**6. Results and discussion**

In this section, simulation results are presented and analyzed. The simulation parameters are presented in Table 2. Channel parameters are obtained from [1], whereas energy consumption parameters are taken as in [18], where measurements are made with 3G communications on the LR, and 802.11 b on the SR using the rate adaptive approach.

In Sections 6.1 to 6.3, we investigate a scenario corresponding to multihop data transmission in a WSN. We consider that each sensor sends its measurement data in a file of size *ST* = 1 Mbits, to be routed to the BS in an energy efficient manner. Two main SN deployment scenarios are investigated:


**Table 2.** Simulation Parameters

12 Will-be-set-by-IN-TECH

• **Step 5:** increment *k* and repeat Steps 3-5 on the SNs whose order is *> k* in the sorted list. • **Step 6:** After all the SNs have been grouped into clusters based on the most energy efficient method, we check if SN *K* can send the data with lower energy than sending on the LR link, since it is still connected to the BS (due to sorting the SNs in decreasing order of LR energy consumption). Hence, if there exists an SN *x* � *K* such that *px* = 0 (i.e. there is another cluster with cluster head other than SN *K*, which means that the link between the BS and SN *K* can be broken while still being able to send the data from the SNs to the BS), then for

> *ST* · ∑ *i*∈D*<sup>K</sup>*

*P*Tx,*K*<sup>0</sup> *RK*<sup>0</sup>

Otherwise, we keep *pK* = 0, i.e., SN *K* is a cluster head sending the data to the BS. If *pK* is

This section presents a complexity analysis that applies to both methods of Sections 5.1 and 5.2. Step 1 of the algorithms is a sorting step, and hence has a worst-case complexity <sup>O</sup>(*K*2). In Step 3, the search involves *<sup>K</sup>* nodes when *<sup>j</sup>* <sup>=</sup> 1, it involves (*<sup>K</sup>* <sup>−</sup> <sup>1</sup>) nodes when *j* = 2, etc., and 2 nodes when *j* = (*K* − 1). Hence, the complexity of Steps 2

<sup>2</sup> <sup>−</sup> 1. This is a quadratic complexity of order <sup>O</sup>(*K*2). Hence, the proposed suboptimal methods are significantly easier to implement than the optimal solution of the NP-hard

In the next section, we compare the methods of Sections 5.1 and 5.2 to each other and to the

In this section, simulation results are presented and analyzed. The simulation parameters are presented in Table 2. Channel parameters are obtained from [1], whereas energy consumption parameters are taken as in [18], where measurements are made with 3G communications on

In Sections 6.1 to 6.3, we investigate a scenario corresponding to multihop data transmission in a WSN. We consider that each sensor sends its measurement data in a file of size *ST* = 1 Mbits, to be routed to the BS in an energy efficient manner. Two main SN deployment scenarios are

+ ∑ *i*∈D*K*,*i*� *K*

*<*

set as the parent of *K*, then we update C*pK* as: C*pK* = C*pK* ∪ D*<sup>K</sup>* = C*pK* ∪ *K* ∪ C*K*.

nodes. Consequently, the worst-case complexity of the algorithms is: *K*<sup>2</sup> + *<sup>K</sup>*(*K*+1)

(*P*Tx,*ij* + *P*Rx,*ij*) *Rij*

> (*P*Tx,*iK* + *P*Rx,*iK*) *RiK*

<sup>2</sup> − 1. In Steps 6-7, the search involves at most *K*

(15)

<sup>2</sup> − 1 + *K* =

all SNs *j < K* such that *pj* = 0, the parent of SN *K* is selected such that:

*pK* <sup>=</sup> arg min *<sup>j</sup>*;*pj*=<sup>0</sup>

(*P*Tx,*ipK* + *P*Rx,*ipK* ) *RipK*

the LR, and 802.11 b on the SR using the rate adaptive approach.

• **Step 7:** We set *pK* as the direct parent of *K* if

∑ *i*∈D*<sup>K</sup>*

to 5 is: *<sup>K</sup>* + (*<sup>K</sup>* <sup>−</sup> <sup>1</sup>) + ··· <sup>+</sup> <sup>2</sup> <sup>=</sup> *<sup>K</sup>*(*K*+1)

**5.3. Complexity analysis**

problem of Section 4.

investigated:

non-cooperative approach.

**6. Results and discussion**

3*K*<sup>2</sup> <sup>2</sup> <sup>+</sup> <sup>3</sup>*<sup>K</sup>*


Results are averaged over 50 iterations. In each iteration, new random SN locations are determined and 50 fading realizations are considered (thus results are averaged over 50 × 50 = 2500 fading realizations). We compare the methods of Sections 5.1 (denoted as "multihop" in the results) and 5.2 (denoted as "clustering" in the results) to the non-cooperative approach.

