4.2 LRNið Þt and LSNið Þt

MCUi¼<sup>1</sup>ð Þt is identified as the main node in the IoTN. In this way, all the embedded systems both IoTNe tð Þ and MCU1ð Þt , enable Wi-Fi Direct mode and a full table with the bandwidth of the nodes are next to them, in order to generate a list of reliable nodes in a certain period of time t (LRNið Þt ). Every MCU generates a particular LRNið Þ<sup>t</sup> ; then the proposed IoTN can generate 4<sup>n</sup> LNRs at the instant <sup>t</sup>. In addition, every LRNið Þt contains the bandwidth of all MCUið Þt with which it establishes connection. In order to belong to the wireless sensor network, every MCUið Þt must connect to at least one link to another MCUið Þt . All LRNs are shared, and ∑<sup>4</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup>MCUið Þ<sup>t</sup> knows the way to any node in the network, namely, everyone knows the topology of the network. In this way it is important to estimate the LRNið Þt , but not all of these nodes are significant to be considered as the best option to establish a link, in which manner we define LSNið Þt as the list of significant nodes (LSN) which is a vector that contains the best bandwidths of a certain MCUið Þt . Algorithm 1 shows the methodology to estimate the LSNið Þt that needs i th MCU and its LRNið Þt . At this moment the IoTN can be considered as a scarce network, since only the <sup>ρ</sup>ið Þ<sup>t</sup> <sup>β</sup> percent of the reliable links is connected.


#### 4.3 Hilbert fractal scanning

The construction of the Hilbert space-filling fractal, under the paper's graphic interpretation framework, was reduced to the replacements of the smallest elements (symbols) or interior replacements. This path can be optimized by means of the correlation of the strength of the signal or the bandwidth of the link by Eq. (1):

$$\phi\_{\mathbf{x}\mathbf{y}} = \frac{\sum\_{i=1}^{n} (\mathbf{x}\_i - \overline{\mathbf{x}}) \left(\mathbf{y}\_i - \overline{\mathbf{y}}\right)}{\sqrt{\sum\_{i=1}^{n} (\mathbf{x}\_i - \overline{\mathbf{x}})^2} \sqrt{\sum\_{i=1}^{n} (\mathbf{y}\_i - \overline{\mathbf{y}})^2}} \tag{1}$$

• R by DRRU

room, and (c) matrix γ

Figure 2.

• D by RDDL

5. Experimental results

5.1 Network simulator

117

more efficient the data parameter sharing.

LBL, Xerox PARC, UCB, USC/ISI, etc.

identify the effectiveness of the proposed algorithm.

(a) First three stages of the Hilbert curve, (b) matrix of dispersion or γ of ∑<sup>4</sup><sup>n</sup>

Optimizing a Centralized Control Topology of an IoT Network Based on Hilbert Space

DOI: http://dx.doi.org/10.5772/intechopen.87206

! ordered as a vector.

The Hilbert curve has the property of remaining in an area as long as possible before moving to a neighboring spatial region. Hence, the correlation between neighbors MCUið Þt is maximized, which is an important property in the optimization of any system. The higher the correlation for estimating the final topology, the

<sup>i</sup>¼<sup>1</sup>MCUið Þ<sup>t</sup> inside a convectional

In this section, first we approach the basic concepts of the network simulator (NS) and then use it to compare the most current SDN topologies, which helps us to

Network simulator is a software to simulate discrete events, and it was designed to aid in the research of telematic networks. NS provides support for the simulation of a multitude of protocols of the application layers (http, ftp, cbr, etc.), transport (TCP, UDP, RTP, SRM), unicast and multicast routing protocols, etc., both for networks wired as local or satellite non-wired and with complex topologies with a large number of traffic generators. NS began in 1989 as a variant to the existing REAL Network Simulator and has evolved substantially in recent years, having been developed by DARPA with the help of several network research institutions, such as

NS is basically an object-oriented simulator, written in C++, whose user interface is presented as an object-oriented Tcl language interpreter or, in other words, OTcl language. The simulator supports a hierarchy of classes written in C++, also

where ϕxy is the sample correlation coefficient, n is the stage 0 level of Hilbert curve (Figure 2a), and xi and yi are, respectively, the best and the second best individual bandwidths in a certain MCUi indexed inside the IoTN as the ith element, while x and y are, respectively, the sample mean of the best and the second best individual bandwidths in a certain MCUi. Thus, we can estimate the best link to a particular MCUi.

All ∑<sup>4</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup>MCUið Þ<sup>t</sup> are distributed randomly giving as a result a shape as shown in Figure 2b. When γ is reordered as vector, it gives as a result the Hilbert curve indexing γ ! which contains the order of the MCUið Þt (see Figure 2c).

#### 4.4 Initial topology

Once the Hilbert curve is defined as L-system, we adapt the production rules of the original work by Hilbert [4], who proposed an axiom with a D trajectory, while we propose to start with an U trajectory. Our proposal is based on the fact that most of the energy is concentrated in the nearest MCU, namely, at the right or left. In this way the production rules of the Hilbert curve are defined by:


Optimizing a Centralized Control Topology of an IoT Network Based on Hilbert Space DOI: http://dx.doi.org/10.5772/intechopen.87206

Figure 2.

(LSN) which is a vector that contains the best bandwidths of a certain MCUið Þt .

its LRNið Þt . At this moment the IoTN can be considered as a scarce network, since

The construction of the Hilbert space-filling fractal, under the paper's graphic interpretation framework, was reduced to the replacements of the smallest elements (symbols) or interior replacements. This path can be optimized by means of the correlation of the strength of the signal or the bandwidth of the link by Eq. (1):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>i</sup>¼<sup>1</sup>ð Þ xi � <sup>x</sup>

Figure 2b. When γ is reordered as vector, it gives as a result the Hilbert curve

! which contains the order of the MCUið Þt (see Figure 2c).

In this way the production rules of the Hilbert curve are defined by:

Once the Hilbert curve is defined as L-system, we adapt the production rules of the original work by Hilbert [4], who proposed an axiom with a D trajectory, while we propose to start with an U trajectory. Our proposal is based on the fact that most of the energy is concentrated in the nearest MCU, namely, at the right or left.

<sup>i</sup>¼<sup>1</sup>ð Þ xi � <sup>x</sup> yi � <sup>y</sup> � �

<sup>i</sup>¼<sup>1</sup>MCUið Þ<sup>t</sup> are distributed randomly giving as a result a shape as shown in

<sup>i</sup>¼<sup>1</sup> yi � <sup>y</sup> � �<sup>2</sup> <sup>q</sup> (1)

2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑<sup>n</sup>

where ϕxy is the sample correlation coefficient, n is the stage 0 level of Hilbert curve (Figure 2a), and xi and yi are, respectively, the best and the second best individual bandwidths in a certain MCUi indexed inside the IoTN as the ith element, while x and y are, respectively, the sample mean of the best and the second best individual bandwidths in a certain MCUi. Thus, we can estimate the best

<sup>ϕ</sup>xy <sup>¼</sup> <sup>∑</sup><sup>n</sup>

∑<sup>n</sup>

th MCU and

Algorithm 1 shows the methodology to estimate the LSNið Þt that needs i

<sup>β</sup> percent of the reliable links is connected.

Internet of Things (IoT) for Automated and Smart Applications

only the <sup>ρ</sup>ið Þ<sup>t</sup>

4.3 Hilbert fractal scanning

link to a particular MCUi.

4.4 Initial topology

• L by ULLD

116

• U is changed by LUUR

All ∑<sup>4</sup><sup>n</sup>

indexing γ

(a) First three stages of the Hilbert curve, (b) matrix of dispersion or γ of ∑<sup>4</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup>MCUið Þ<sup>t</sup> inside a convectional room, and (c) matrix γ ! ordered as a vector.


The Hilbert curve has the property of remaining in an area as long as possible before moving to a neighboring spatial region. Hence, the correlation between neighbors MCUið Þt is maximized, which is an important property in the optimization of any system. The higher the correlation for estimating the final topology, the more efficient the data parameter sharing.

#### 5. Experimental results

In this section, first we approach the basic concepts of the network simulator (NS) and then use it to compare the most current SDN topologies, which helps us to identify the effectiveness of the proposed algorithm.

#### 5.1 Network simulator

Network simulator is a software to simulate discrete events, and it was designed to aid in the research of telematic networks. NS provides support for the simulation of a multitude of protocols of the application layers (http, ftp, cbr, etc.), transport (TCP, UDP, RTP, SRM), unicast and multicast routing protocols, etc., both for networks wired as local or satellite non-wired and with complex topologies with a large number of traffic generators. NS began in 1989 as a variant to the existing REAL Network Simulator and has evolved substantially in recent years, having been developed by DARPA with the help of several network research institutions, such as LBL, Xerox PARC, UCB, USC/ISI, etc.

NS is basically an object-oriented simulator, written in C++, whose user interface is presented as an object-oriented Tcl language interpreter or, in other words, OTcl language. The simulator supports a hierarchy of classes written in C++, also

\$traffic set idle-time \$idle \$traffic set rate \$rate

DOI: http://dx.doi.org/10.5772/intechopen.87206

\$traffic attach-agent \$source #Connect the source and the sink

set sink0 [new Agent/LossMonitor] set sink1 [new Agent/LossMonitor] set sink2 [new Agent/LossMonitor] \$ns attach-agent \$n4 \$sink0 \$ns attach-agent \$n4 \$sink1 \$ns attach-agent \$n4 \$sink2

\$ns connect \$source \$sink

return \$traffic

of the next section.

5.2 Comparing SDN performance

Mostafaei in [10]

119

algorithms similar to the latest technology:

Karp and Kung in our experiments [12].
