in the network. The inference strategy works on the common path

0 2

4

3

;


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2igure ;.;I The logical tree is composed of three links. The goal is to estimate

the probability delay distribution 6 , % and 4 .

the packet pairs crosses. $n fact, the algorithm cannot distinguish between the differentlinks from root until node number 6, and considers the global link from the root to the host

number 6. The branch nodes play a very important role in defining, which link can be

analy3ed by the algorithm. The key sentence is that the inference strategy works on the

common path of the packet pairs. The measurement represents the one way delay from the

0 2

4

312

#"$$"n %ath

2igure ;.


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2igure ;.BI The red line show the one way direction, the black, show the coming

back. $t is easy to note the same shared path between the two directions.

2igure ;.B helps to understand how the problem can be solved by means of applying the0TT measurement. The first and second members cover the same path to go and to come

back. 2or e#ample, the first member is directed to the end host 4, makes the trip /, 6,4 and

comes back covering the same path, 4, 6 ,/. There is no difference between the application

of the inference to the one way measurement and the 0TT, because the member makes the

same way. $t is a specific case for this application, when there is the certainty that the

packet member travels along the same path. This simplification makes the 0TT calculation

preferable to calculating the one way delay. This computation can be made more easily

applying one of the tools mentioned in the Section 4.

5.5 The #hoice of !ing

$n the Section 4 some tools to measure the 0TT of the packets pair along a path have been

described.

Ping, Traceroute and Pathchar are equivalent in the 0TT computation where the ambient to

measure is a small area network, such as &"'. The path is relatively short and the

computation of the 0TT is easy to conduct and its precision is reliable enough.

The goal is to choose one of these three programs. The aim of the present work is to offer

an inference instrument able to operate in all the &"'s. To obtain a universal instrument, itis preferable to focus on the most simple solutions.

$n this case, Ping represents this simplest choice. Ping is an administrator(s tool that all

hosts should implement. $t does not have any restrictions to work. Ping is a common tool

and is really easy in use. Traceroute and Pathchar are focused on other specific goals apart

from 0TT, such as the record of the routes, latency and bandwidth estimate. This

information is not the subect of interest to the inference algorithm this work is devised to

implement. )esides, Traceroute and Pathchar depend more heavily on the operative system

of the machine in which they run. Ping, on the contrary, is a more transparent tool. $n a

&"', the 0TT of Ping is reliable, because the path to test is short and is composed of few

routers.

The precision of the measurement is linked to the dimension of the bin si3e of the &o Presti

algorithm. $n fact, why should be a higher degree of precision of the measurement required,

if it will be eliminated during the discreti3ation processQ This e#ample shows, why

measurements and inference strategies should be chosen ointly.

;=


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The 2igure 4.6 in the Section 4 shows an application of Ping. $t works simply by typing the

destination host to ping and the number and the dimension of the packet to send D%/F. The

goal is to obtain the 0TT for a packet pair described in the Section %. Two $*@P packets

should be sent consequently to the respective end host destinations as 2igure ;.= e#plains.

The 2igure %.6 in the Section % shows the trip of a packet pair along the path. The Ping

program iputils-6//6/E6B is able to ping one host at a time. $t must ping two hosts at thesame time. " first packet should be sent to the address of the first end node, and the

second packet should be sent immediately after the first one to the address of the second

node. The 2igure ;.= shows this request.

2 1

t

&''ress 2

&''ress 1

&''ress 1 &''ress 2

"ur#e

"ur#e

IM

2igure ;.=I Two $*@P packet should be sent in the time :t and 6t to the address

: and 6.

$t is vital to define the processing gap : 6G GP# t t = as the current time between the first

and the second $*@P. The inference strategy requires a P# / that is the second $*@P

leaves immediately after the living of the first. )oth packets should travel one after another

and come back in the same order to the source. "n P# / allows the packets to test thesame state of the traffic network and is essential for the reliability of the measurement. The

algorithm works on the common path the two packets cover. The second $*@P tests the

congestion of the first one. This permits the second packet to be behind the first one with a

high probability. This operation should be implemented by Ping.

&et us focus the attention on the Ping iputils and its processing gap. Ping iputils allows to

ping one host at a time and listen to the $*@P 1cho 0eply coming back. !nly after the

answer is received, another $*@P 1cho 0equest is sent. This is a serious problem, becausethe Ping iputils allows to work only with one end host. There is not the possibility to

initiali3e a sequence of two $*@P sending process to two different end hosts.

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1manuele !rlando

1 1 2 2

*

E#h" Re+1 E#h" Re%1E#h" Re+2 E#h" Re%2

,ar-et h"st 1 ,ar-et h"st 21 1 2 2

*

E#h" Re+1 E#h" Re%1E#h" Re+2 E#h" Re%2

,ar-et h"st 1 ,ar-et h"st 2

2igure ;.EI The normal operation of Ping iputils. The second packet can be sent

only after the source has received the answer of the first one.

The implementation shown in 2igure ;.= must be reached by modifying the original

program. Ping iputils is able to implement the operation shown in the 2igure ;.E. The

source sends the first packet and then, only after the 1cho 0eply is received, Ping sends

another packet to the second host. This operation is rather difficult. $n practice, to change

the address of the first end host in the address of the second end host means to initiali3e

another Ping process. The 2igure ;.E shows how large is the processing gap. This cannot

be a reliable measurement and it cannot be applied to the inference strategy. $t is necessary

hence to modify the Ping iputils in a program able to work like the pseudo code depicted in

the 2igure ;.:/. This pseudo code shows how the new modified Ping works. The icmp_se$

represents an inde# which will play an important role in the modified Ping. nrepresents the

number of the packets pairs to send. The result of the measurements is a n # 6 vector Rx .

The original Ping program is replaced with the modified Ping program.

for icmpLseq+: to n

send Packet :icmpLseq to host :

send Packet 6icmpLseq to host 6

print timestamp Packet :returned icmpLseq from host :

print timestamp Packet 6returned icmpLseq from host 6

end

2igure ;.:/I The pseudo code represents the functioning of the modified Ping

program.

Modified Ping

The original Ping is the version iputils-6//6/E6B, written by @ike @uuss, ?.S. "rmy

)allistic 0esearch &aboratory, in Necember :E=%.


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The most important modifications conducted in the process of the present work areI

:. Nuplicating the buffers, the structures and the necessary variables to describe an

$*@P packet. The task of a second buffer is to allocate the packet direct to the second

host.

6. Nuplicating the necessary variable to count the $*@P packet sent.

%. "dding a control to be able to specify in the command line the second host which has

to be pinged.

4. "dding a code responsible to traduce the hostname string in the format network-byte-

ordered.

;. @aking a second socket icmp_sock%.


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1manuele !rlando

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2igure ;.::I "n application of the modified Ping program. The end host &atona

and enus are pinged. '+4 and the dimension of the packet is ;< byte.

The icmp_se$ is vital for the right functioning of this new tool of measurement. $t

represents the current packet pair sent. $t can check the right order of each member of thepacket pair. 2or e#ample, the 2igure ;.:: shows how the right order of the two members is

achieved. This is really an important inde#, because it allows to notice if the second

member goes past the first during the trip. The icmp_se$ Rlabels the member of the packet

pair, making recogni3able its arrival to the source.

"nother advantage is that the icmp_se$ can be used as a pointer in the construction of

the vector Rx . 1ach line of Rx corresponds to a same icmp_se$, and the columns, to the

first and second member of the same packet pair. "ll the measurements in this work are

obtained by applying the modified Ping.

5.& 'mplementation

This section e#plains how the modeling delay can be applied to the logical tree depicted in

the 2igure ;.;. The set of the nodes k is composed of node 6, % and 4. $t is possible to

define only a 06, because the nodes % and 4 do not have children nodes. That is why the

only node satisfying the condition k0 is the node number 6. The 1quations ;.% and ;.4

can be written in the following wayI

% %% % A

R Rz z = k+%0 ;.E

4 44 4 A

R Rz z = k+40 ;.:/


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6 6 66 66

A A R R

d Q j dz d z d

= k+60 ;.::

2rom the 1quation ;.: it is possible to estimate the value of the delay probability for

each value of d as

> > kk

n dd

n = k+56,%,48 ;.:6

Choice of initial distributions/>

k

The logical tree is of a small dimension. The typical initial delay probability

distribution/

6> ,

/%

> , and /4> can be chosen. The most advisable choice is the

uniform probability distribution depicted in the 2igure ;.:6. 1ach value d at the step

3ero has the same probability.

1/(B61)

+ 2+ 3+ B+

i+7+/2 i+6+/2

i+ '

2igure ;.:6I 1ach value belonging to the set has the same probability.

The algorithm works iteratively, and the distribution will change until the stationary

probability distribution is reached. The algorithm models the distributions, giving more

weight to the event of delay with more probability to be verified. The value of delay with a

null probability represents a delay that cannot ever be tested on the link. "fter a number of

iterations, the algorithm reaches the steady solution and the distributions can be analy3ed.


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1manuele !rlando

Computation of >D F

lP X x

R R=

This computation is obtained by using the 1quation ;.< and the recursion of the 1quation;.::. The 2igure ;.:% shows the graphical procedure of the computation.

/ 6 % 4 6 % 4>D 6 ,% F / 6 % 6

RP X d d d d d d d

= = + +

6 % 46 / d d + ;.:%

1/5

1/5

0 ' 2' 3' 4'

0'2'

0'2'3'

d

d

d

1/5

1/5

1/5

0 ' 2' 3' 4'

0'2'

0'2'3'

d

d

d

1/5

2igure ;.:%I " graphical approach to the computation of >D FR RlP X x

=

>D FR RlP X x

= is calculated for each measurement Rx , and it does not depend on the value

d, but only on the specific value of Rx . )esides, the computation depends on the step l.

1ach iteration of the algorithm changes the previous distributions, and a new computation

of >D F

R Rl

P X x

=must be calculated with the new distributions.

Computation of >D G F

l R R kP X x D d

= =

This computation is the most complicated. There is not only the dependence by the

measurement, but also by the value of d. This is a conditioned probability, which is


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the probability that a measurement assumes the value Rx along the path, by conditioning to

a measured delay equal to dover the link kof the path. $t is vital then to compute this

probability d and Rx measured. This is the most difficult part of the algorithm,

because the comple#ity increases if the bin si3e decreases and ) augments.

The computation is obtained by using the same strategy as for the computation of

>D FR RlP X x

= . $t is necessary to compute the new probability distribution

> ,

D FR Rlk d

P X x

=mentioned in the 1quation ;.B . $t is obtained by setting the distribution

> J :k d = , where d(+d and 6> 6 :d = > J /k d = otherwise.

1

1/5

0 ' 2' 3' 4'

0'2'

0'2'3'

d

d

d

1/5

2igure ;.:4. The 6> is set for d(+6d and the 6 >D 6 ,% G 6 FRlP X d d D d

= =

can be found.

2or e#ample, if the goal is to estimate >D G 6 FR Rl kP X x D d

= = , the distribution

> J 6 :k d d = = and the 1quation ;.:: can be applied. The 2igure ;.:4 shows the graphical

approach to understand the computation. $n this case, given the measurement RX +6d,%d

the probability 6 >D 6 ,% G 6 FRlP X d d D d

= = is obtained by setting the distribution

6> 6 :d = .

2igure ;.:4 shows a particular computation for d(+6d, given the measurement Rx . $n this

case the probability is e#pressed by the 1quation ;.:4.


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1manuele !rlando

/ /6,> >

F / 6 % 4

D 6 ,% G 6 F Dd

R R R dP X d d D d P X x

== = = =

;.:4

Ohen all the probabilities are computed d, the set of > ,D FR Rl

k d

P X x

=is obtained

and can be used in the 1quation ;.6. )y changing the node k the />%,D FR R

d

P X x

=and

/>4,

D FR Rd

P X x

=are computed.

The pseudo code described above makes the computation of > kn d easy.

The simplest example

The following is a theoretical e#ample that shows the computation of > kn d given the one

way measurement. n+4 , the bin si3e q+/.: ms and )+;. The case of value is omitted.

The algorithm is implemented in the @atlab language.

Nefine a vector of the measurement as

me*sX +D/.:6%,/.6:A/.64,/.6=:A/.:6%,/.6%A/.6B,/.%


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3'2''0 d4'

2igure ;.:;. The initial delay probability distribution is an uniform

discrete probability distribution.

Computation of , >D F

lP X x

R R=

"pplying the 1quation ;.D FP X x

R R

= is

0

: 6

6 %

: 6

% 4

X

= ,/>

/./:


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1manuele !rlando

/

/

6 />6 6

>

D G /F>> / /

D FR

R R

Rx X R RR

P X x Dn n x

P X x

= ==

=+:.;=%% ;.:B

?sing the 1quation ;.:< and ;.:B to calculate 6, >n id for / U i U ), the vector 6> , n d will

be obtained in one step.

6

:.;=%%

:.;=%%

> , /.;=%%

/.6;//

/

n d

= ;.:=

The new delay probability distribution after one step is obtained by applying the 1quation

;.:6.

,:

6

/.%E;=

/.%E;=

> , /.:4;=

/./ and ,:

6> shows how in the process of the iteration of the

algorithm the distributions trying to reach the steady solution are modeled.

"ll the computations are made for the nodes k+%, 4 and the distributions,:

%> and ,:

4>

are the followingI


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,:

%

/.%E;=

/.%E;=

> /.:4;=

/./ /.%E;=

/.:4;=/./ /.6;/

/

/

=

,B

%

/.//:

/.EEE

> /

/

/

=

,B

4

/

/.//6

> /.EE=

/

/

=

;.6:

These results represent the steady solution of the delay distribution probability along the

links 6,%, and 4. *ertainly, it is a really simplified e#ample, and the shape of thedistributions is readable enough. 'e#t *hapter shows some e#periments on a large time

interval with a high number of bin. $n this case the number of necessary iterations

increases. The 2igures ;.:< ,;.:B and ;.:= show the probability delay distributions obtained

by @atlab.


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1manuele !rlando

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

Delay Probability Distribution - Link 2

Delay d (ms)

Probability

a2(d)

0.5

0.25 0.25

2igure ;.:


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-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Delay Probability Distribution - Link 4

Delay d (ms)

Probability

a4(d)

2igure ;.:=I Nelay Probability Nistribution along link 6

B:

5 Application of Network Tomography

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