### 4.1 Energy consumption model

#### 4.1.1 Total number of hops between the cluster-heads and the sink node

The average number of hops between the cluster-heads and the sink node depends on the sink node's location. We consider a disc-shaped sensing terrain with radius *r*. According to AS 4, the sink node is placed at the center of the sensing terrain. Any cluster-head having one hop to the sink node may be placed to an area which is disc-shaped with radius *R*. In the same manner, any cluster-head having two hops to the sink node may be placed to a ring-shaped area whose outer radius is *2R* and inner radius is *R*. Consequently, cluster-heads with *k* hops to the sink node may be placed in a ring-shaped area whose outer radius is *kR* and whose inner radius is *(k - 1)R*. We depict this approach in Figure 1a.

Let *λ*_{CH} denote the density of the cluster-heads in the network. Define *I*_{
k
}to be the number of cluster-heads with *k* hops from the sink node. Then,

E\left[{I}_{k}\right]={\lambda}_{\mathsf{\text{CH}}}{\int}_{\left(k-1\right)R}^{kR}2\pi r\phantom{\rule{0.3em}{0ex}}dr.

(5)

Let *X* be the total number of hops from all the cluster-heads to the sink node in the network. Since the total number of hops of cluster-heads with *k* hops to the sink node is given as *kI*_{
k
}, we have

\begin{array}{ll}\hfill E\left[X\right]& ={\lambda}_{\mathsf{\text{CH}}}\sum _{k=1}^{u}k{\int}_{\left(k-1\right)R}^{kR}2\pi r\phantom{\rule{0.3em}{0ex}}dr\phantom{\rule{2em}{0ex}}\\ =\pi {\lambda}_{\mathsf{\text{CH}}}\sum _{k=1}^{u}k\left(2k-1\right){R}^{2}\phantom{\rule{2em}{0ex}}\\ =\pi {\lambda}_{\mathsf{\text{CH}}}{R}^{2}\frac{u\left(u+1\right)\left(4u-1\right)}{6},\phantom{\rule{2em}{0ex}}\end{array}

(6)

where *u = r/R*.

Our modeling approach can be applied to an arbitrary-shaped network with a mathematical modification though the model becomes more complicated. For example, in the case of a rectangular-shaped sensing terrain with side *2r'* as shown in Figure 1b, deriving the number of hops from the cluster-heads inside a circle with radius *r'* to the sink node is referred to as the modeling approach of a disc-shaped network. Then, the number of hops from the other cluster-heads located outside the circle to the sink node is derived in a mathematical modification which considers the area of the outside region and the distance from the cluster-heads in the outside to the sink node. In this article, we deal with a disc-shaped network for mathematical simplicity.

#### 4.1.2 Total number of hops between the cluster-members and their respective cluster-heads

Generally, as the distance between a sensor node and a cluster-head increases, a possibility that the sensor node becomes a member of a cluster with the cluster-head decreases. This is because as two nodes become more distant, the number of hops between them is likely to become larger and in the clustering algorithm, any node that is not a cluster-head joins a cluster with a cluster-head that has the smallest number of hops from it.

Now, we will derive a probability a node to join a cluster with a cluster-head when the distance between the node and the cluster-head is given. Let *x* be the distance from a node to a cluster-head, and it can be ranged from *a* to *b*, i.e, *a* ≤ *x* ≤ *b*. If *a* is not zero, then a region where the node can be placed is ring-shaped as shown in Figure 2. Let *A*^{[a,b]}be an area of the ring-shaped region. We can divide the interval [*a, b*] into *m* subintervals of equal length Δ*x* = (*b - a*)*/m*. Let *x*_{0}(= *a*)*, x*_{1}, *x*_{2}*,..., x*_{
m
}(*= b*) be the end point of these subintervals. Then, *A*^{[a,b]}is equivalent to \underset{m\to \infty}{lim}{\sum}_{i=0}^{m}\pi \left({x}_{i+1}^{2}-{x}_{i}^{2}\right). Since the density of the nodes is given as *λ*, the average number of nodes in the ring-shaped region can be approximated to *λA*^{[ a,b]}.

Though two sensor nodes are placed within the same distance *x* from a cluster-head, they can be members of different clusters as illustrated in Figure 3. This shows that the probability that a sensor node joins a cluster with a cluster-head is influenced by the existence of other cluster-heads as well. To deal with such problem, we employ a probability that a node becomes a member of a certain cluster with the consideration of cluster-head density. Let CH(1) be the cluster-head of the cluster(1). As shown in Figure 3, when the distance from a node to CH(1) is *x*, let *P* {(*x*, CH(1)) ∈ cluster(1)} be the probability that the node becomes a member of cluster(1). Then, let *M*^{[a,b]}be the number of the member nodes which belong to cluster(1) and are located in a ring-shaped area whose the inner radius is *a* and the outer radius is *b*. Then, we have

\begin{array}{ll}\hfill E\left[{M}^{\left[a,b\right]}\right]& ={\lambda}_{\mathsf{\text{CM}}}\underset{x\to \infty}{lim}\sum _{i=0}^{m}\pi \left({x}_{i+1}^{2}-{x}_{i}^{2}\right)\cdot P\left\{\left({x}_{i}+\frac{\Delta x}{2},\mathsf{\text{CH}}\left(1\right)\right)\in \mathsf{\text{cluster}}\left(1\right)\right\}\phantom{\rule{2em}{0ex}}\\ ={\lambda}_{\mathsf{\text{CM}}}\underset{x\to \infty}{lim}\sum _{i=0}^{m}\pi \left(2{x}_{i}\Delta x+\Delta {x}^{2}\right)\cdot P\left\{\left({x}_{i}+\frac{\Delta x}{2},\mathsf{\text{CH}}\left(1\right)\right)\in \mathsf{\text{cluster}}\left(1\right)\right\},\phantom{\rule{2em}{0ex}}\end{array}

(7)

where *λ*_{CM} denotes the density of the cluster-members in the network. As *m* goes to infinity, Δ*x* becomes extremely small, and we can ignore Δ*x*^{2}. Similarly, we can regard *x*_{
i
}*+* Δ*x/* 2 as *x*_{
i
}. Then,

E\left[{M}^{\left[a,b\right]}\right]=2\pi {\lambda}_{\mathsf{\text{CM}}}\underset{x\to \infty}{lim}\sum _{i=0}^{m}{x}_{i}\Delta x\cdot P\left\{\left({x}_{i},\mathsf{\text{CH}}\left(1\right)\right)\in \mathsf{\text{cluster}}\left(1\right)\right\}.

(8)

The probability *P*{(*x*,CH(1)) ∈ cluster(1)} can be approximated to the probability that any cluster-head does not exist within distance *x* from a non-cluster-head node. Since the area of the sensing terrain is *A*, the number of cluster-heads can be approximated to *λ*_{CH}*A* According to Campbell's theorem and the results in [17], we get

P\left\{\left(x,\mathsf{\text{CH}}\left(1\right)\right)\in \mathsf{\text{cluster}}\left(1\right)\right\}={{\left(1-\frac{\pi {x}^{2}}{A}\right)}^{\lambda}}^{{}_{\mathsf{\text{CH}}}A}.

(9)

When the sensor field is large, we approximately have

P\left\{\left(x,\mathsf{\text{CH}}\left(1\right)\right)\in \mathsf{\text{cluster}}\left(1\right)\right\}\approx \underset{A\to \infty}{lim}{\left(1-\frac{\pi {x}^{2}}{A}\right)}^{{\lambda}_{\mathsf{\text{CH}}}A}={e}^{-\pi {\lambda}_{\mathsf{\text{CH}}}{x}^{2}}.

(10)

From Equations 8 and 10, we have

E\left[{M}^{\left[a,b\right]}\right]=2\pi {\lambda}_{\mathsf{\text{CM}}}{\int}_{a}^{b}x\cdot {e}^{-\pi {\lambda}_{\mathsf{\text{CH}}}{x}^{2}}dx.

(11)

If we set *a* = 0 and *b* = ∞, then *M*^{[0,∞]} is the number of member nodes in an arbitrary-shaped cluster.

The total number of the member nodes having *k* hops from a cluster-head can be expressed as *M*^{[(k-1)R,kR]}, and thus, the total number of hops between the member nodes and the cluster-head is approximated to *kM*^{[(k-1)R,kR]}. Since *p* is the probability of being a cluster-head, the density of cluster-heads and cluster-members can be expressed as *pλ* and (1 *-p)λ*, respectively. Let *Y*_{
0
}be the total number of hops between all member nodes and the cluster-head in a cluster. Then, we have

\begin{array}{ll}\hfill E\left[{Y}_{0}\right]& =2\pi \left(1-p\right)\lambda \sum _{k=1}^{\infty}k{\int}_{\left(k-1\right)R}^{kR}x\cdot {e}^{-\pi p\lambda {x}^{2}}dx\phantom{\rule{2em}{0ex}}\\ =\frac{\left(1-p\right)}{p}\sum _{k=0}^{\infty}{e}^{-\pi \lambda {\left(kR\right)}^{2}p}.\phantom{\rule{2em}{0ex}}\end{array}

(12)

Let *Y* be the total number of hops between all the cluster-members and their respective cluster-heads in a network. Since there are *λAp* clusters on average, the expected value of *Y* is as follows:

\begin{array}{ll}\hfill E\left[Y\right]& =\lambda Ap\cdot E\left[{Y}_{0}\right]\phantom{\rule{2em}{0ex}}\\ =\lambda A\left(1-p\right)\sum _{k=0}^{\infty}{e}^{-\pi \lambda {\left(kR\right)}^{2}p}.\phantom{\rule{2em}{0ex}}\end{array}

(13)

#### 4.1.3 MAC inefficiency and signaling overhead

The energy loss due to inefficient operations in MAC, such as idle listening or overhearing, and clustering overhead may depend on the MAC protocol, the routing protocol and the clustering algorithm that are used [12]. We define *e*_{wt} and *e*_{wr} as the energy wasted by a transmitter due to MAC inefficiency for transmitting a bit and the energy wasted by a receiver due to MAC inefficiency for receiving a bit in one-hop communication, respectively. Then, we replace *α*_{
0
}and *α*_{1} with {\alpha}_{0}^{\prime} and {\alpha}_{1}^{\prime}, where {\alpha}_{0}^{\prime}={\alpha}_{0}+{e}_{\mathsf{\text{wt}}} and {\alpha}_{1}^{\prime}={\alpha}_{1}+{e}_{\mathsf{\text{rt}}}.

The signaling overhead associated with clustering consists of two major factors: one for the cluster-head selection and another for the distribution of TDMA schedules. To select the cluster-head, each sensor node receives advertisement messages from its neighboring nodes and the node forwards a message which advertises its cluster-head to the other nodes. Let the length of an advertisement message be *l*_{1} bits. Then, the energy consumed for the cluster-head selection in a network, *S*_{1}, is defined as follows:

E\left[{S}_{1}\right]={\phi}_{1}\lambda A,

(14)

where {\phi}_{1}=\left({{\alpha}^{\prime}}_{0}+\lambda \pi {R}^{2}{{\alpha}^{\prime}}_{1}+\beta {R}^{t}\right){l}_{1}*. φ*_{1} represents the energy consumed for data processing, receiving an *l*_{1} bits message from the neighboring nodes, and transmitting *l*_{1} bits message over the radio range *(R)*.

To avoid data collision, the cluster-heads and the sink node set up TDMA schedules for each node in their respective clusters and for the cluster-heads, respectively. Then, the cluster-heads and the sink node distribute the schedules to their clustered nodes and the cluster-heads in the network, respectively. In the case of the cluster-heads, the schedules are distributed twice; one for data collection and another for aggregated data report. Let the length of a TDMA schedule message be *l*_{2} bits. Then, the energy consumed for the distribution of the TDMA schedules in a network, *S*_{2}, can be expressed using the total energy consumed to collect the sensed information. Then, we have

E\left[{S}_{2}\right]={\phi}_{2}\left(E\left[X\right]+2E\left[Y\right]\right),

(15)

where {\phi}_{2}=\left({\alpha}_{0}^{\prime}+{\alpha}_{1}^{\prime}+\beta {R}^{t}\right){l}_{2}*. φ*_{
2
}represents the energy consumed for data processing, transmitting, and receiving an *l*_{2} bits message over the radio range *(R)*.

#### 4.1.4 Total energy consumption in the network

In Equation 3, we showed that the energy required for data transmission and reception depends on the number of hops between the end-to-end nodes. In addition, the cluster-heads consume additional energy due to data aggregation. Since we are interested in the total energy consumption in a network, we need to derive the total number of data streams to be aggregated by all cluster-heads, which equals to the number of nodes, i.e., *λA*. Let *Z* be the total energy consumed by the cluster-heads for aggregating *l* bits messages in a network. Then, from Equation 2, we have

E\left[Z\right]=\gamma \lambda Al.

(16)

Then, we can derive the total energy consumed by all nodes in a single round as the sum of the energy consumed for processing, transmitting, receiving, aggregating, and signaling. From Equations 3, 6, 13-16, we can derive *C(p)* as follows:

\begin{array}{ll}\hfill C\left(p\right)& ={\phi}_{0}E\left[X\right]+{\phi}_{0}E\left[Y\right]+E\left[Z\right]+E\left[{S}_{1}\right]+E\left[{S}_{2}\right]\phantom{\rule{2em}{0ex}}\\ =\pi \lambda {R}^{2}\left({\phi}_{0}+{\phi}_{2}\right)\frac{u\left(u+1\right)\left(4u-1\right)}{6}p+\lambda A\left(1-p\right)\left({\phi}_{0}+2{\phi}_{2}\right)\sum _{k=0}^{\infty}{e}^{-\lambda \pi {\left(kR\right)}^{2}p}\phantom{\rule{2em}{0ex}}\\ +{\phi}_{1}\lambda A+\mu ,\phantom{\rule{2em}{0ex}}\end{array}

(17)

where {\phi}_{0}=\left({{\alpha}^{\prime}}_{0}+{{\alpha}^{\prime}}_{1}+\beta {d}^{t}\right)l and *μ* = *γλAl. φ*_{0} and *μ* represent the energy consumed for data processing, transmitting, and receiving an *l* bits message, and the total energy consumption by the cluster-heads for aggregating information in a network, respectively.

### 4.2 Optimal clustering

From Equations 4 and 17, we can determine the optimal probability *p** to minimize the total energy consumption. According to the Galois Theory [18], *p** cannot be obtained by elementary algebra. However, we can use numerical methods to solve a general polynomial equation [9]. Since *C(p)* is a convex function, we use Newton's method to find a minimum of *C(p)*. The proof of the convexity is shown in the Appendix.

Though we assume a disc-shaped sensing terrain for mathematical simplicity, our model enables to simply determine the optimal number of clusters because it only requires information on the node density, the area of sensing terrain, and the radio range to find a solution.