6 Results
6.1 The LAN Tomography
The Internet Tomography is generally applied to large scale wide area networks (WANs).
The aim of the present work, in contrast, is to conduct a link-level delay estimation on a
LAN.
First of all, a subclass of the Internet Tomography called LAN Tomography should be
defined. A LAN Tomography conducts its estimations on a small and bordered network.
This provides the ability to easily manage the network and to keep contact with the
estimation process. The LAN Tomography offers some tools to infer interesting
characteristics of the network.
The results of the inference analyses on the LAN are more easy to be measured and visible
than a large scale network. The LAN, in fact, is manageable and usually belongs to the
same domain. The application of the LAN Tomography is important not only to test the
reliability of the Internet Tomography, but also to offer another solution to investigate the
state of the LAN.
The work also presents the results of the experiments that have been conducted in the
Router Lab of the Department of Distributed System of the University of Wuerzburg
(Germany). The goal of the experiments was to test the reliability of the algorithm and to
find the optimum conditions for its operation.
6.2 The experiments and results
The results of the experiment present the inference of the link delay distributions in a path
of the Router Lab local area network which is shown in the Figure 5.3 in the Section 5. The
modified Ping application of Section 5.5, supplies the sequence of RTT from the host Ull to
Latona and Venus. The vector of the obtained measurements represent the sole data to feed
into the Lo Presti algorithm of Section 3.5. The algorithm is implemented by Matlab
program language.
An experiment is consists of two stages. The first stage is the measurement of the RTTs,
and the second one is the application of the algorithm. Let us define the most important
Emanuele Orlando
variables of the experiment:
 The number of packet pairs sent to measure the RTT (n).
 The length (byte) of each packet in the packet pair sent to measure the RTT (s).
 The bin size of the Lo Presti algorithm (d).
 The number of bins of the Lo Presti algorithm (B).
 The parameter which allows the algorithm to know if the maximum is reached
(threshold).
 The number of necessary iterations of the algorithm to reach the steady solution (l).
 The measured RTT from source (Ull) to end host 1 (Latona) of the first packet in the
packet pair (measurement 1). The measured RTT from Ull to end host 2 (Venus) of the
second packet in the packet pair (measurement 2).
Experiment 1
Let us define the common parameters used in the experiment 1. The missing parameters in
this list, are used to test the experiment 1 in different cases. Each case represents a
particular application of the experiment 1.
-
Link Hera-Zeus Ethernet V.2.1 Serial Link with 2 Mbit/sec ;
s = 56 Byte ;
d = 0.1 ms ;
B = 20 ;
Threshold = 10e-2 ;
Case 1
In the first case, the source sends five packet pairs to the end hosts. The RTTs measured are
mentioned as following:
- n= 5;
- measurement1=[1.66;1.26;1.26;1.36;1.24] , measurement2=[1.8;1.57;1.56;1.62;1.54];
The solution of the algorithm is obtained by a number of the iterations l= 13. The vector
a2(d), a3(d) and a4(d) represent the numerical results. The i-th component of a vector
provide the probability of the i-th bin size of the set Q.
72
6 Results
a2(d)=[0,0,1.0319e-318,2.6847e-134,1.1356e-038,1.3687e-009,0.20055,0.59945,
1.1346e-018, 0.2, 1.3218e-020, 9.4317e-057, 1.8829e-197,0,0,0,0,0,0,0];
a3(d)=[0,0,0,8.3214e-177, 4.192e-062 ,4.1965e-022, 0.59978 , 0.19988, 0.20033,
1.4066e-020, 3.5397e-032,1.7487e-113,1.7754e-286,0,0,0,0,0,0,0];
a4(d)=[0,0,0,0,0, 2.5741e-321, 3.3389e-123, 3.1357e-035, 1.1342e-018, 0.99945,
0.00054969,2.1129e-020, 8.8e-083,2.1445e-219,0,0,0,0,0,0];
The graphical results are shown in the following Figures 6.1, 6.2, and 6.3.
Delay Probability Distribution - Link 2
0.7
0.59945
0.6
Probability a2(d)
0.5
0.4
0.3
0.20055
0.2
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.1: Delay Probability Distribution - Link 2
Figure 6.1 shows the delay probability distribution computed along link 2. It represents, in
particular, the probability distribution of the experienced delays over a common path.
Analyzing the Figure 6.1, d=0.7 ms represents the delay with the higher probability value
(0.59945). d=0.7 represents then the value of delay which should be experienced by a
packet pairs along link 2. The shape of the distribution shows the narrow range of possible
delays. From d=0.6 ms to d=0.9 ms the delay probability are not null. It is important to
analyze the missing value d=0.8.
Figure 6.2 depicts the delay probability distribution over link 4. The estimated delay is
d=0.9 with an unitary probability. The sum of two delays experienced over link 2 and link
4 should provide at least one of the delays measured along the path. The case d=0.8 and
d=0.9 cannot exist. The components of the vector measurement2 , in fact, show how the
value d=1.7 does not belong to the set of dicretized measurement2.
73
Emanuele Orlando
Finally, Figure 6.3 shows d=0.7 as the delay with the higher value of probability (0.59978).
After each experiment the congruence property has be verified. In this experiment it was
verified for each distributions.
Delay Probabability Distribution - Link 4
1
0.9
0.8
Probability a4(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.2: Delay Probability Distribution - Link 4
Delay Probability Distribution - Link 3
0.7
0.59978
0.6
Probability a3(d)
0.5
0.4
0.3
0.19988
0.20033
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.3: Delay Probability Distribution - Link 3
74
6 Results
Case 2.
The application of the first case is repeated in the the second case by increasing the number
of sent packet pairs. The goal is to understand how the results are dependent on the number
of probe sent.
- n = 10 ;
- measurement1=[1.57;1.26;1.26;1.26;1.27;1.28;1.39;1.29;1.26;1.24],
measurement2= [1.84;1.54;1.61;1.57;1.59;1.54;1.56;1.62;1.56;1.54];
The solution of the algorithm is obtained by a number of the iterations l= 16. An increase
of the number of the iterations is experienced respect the previous case.
The following are the numerical results.
a2(d)=[0, 0, 0, 4.0007e-151, 3.1324e-040, 3.3682e-012, 0.29778, 0.70222, 3.0669e-006,
2.1825e-033, 9.3275e-166, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ;
a3(d)=[0, 0, 0, 0, 2.1653e-098, 1.3224e-014, 0.60222, 0.39778, 3.0632e-013, 2.6024e039,3.9767e-152,0,0,0,0,0,0,0,0,0,0] ;
a4(d)=[0,0,0,0,0,0,2.0951e-215,2.0615e-046,0.0029318,0.89636,0.00071253,
0.099997,4.8242e-030,1.5062e-114,0,0,0,0,0,0,0] ;
The graphical results are shown in the following Figures 6.4, 6.5, and 6.6.
Delay Probability Distribution - Link 2
0.8
0.70222
0.7
Probability a2(d)
0.6
0.5
0.4
0.29778
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.4: Delay Probability Distribution - Link 2
75
Emanuele Orlando
Delay Probability Distribution - Link 3
0.7
0.60222
0.6
Probability a3(d)
0.5
0.39778
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.5: Delay Probability Distribution - Link 3
Delay Probability Distribution - Link 4
1
0.89636
0.9
0.8
Probability a4(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.099997
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.6: Delay Probability Distribution - Link 4
The case 2 allows to understand the importance of the measurement to solve an inference
problem. A comparison with the case 1 can show the difference between the distributions
depicted in the Figure 6.1,6.2 and 6.3 and those shown in the Figure 6.4,6.5, and 6.6. The
76
6 Results
average shape is substantially the same but the probability values change. An increase of
the number of the sent probe provides more information to feed to the estimation process.
For example in the Figure 6.6 at difference of the Figure 6.3, the value of delay d=1.1 has a
not null value probability (0.99997). An increase of measurements allows a more detailed
estimation, because a the estimation is computed with an higher number of samples.
Case 3.
The third case focus, as the second case, on the increase of the packet pairs sent.
- n=15 ;
- measurement1=[1.57;1.26;1.26;1.26;1.27;1.28;1.39;1.29;1.26;1.24;1.24;1.27;
1.26;1.41;1.26] ,
measurement2=[1.84;1.54;1.61;1.57;1.59;1.54;1.56;1.62;1.56;1.54;1.54;1.63;
1.54;1.53;1.63] ;
The graphical results are shown in the following Figures 6.7, 6.8 and 6.9.
Delay Probability Distribution - Link 2
0.9
0.82751
0.8
Probability a2(d)
0.7
0.6
0.5
0.4
0.3
0.17249
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.7: Delay Probability Distribution - Link 2
77
Emanuele Orlando
Delay Probability Dostribution - Link 3
0.9
0.8
0.76086
Probability a3(d)
0.7
0.6
0.5
0.4
0.3
0.23912
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.8: Delay Probability Distribution - Link 3
Delay Probability Distribution - Link 4
0.8
0.70582
0.7
Probability a4(d)
0.6
0.5
0.4
0.3
0.22751
0.2
0.1
0
0.066667
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Delay d (ms)
Figure 6.9: Delay Probability Distribution - Link 4
The following are the numerical results.
78
6 Results
The solution of the algorithm is obtained by a number of the iterations l= 19. Once again an
increase of the number of the iterations is experienced compered the previous two cases.
a2(d)=[0,0,0,0,2.7913e-038,3.3377e-027,0.17249,0.82751,7.6175e-010,1.2494e-050,
0,0,0,0,0,0,0,0,0,0,0] ;
a3(d)=[0,0,0,0,1.7203e-135,5.5582e-011,0.76086,0.23912,1.5179e-005,9.3104e-093,
5.6394e-1730,0,0,0,0,0,0,0,0,0] ;
a4(d)=[0,0,0,0,0,0,1.1347e-205,1.8386e-035,0.22751,0.70582,1.3841e-008, 0.066667,
1.1034e-024,2.1892e-309,0,0,0,0,0,0,0];
Increase of input of sending Packet Pairs
Let us compare the Figures 6.1, 6.4, 6.7. They are the inferred distributions of the link 2 by
probing the network with a different number of packet pairs n. The results obtained allow
to see if the consistence property is verified. Analyzing the distributions, the probability
1
0.9
0.8
0.7
n=5
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
1
0.9
0.8
0.7
0.6
n=10
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
1
0.9
0.8
0.7
0.6
n=15
0.5
0.4
0.3
0.2
0.1
0
d=0.7ms
Figure 6.10: The consistence property is verified. The augment of n gives more
weight to the most probable delay d.
value of d = 0.7 ms prevails by increasing n. This corresponds to the forecast, because the
algorithm is built by maximizing the MLE function. The MLE verifies the Consistence
property, and the estimated value gets closer to the true value by increasing the number of
79
Emanuele Orlando
samples used on a process of estimation. Figure 6.10 shows how the distribution converges
to a specific value.
Another important aspect is the number of the necessary iterations of the algorithm to reach
the steady solution. The cases 1,2 and 3 show how the number of steps l increases if the
number of packet pair n increases. In particular:
n=5
n=10
n=15
l=13
l=16
l=19
For these reasons a strong increase of n results in an inevitable increase of the number of
necessary iterations to maximize the MLE. The computational process contains an
inevitable error of approximation, because it is impossible to achieve the infinite precision.
It is preferable to get the solution with the low number of step l. It is vital to find a trade-off
between the number of packet pairs sent n and the number of
steps l.
Case 4.
This case is the same as the experiment of the Case 2, but is conducted by changing the bin
size from d=0.1 to d=0.05. A decrease of bin size generates an increase of the number of
bin to cover the same interval.
B= 40 ;
The graphical results are shown in the following Figures 6.11, 6.12, 6.13. The following
are the numerical results.
The solution of the algorithm is obtained by a number of the iterations l= 16. It is the same
number of the case 2. The increase of bin size does not provide an increase the number of
steps l . The Pseudo Code in the Figure 5.2 in the Section 5 shows how the comparison
between the current and previous estimations are made on the probability values and not on
the bin size.
a2(d)=[0,0,0,0,0,0,0,0,5.879e-286,3.2071e-169,8.922e-085,1.0143e-012,0.83438,0.065622,
1.633e- 025, 5.8999e-163, 1.7058e-109, 6.9105e-012, 0.1, 3.1622e-024,1.452e-212,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
a3(d)=[0, 0, 0, 0, 0, 0, 0, 1.9034e-244, 2.7781e-225, 7.0648e-225,6.782e-279, 1.5148e-065,
2.8166e-005,0.76557,0.13441,6.4054e-017,0.1,2.3849e-041,1.1464e-199,0,0,0, 0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
80
6 Results
a4(d)=[0,0,0,0,0,0,0, 0, 0, 0, 0, 0, 0, 2.293e-292, 2.2033e-149, 1.2966e-159, 1.2118e-179,
1.9668e-058,1.2242e-005,0.7656,0.23439,6.7694e-042,2.1703e-128,5.5382e-134,
3.5743e-251,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
Delay Probability Distribution - Link 2
1
0.9
0.83438
0.8
Probability a2(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.065622
0.1
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Delay d (ms)
Figure 6.11: Delay Probability Distribution - Link 2
Delay Probability Distribution - Link 3
0.9
0.8
0.76557
Probability a3(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.13441
0.1
0.1
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
Delay d (ms)
Figure 6.12: Delay Probability Distribution - Link 3
81
2
Emanuele Orlando
Delay Probability Distribution - Link 4
0.9
0.7656
0.8
Probability a4(d)
0.7
0.6
0.5
0.4
0.3
0.23439
0.2
0.1
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Delay d (ms)
Figure 6.13: Delay Probability Distribution - Link 4
Decrease of the bin size.
A comparison between Figure 6.4,6.5,and 6.6 of the case 2 and the current results depicted
in the Figure 6.11,6.12, and 6.13, show that the accuracy of the distributions depends on the
choice of d. If the bin size is decreased, the discretization error is also reduced. The
algorithm can therefore use the values which were discretizated by a larger bin size. The
average steady solution is the same for the Case 2 and the Case 4. The accuracy of the
new distribution is increased. The following tables depicted in the Figure 6.14 shows how
the distributions maintain the same average characteristics and at new values of delays are
assigned a not null probability.
An inevitable consequence of a decrease of the bin size is an increase of the number B. The
Figures 6.11, 6.12, 6.13 show how the distributions are concentrated in a small region in
respect to an empty region of probability values. There are too much “dead” regions
analyzed pointlessly by the algorithm. A solution of this problem is to apply a variant of the
Lo Presti algorithm called Variable Bin Size Discrete Model [6]. In this discrete
model, Dk takes values in a more general finite set Q. This is motivated by the observation
that the use of a fixed bin size may be too restrictive in a network where delay
characteristics vary significantly. The Variable Bin Size Discrete Model defines a set of Q,
82
6 Results
which guarantees the desired resolution in the delay range of interest [6].
Delay
(ms)
0.65
0.7
0.95
Delay
(ms)
0.7
0.75
0.85
Delay
(ms)
1
1.05
*
Link 2 d=0.05
Probability
Link 2 d=0.1
Delay
Probability
(ms)
0.7
0.29
0.8
0.7
*
*
0.83
0.065622
0.1
Link 3 d=0.05
Probability
Link 3 d=0.1
Delay
Probability
(ms)
0.7
0.6
0.8
0.39778
*
*
0.76557
0.13441
0.1
Link 4 d=0.05
Probability
Link 4 d=0.1
Delay
Probability
(ms)
0.9
0.0029318
1
0.89636
1.2
0.0999
0.7656
0.23439
*
* negligible
Figure 6.14: Decreasing the bin size, the average solution is the same but the
shape of the distributions have a major accuracy.
Case 5
This case is the same as the experiment of the Case 2, but is conducted by changing the size
of the packets sent from s=56Byte to s=256Byte. The measurements are obtained by testing
the network with a larger packet pairs. All the other common characteristics of the
experiment 1 are maintained.
-
n=10 ;
s=256 ;
d=0.1 ;
B=40 ;
83
Emanuele Orlando
- measurement1=[2.74;2.71;2.64;2.71;2.69;2.76;2.68;2.7;2.69;2.64] ,
measurement2=[3.41;3.36;3.33;3.36;3.33;3.38;3.32;3.36;3.36;3.38] ;
The following are the numerical results.
The solution of the algorithm is obtained by a number of the iterations l= 33.
a2(d)=[0,0,0,0,0,0,0,0,0,0,0,2.7801e-266,1.5318e-009,0.88708,0.11292,2.0391e-058,
0,0,0,0,0,0,0 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
a3(d)=[0,0,0,0,0,0,0,0,0,0,0,4.9354e-245,5.5884e-011,0.21292,0.78708,6.9427e-007,
3.2409e-274,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
a4(d)=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.011e-239,4.0981e-013,0.41292,0.58708,
1.4906e-060,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
The Case 5 does not provide with much information. This represents an analogy of the
Case 2. The particularity is represented by the measurements, which are much different
from the Case 2. The reason for the difference is the different size of the packets traveling
on the path which has the same properties in both the Cases. This augment of the
measurements requires the increase of the number of the bin size B and, consequently, the
increase of the number of the iterations l . It is important to find the trade-off between d and
B to obtain better results. Even in this Case the consistence property is verified by
increasing the number of the packet sent n.
The graphical results are shown in the following Figures 6.15, 6.16 and 6.17.
Delay Probability Distribution - Link 2
1
0.88708
0.9
0.8
Probability a2(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.11292
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
Delay d (ms)
Figure 6.15: Delay Probability Distribution - Link 2
84
6 Results
Delay Probability Distribution - Link 3
0.9
0.78708
0.8
Probability a3(d)
0.7
0.6
0.5
0.4
0.3
0.21292
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
Delay d (ms)
Figure 6.16: Delay Probability Distribution - Link 3
Delay Probability Distribution - Link 4
0.58708
0.6
Probability a4(d)
0.5
0.41292
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
Delay d (ms)
Figure 6.17: Delay Probability Distribution - Link 4
85
Emanuele Orlando
Figure 6.15, 6.16 and 6.17 show the increase of the number of bin. The estimate does not
loose its accuracy, but the algorithm paid useless efforts to compute the probability values
for almost the totality of the set Q.
Experiment 2
Experiment 2 tests the network of the experiment 1 in which the speed on the link HeraZeus is decreased. In particular the speed from 2 Mbit/sec to 0.5 Mbit/sec is decreased. The
goal is to test the algorithm on a link with a lower speed and to compare the obtained
results with the previous experiment 1. The following are the common parameters of the
experiment 2. Once again, the missing parameters in this list, are used to test the
experiment 2 in different cases. Each case represents a particular application of the
experiment 2.
-
Link Hera-Zeus V.21 Serial Link with 0.5 Mbit/sec ;
s = 56 Byte ;
d = 0.1 ms ;
B = 70 ;
Threshold = 10e-2 ;
Case1
The present application describes an experience with a number of packet pairs sent n=10.
The following are the obtained measurements.
- n=10 ;
- measurement1= [4.24;5.49;3.91;3.87;3.93;3.94;3.93;3.96;3.91;3.91] ,
measurement2= [5.67;6.94;5.36;5.37;5.37;5.32;5.33;5.36;5.36;5.34] ;
The following are the numerical results.
The solution of the algorithm is obtained by a number of the iterations l= 17.
a2(d)=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.5737e-019, 0.7022, 0.097797,1.6536e-127, 0.1,
5.1705e-034,0,0,0,0,0,0,0,0,0,0,2.781e-017,0.1,1.1175e-128,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] ;
a3(d)=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4.575e-320,3.2416e-010, 0.9978,0.0022027,
2.4151e-221,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0] ;
86
6 Results
a4(d)=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2.1812e041,0.4978,
0.5022,9.4424e-076,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0] ;
The graphical results are shown in the following Figures 6.18, 6.19 and 6.20.
Delay Probability Distribution - Link 2
0.8
0.7022
0.7
Probability a2(d)
0.6
0.5
0.4
0.3
0.2
0.097797
0.1
0.1
0
0
0.5
1
1.5
2
0.1
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Delay d (ms)
Figure 6.18: Delay Probability Distribution - Link 2
Delay Probability Distribution - Link 3
1
0.9
0.8
Probability a3(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Delay d (ms)
Figure 6.19: Delay Probability Distribution - Link 3
87
Emanuele Orlando
Delay Probability Distribution - Link 4
0.4978
Probability a4(d)
0.5
0.5022
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Delay d (ms)
Figure 6.20: Delay Probability Distribution - Link 4
Testing the network with a same packet length s, a decrease of the speed on the link HeraZeus results in an increase of the measured delay. The experiment decrease, in particular,
for times the speed of the link Hera-Zeus respect the experiment 1. The experienced delay
of the vectors measurement1 and measurement2 are increased compared the case1 in
experiment 1. Let us focus the attention on the difference between the measurement1 and
measurement2. In a path where the dimension of the packets size is comparable to its
capability, the second member of the packet pairs experiences higher congestion from the
first packet. This can be noticed by the high gap between measurement1 and
measurement2. The algorithm does not make difference to compute its estimate. The
information is contained in the measurements.
By increasing the number of the packet pairs n, the Consistence property is verified.
Case 2.
This case is the same as the experiment of the Case 1, but is conducted by changing the bin
size from d=0.1 to d=0.05. A decrease of bin size generates an increase of the number of
bin to cover the same interval.
B= 140 ;
The graphical results are shown in the following Figures 6.21, 6.22 and 6.23.
88
6 Results
The solution of the algorithm is obtained by a number of the iterations l= 17 ;
Delay Probability Distribution - Link 2
0.9
0.8
0.8
Probability a2(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.099984
0.1
0
0
0.5
1
1.5
2
2.5
0.1
3
3.5
4
4.5
5
5.5
6
6.5
7
Delay d (ms)
Figure 6.21: Delay Probability Distribution - Link 2
The distributions provide a higher accuracy grade than in the Case 1. A comparison of the
Figure 6.21 with Figure 6.18 shows for example the increase of accuracy in the range from
d=1,9ms to d=2ms. Using a smaller bin size, the value d=2ms disappears from the Figure
6.18 to become d=1,95ms in the Figure 6.21. In the case 1, in fact, the value d=1,95ms
belonged to the bin d=2ms and the discretization provided a loss of information.
The algorithm carries out optimal results both in Case 1 and Case 2. The requirements of
particular accuracy depend on the needs of the user. An increase of accuracy does not
provide an increase of the number of iterations. Both the cases experienced the same l =17.
The substantial difference is represented by the processing time of the algorithm to carry
out the results. An increase of bins needs an higher number of operations even if the
number of steps to reach the solution is the same.
89
Emanuele Orlando
Delay Probability Distribution - Link 3
0.49998
Probability a3(d)
0.5
0.40002
0.4
0.3
0.2
0.1
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Delay d (ms)
Figure 6.22: Delay Probability distribution - Link 3
Delay Probability Distribution - Link 4
1
0.89998
0.9
0.8
Probability a4(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.10002
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Delay d (ms)
Figure 6.23: Delay Probability Distribution - Link 4
The present case 2 it is a proof that with high values of measurement the Variable Bin Size
Model Fixed [6] is preferable to the Bin Size Model by Lo Presti described in Section 3.5.
The Variable Bin Size Model allows to estimate the distribution by using a different set Q.
90
6 Results
The “dead” regions of the experienced delay can be substituted with a set Q with an higher
bin size. The interest region will be reached faster and without loss of information. In the
range where the delay is experienced, a set Q with smaller bin size can be used. An
increase of accuracy is provided only in the range where the probability distribution is not
null.
The average distributions and the congruence property are maintained.
Congruence Property
The Congruence Property is verified in each Case of the Experiment 1 and 2. This aspect
is very important for testing the reliability of a distribution.
Different kinds of experiments
A lot of experiments have been conducted to test the characteristics of the Lo Presti Fixed
Bin Size Discrete Model algorithm.
The algorithm is tested with a maximum number of packet pairs n= 30. The results are
highly satisfying, although the number of the iterations is significantly increased.
The algorithm is tested with a bin size d=0.2ms, but the results are not reliable particularly
for the measurements with smaller values. The loss of accuracy influences significantly the
shape of the distributions .
Finally, the algorithm works sufficiently well until the values of the measurements are not
high. In order not to loose accuracy, it is advisable to choose a d=0.1 and to increase the
number of bin to reach the required zone of delays, which is a problem because the
algorithm has an higher computational complexity in terms of time processing. In a LAN
the delays are quite limited and this property allows the algorithm to be able to carry out
good results.
91
Emanuele Orlando
92