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Abstract

In this article, the inhomogeneous energy consumption is characterized by the concept of information density, which is defined as the number of bits per unit time passing through a specific region. With information density, it is possible to derive the energy consumption of each region and determine the energy configuration scheme to maximize the network lifetime. The information density of pure ad hoc network and hybrid ad hoc network is derived. It is discovered that the information density of pure ad hoc network is inhomogeneous and the information density of hybrid ad hoc network is homogeneous, except for the regions near the edge of the entire area. With information density, the energy-limited capacity of pure and hybrid ad hoc networks is derived. The information density introduced in this article provides more insights into the information transfer of ad hoc networks, which may be applied in the energy configuration of ad hoc networks.

Keywords

Information density scaling laws ad hoc networks energy efficiency routing

Introduction

Ad hoc networks are widely applied in the civil and military domains, since they can be deployed flexibly. Due to the fact that wireless nodes in ad hoc networks generally contain limited energy supply, the energy-limited performance analysis and energy-efficient protocol design have been widely studied.

The capacity of wireless networks was extensively studied in the area of network information theory.^1,2 The energy-limited ad hoc networks were introduced by Giovanardi and Mazzini,³ where the impact of limited battery energy on the performance of ad hoc networks was evaluated. Besides, the routing protocols were investigated by Giovanardi and Mazzini.³ Zhao et al.⁴ studied the energy-limited capacity of wireless networks, where two types of traffic models were considered: the sensor-type traffic and ad hoc-type traffic. Rodoplu and Meng⁵ defined a new concept called bits-per-Joule capacity, which is the number of bits that can be delivered per Joule of energy. Then they studied the bits-per-Joule capacity of energy-limited ad hoc network. Negi and Rajeswaran⁶ studied the capacity of power-constrained ad hoc network with arbitrarily large bandwidth, such as ultra-wide hand (UWB)-enabled ad hoc network. The analysis results of Negi and Rajeswaran⁶ showed that the throughput of power-constrained ad hoc network increases with the number of nodes. Tang and Hua⁷ derived a tighter capacity bound of power-constrained ad hoc networks than Negi and Rajeswaran. Liu et al.⁸ studied scaling laws of energy efficiency in massive multiple-input multiple-output (MIMO) system. Urgaonkar and Neely⁹ studied the energy efficiency of delay-tolerant mobile ad hoc network (MANET). They found that besides pure energy constraint, the energy consumption of ad hoc network is also related with the delay. Byun et al.¹⁰ derived delay and energy-constraint capacity of the ad hoc network.

Energy efficiency is critical in the protocol design of ad hoc network. Jin et al.¹¹ proposed an energy-efficient multiple access control (MAC) protocol based on reservation and scheduling ad hoc network. Malekshan et al.¹² reduced the energy consumption of ad hoc network by adding a sleep state and reducing the transmission collisions.¹¹ Huang et al.¹³ proposed delay and power aware routing scheme, where the routing was established to minimize the end-to-end delay and power consumption. Malekshan and Zhuang¹⁴ studied the joint MAC design and power control to optimize spectrum efficiency and energy efficiency of ad hoc network simultaneously.¹⁴ The energy-efficient design was also studied in MANET. Taha et al.¹⁵ applied the fitness function technique in MANET to minimize the energy consumption of multi-path routing protocol. Specifically, in cognitive radio ad hoc networks, the energy consumption is more severe than that in traditional ad hoc networks. Zhang et al.¹⁶ proposed a cooperative spectrum access scheme for secure information transmission. Zhang et al.¹⁷ studied spectrum sensing and spectrum sharing in multi-channel cognitive radio networks (CRNs). Overall, spectrum sensing, spectrum access, spectrum handover, cooperative spectrum detection, and access schemes in CRNs will result in more energy consumption. Hence, there were also literatures studying the energy-efficient CRNs. Ren et al.¹⁸ proposed an enhanced dynamic spectrum access method to improve the energy efficiency of cognitive radio sensor networks (CRSNs).

However, the traffic pattern of ad hoc network is rarely considered in these literatures. Actually, there is a trade-off between the delay and energy consumption. The routing adopted by Gupta and Kumar,¹ Zhao et al.,⁴ and Zhang et al.¹⁹ will result in inhomogeneous routing distribution and inhomogeneous energy consumption, which will further determine the energy distribution of ad hoc network. In routings that minimize the delay of ad hoc network, energy consumption of ad hoc network is inhomogeneous. There exist hot spots where the number of routings is large and the energy consumption is heavy. However, these issues are rarely addressed. In this article, the concept of information density is proposed. Information density is defined as the number of bits passing through a specific region per unit time. Notice that the information density is related with the energy consumption, which will further determine the energy configuration of ad hoc network. In order to reveal the feature of information density, we address two types of ad hoc network: pure ad hoc network and hybrid ad hoc network. It is derived that the information density of pure ad hoc network is inhomogeneous and there exists a hot spot where the information density is maximum. However, the information density of hybrid ad hoc network is homogeneous, except for the regions near the edge of the entire area. Hence, the energy consumption of hybrid ad hoc network is less than that of pure ad hoc network because the infrastructure layer will offload traffic from the ad hoc layer. Finally, the energy distribution of ad hoc network is designed to match the information density to maximize the network lifetime.

The remainder of this article is organized as follows. In section “System model,” the system model is introduced. In section “Distribution of routings and information density,” the distribution of routings and the information density are derived. In section “Energy-limited capacity,” the energy consumption is analyzed. The energy configuration scheme to maximize network lifetime is proposed in Section “Network lifetime maximization via energy configuration.” In section “Discussions,” discussions on theoretical results are provided. Finally, we summarize this article in section “Conclusion.”

System model

A network with $n$ nodes uniformly distributed in a unit square is considered. Two types of networks are considered: pure ad hoc network and hybrid ad hoc network.

For pure ad hoc network, source and destination are uniformly selected. According to Gupta and Kumar,¹ the formation of lattice-based network utilizes Voronoi polygon network division. In order to coordinate the possible transmission conflicts and ensure network connectivity, the unit square is divided into small squares with area $s_{n} = ((c_{0} \log n) / n), c_{0} > 1$ , which is called a cell. The side length of a cell is $r_{n} = \sqrt{s_{n}}$ . 9-TDMA transmission scheme is employed where the small cells take turns being active in a round-robin fashion. When a cell is active, it chooses a transmitter (TX) to transmit to a receiver (RX) in the adjacent cell.

As to the routing scheme, the data are transmitted by first hopping to an adjacent cell on the horizontal data path (HDP), then on the vertical data path (VDP), which is denoted as HDP-VDP routing²⁰ as illustrated in Figure 1. Conversely, the data can also be transmitted first on the VDP, then on the HDP, which is denoted as VDP-HDP routing. A data source selects the HDP-VDP routing and VDP-HDP routing with equal probability. Each node in a cell transmits in turn when the cell is active. The transmit power of a TX is $P_{0} r_{n}^{α}$ , where $P_{0}$ is a constant.

Figure 1.

The HDP and VDP.

Network protocols of hybrid ad hoc network are the same to Zhang et al.¹⁹ In hybrid ad hoc network, there are $m$ base stations, which are evenly distributed and divide the entire region into $m$ base station cells. It is noted that the vehicular networks with road-side units (RSUs) can also be modelled as hybrid networks.²¹ If the source can reach the destination within $L$ hops, the source will communicate with the destination with multi-hop transmission manner, namely, the source transmits data to the destination with ad hoc mode. In these circumstances, the protocols of ad hoc mode in hybrid ad hoc network are the same to that of pure ad hoc network. On the contrary, if the source cannot reach the destination within $L$ hops, the source will forward the data using the base stations. As illustrated in Figure 2, the source node A cannot reach the destination node B within $L$ hops. Therefore, node A transmits its data to the base station near it and then the data is forwarded to the base station near node B. Finally, the data are transmitted to node B via the nearby base station.

Figure 2.

The hybrid network model.

In this article, $λ (n)$ is defined as per-node capacity, that is, the number of bits per second that a node can transmit.

Distribution of routings and information density

In this section, we derive the distribution of routings for pure ad hoc network and hybrid ad hoc network. Moreover, the information density is derived.

Pure ad hoc network

In ad hoc network, there are multiple sources and destinations. Each node has a chance to be a source or destination. We analyze the probability that a routing passes through cell $(i, j)$ , which is denoted as $p (i, j)$ . A theorem as follows is derived.

Lemma 1

The probability that a routing passes through the cell $(i, j)$ is

\begin{matrix} p (i, j) = \frac{1}{2} \frac{i}{N^{2}} (\frac{(N - i) (N - 1) - 1}{N^{2}} + \frac{2 (N - i - 1)}{N^{2}}) \\ + \frac{1}{2} \frac{j}{N^{2}} (\frac{(N - j) (N - 1) - 1}{N^{2}} + \frac{2 (N - j - 1)}{N^{2}}) \\ + \frac{1}{2} \frac{N - i - 1}{N^{2}} (\frac{(i + 1) (N - 1) - 1}{N^{2}} + \frac{2 i}{N^{2}}) \\ + \frac{1}{2} \frac{N - j - 1}{N^{2}} (\frac{(j + 1) (N - 1) - 1}{N^{2}} + \frac{2 j}{N^{2}}) \\ \overset{Δ}{=} \frac{1}{2} f (i) + \frac{1}{2} f (j) \end{matrix}

(1)

where $N = 1 / r_{n} = \sqrt{n / (c_{0} \log n)}$ , and the function $f (x)$ is a quadratic function as follows

\begin{matrix} f (x) = \frac{1}{2 N^{2}} - \frac{3}{2 N^{3}} + \frac{1}{N^{4}} \\ + (\frac{1}{N^{2}} - \frac{1}{N^{4}}) x - (\frac{1}{N^{3}} + \frac{1}{N^{4}}) x^{2} \end{matrix}

(2)

Proof

According to the system model in section “System model,” the square of a unit area is divided into $N \times N$ grids. Nodes are uniformly distributed. Hollow dots represent TXs, and solid dots represent RXs. As shown in Figure 3(a), a TX falls into Block A, and its corresponding RX falls into Block B. There are two situations as follows:

The cell containing the TX and the cell containing the RX are not in the same row. In this case, according to the routing scheme in section “System model,” TX selects HDP-VDP routings and VDP-HDP routings with the same probability. The routing connecting TX and RX has a probability of $1 / 2$ crossing the cell $(i, j)$ . The probability that TX falls into the corresponding shaded region of Block A is $(((N - i) (N - 1)) - 1) / N^{2}$ . The RX falls into the shaded region of block B with probability $i / N^{2}$ . Hence a routing passes through cell $(i, j)$ with probability

\frac{1}{2} \frac{i}{N^{2}} \frac{(N - i) (N - 1) - 1}{N^{2}}

2. The cell containing the TX and the cell containing the RX are in the same row. There is only one type of straight-line routing and the routing connecting TX and RX has a probability of $1$ passing through the cell $(i, j)$ . Correspondingly, the probability that a TX falls into the same row with RX is $(N - i - 1) / N^{2}$ . Therefore, the probability that a routing passes through cell $(i, j)$ is

\frac{i}{N^{2}} \frac{N - i - 1}{N^{2}}

Figure 3.

The routings passing through cell $(i, j)$ in pure ad hoc network. The horizontal and vertical ordinates are the indexes of the cells.

Overall, the probability that a routing passes through cell $(i, j)$ in the state shown in Figure 3(a) is

\frac{1}{2} \frac{i}{N^{2}} \frac{(N - i) (N - 1) - 1}{N^{2}} + \frac{i}{N^{2}} \frac{N - i - 1}{N^{2}}

(3)

Notice that (3) is a quadratic function with respect of $i$ . Similarly, Figure 3(b)–(d) illustrate the remaining three cases that a routing has a probability to pass through cell $(i, j)$ . The conclusions are as follows:

In the case of Figure 3(b), a routing passes through cell $(i, j)$ with probability

\frac{1}{2} \frac{j}{N^{2}} \frac{(N - j) (N - 1) - 1}{N^{2}} + \frac{j}{N^{2}} \frac{N - j - 1}{N^{2}}

(4)

In the case of Figure 3(c), a routing passes through cell $(i, j)$ with probability

\frac{1}{2} \frac{N - i - 1}{N^{2}} \frac{(i + 1) (N - 1) - 1}{N^{2}} + \frac{N - i - 1}{N^{2}} \frac{i}{N^{2}}

(5)

In the case of Figure 3(d), a routing passes through cell $(i, j)$ with probability

\frac{1}{2} \frac{N - j - 1}{N^{2}} \frac{(j + 1) (N - 1) - 1}{N^{2}} + \frac{N - j - 1}{N^{2}} \frac{j}{N^{2}}

(6)

Finally, we can derive (1) in Lemma 1.

According to Lemma 1, when $x = (N - 1) / 2$ , $f (x)$ has maximum value, which is

\begin{matrix} f_{\max} = f (\frac{N - 1}{2}) \\ = \frac{1}{4 N} - \frac{1}{4 N^{2}} - \frac{7}{4 N^{3}} + \frac{5}{4 N^{4}} \equiv \frac{1}{4 N} \end{matrix}

(7)

Thus, when $i = (N - 1) / 2$ and $j = (N - 1) / 2$ , $p (i, j)$ has maximum value, which is

p_{max} = \frac{f_{max} (i) + f_{max} (j)}{2} = f_{max}

(8)

With Lemma 1, we further investigate the number of routings passing through cell $(i, j)$ . The following lemma is needed.

Lemma 2²²

$K$ is a binomially distributed random variable with parameters $(n, p)$ , namely, $K ~ B (n, p)$ . Then when $n \to \infty$ , we have $K = Θ (np)$ with high probability, namely, $K$ and its expectation $np$ are in the same order when $n \to \infty$ .

According to Lemma 2, the number of routings passing through cell $(i, j)$ , which is defined as $M (i, j)$ , is a binomially distributed random variable with parameters $(n, p (i, j))$ . According to Lemma 1 and Lemma 2, the number of routings passing through cell $(i, j)$ is

M (i, j) = np (i, j) = \frac{1}{2} n (f (i) + f (j))

(9)

where the function $f (^{*})$ is defined in (2).

When $i = j = (\sqrt{(n / (c \log n))} - 1) / 2$ , $M (i, j)$ is maximized, which is

\begin{matrix} M_{max} = (\frac{1}{2} n (f (\frac{N - 1}{2}) + f (\frac{N - 1}{2}))) \\ = Θ (\sqrt{n \log n}) \end{matrix}

(10)

The number of routings generated from cell $(i, j)$ is also considered. It is the same to the number of nodes in cell $(i, j)$ , which is $c_{1} \log n$ with $c_{1}$ a constant.

With the per-node capacity of ad hoc network $λ (n)$ , the number of bits passing through cell $(i, j)$ per unit time, namely, the information density of cell $(i, j)$ is

I_{p} (i, j) = λ (n) (M (i, j) + c_{1} \log n)

(11)

where $λ (n)$ is the per-node capacity and $M (i, j) + c_{1} \log n$ is the number of routings passing through cell $(i, j)$ . Hence $I_{p} (i, j)$ is the number of bits passing through cell $(i, j)$ per unit time.

Hybrid ad hoc network

In hybrid ad hoc network, the traffic can be offloaded to the base station layer. Therefore, the traffic load of ad hoc layer is relieved. For hybrid ad hoc network, we consider the ad hoc layer and have a lemma as follows:

Lemma 3

When $L < i < N - L$ and $L < j < N - L$ , which can be satisfied when $L << N$ , the probability that a routing passes through the cell $(i, j)$ is

p (i, j) = \frac{12 + 22 L + 12 L^{2} + 2 L^{3}}{3 N^{4}}

(12)

where $L$ is the maximum number of hops in ad hoc layer and $N = 1 / r_{n} = \sqrt{n / (c_{0} \log n)}$ .

Proof

The maximum number of hops in ad hoc layer is $L$ . In the case of $L < i < N - L$ and $L < j < N - L$ , TX and RX are deployed as in Figure 4. The lattice distance between the cell $(i, j)$ and the RX is denoted as $k$ . The RX is in the same row or column with the cell $(i, j)$ and is $k \leq L$ hops away from the cell $(i, j)$ . The probability that a routing passes through cell $(i, j)$ is divided into two cases, which is similar to the pure ad hoc network.

The cell containing the TX and the cell containing the RX are not in the same row or column. As a result, the routing connecting TX and RX has a probability of $1 / 2$ crossing the cell $(i, j)$ .

The cell containing the TX and the cell containing the RX are in the same row or column. Then the routing connecting TX and RX has a probability of $1$ passing through the cell $(i, j)$ .

Figure 4.

The routings passing through cell $(i, j)$ in hybrid ad hoc network. The horizontal and vertical ordinates are the indexes of the cells.

As illustrated in Figure 4(a), the RX falls into the cell with lattice distance $k$ from the cell $(i, j)$ with probability $1 / N^{2}$ . In the two cases above, the number of cells in the shaded regions are $(L + 1 - k) (L - k)$ and $(L + 1 - k)$ , respectively. Thus the two probabilities that the TX falls into the triangular shaded region are $((L + 1 - k) (L - k)) / N^{2}$ and $(L + 1 - k) / N^{2}$ , respectively. Hence in the case of Figure 4(a), the probability that a routing passes through cell $(i, j)$ is

\frac{1}{2} \frac{1}{N^{2}} \frac{(L + 1 - k) (L - k)}{N^{2}} + \frac{1}{N^{2}} \frac{L + 1 - k}{N^{2}}

(13)

Similar to the case of Figure 4(a), in Figure 4(b)–(d), the probability that a routing passes through cell $(i, j)$ equals to equation (13). According to the four cases, the probability that a routing passes through cell $(i, j)$ is

p (i, j) = \sum_{k = 0}^{L} \frac{2}{N^{2}} \frac{(L + 1 - k) (L - k) + 2 (L + 1 - k)}{N^{2}}

(14)

With some manipulations, equation (12) in Lemma 3 can be derived.

According to Lemma 3, the probability $p (i, j)$ is increasing with $L$ . Specifically, when $L = Θ (N)$ , equation (14) becomes

p (i, j) = \frac{12 + 22 L + 12 L^{2} + 2 L^{3}}{3 N^{4}} = Θ (\frac{1}{N})

(15)

which is in the same order with the $p (i, j)$ of pure ad hoc network.

Similarly, the number of routings passing through cell $(i, j)$ , which is defined as $M (i, j)$ , is a binomially distributed random variable with parameters $(n, p (i, j))$ . According to Lemma 2, the number of routings passing through cell $(i, j)$ is $M (i, j) = np (i, j)$ . In the derivation of equation (12), we have already counted the number of routings generated from cell $(i, j)$ . However, the number of routings in base station layer is not counted. The bound effect is still ignored, namely, we consider the case of $L < i < N - L$ and $L < j < N - L$ . In this case, the number of routings in base station layer in cell $(i, j)$ is

\frac{N^{2} - 4 L^{2}}{N^{2}} c_{1} \log n = \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n

(16)

Assume that the per-node capacity of ad hoc network is $λ (n)$ . Then the number of bits passing through cell $(i, j)$ per unit time, namely, the information density is

I_{h} (i, j) = λ (n) (M (i, j) + \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n)

(17)

Energy-limited capacity

The network lifetime is defined as the time period before a cell runs out its energy. If nodes are uniformly distributed in the network and the battery capacity is also uniform, the cell at the center will run out of energy first. Thus the cell at the center is the bottleneck of the network lifetime.

The energy consumption to deliver a bit is $e (r_{n})$ , which is as follows⁴

e (r_{n}) = a + {br}_{n}^{α}

(18)

where $r_{n}$ is the transmission distance, $a$ is the basic energy consumption, ${br}_{n}^{α}$ is the transmit energy consumption, and $α$ is the path loss factor. With these parameters, we study the network lifetime of pure ad hoc network and hybrid ad hoc network in the following sections.

Pure ad hoc network

A cell contains $c_{1} \log n$ nodes averagely. Assume that the energy of a node is $c_{2}$ . Then the energy of a cell, such as cell $(i, j)$ , is $c_{1} c_{2} \log n$ . Cell $(i, j)$ has to relay $I_{p} (i, j)$ bits per unit time. Before the energy is exhausted, the number of bits that cell $(i, j)$ can deliver is at most $(c_{1} c_{2} \log n) / e (r_{n})$ . Thus the network lifetime $T_{p}$ must satisfy the following relation

I_{p} (i, j) T_{p} \leq \frac{c_{1} c_{2} \log n}{e (r_{n})}

(19)

Therefore, we have

T_{p} \leq \frac{c_{1} c_{2} \log n}{e (r_{n}) I_{p} (i, j)}

(20)

It can be observed that the network lifetime is a decreasing function of the number of routings and the per-node capacity. The number of bits that a node can transmit before a cell runs out of its energy is

λ_{e} (n) = \frac{c_{1} c_{2} \log n}{e (r_{n}) (M (i, j) + c_{1} \log n)}

(21)

which is defined as the energy-limited capacity.⁴ According to equation (21), $λ_{e} (n)$ is a decreasing function of the number of routings.

For pure ad hoc network, when $i = j = (\sqrt{(n / (c_{1} \log n)} - 1) / 2$ , $M (i, j)$ is maximized and the upper bound of per-node capacity is tightest because of equation (20). Substituting the value of $M_{max}$ in equation (10) into equation (20), we have

T_{p} (n) \leq \frac{c_{1} c_{2} \log n}{e (r_{n}) λ (n) (Θ (\sqrt{n \log n}) + c_{1} \log n)}

(22)

Thus the network lifetime is mainly impacted by the number of routings passing through a cell. Because $α$ is the path loss factor and $α > 2$ , the per-node energy-limited capacity is

λ_{e} (n) \leq Θ (\sqrt{\frac{\log n}{n}})

(23)

Hybrid ad hoc network

For hybrid ad hoc network, when $L < i < N - L$ and $L < j < N - L$ , the boundary effect can be ignored, and the number of routings passing through cell $(i, j)$ is $np (i, j)$ .

Similar to the derivation in pure ad hoc network, before a cell runs out of its energy, the number of bits that cell $(i, j)$ can deliver is at most $(c_{1} c_{2} \log n) / e (r_{n})$ . Thus the network lifetime of hybrid ad hoc network is

\begin{matrix} T_{h} \leq \frac{c_{1} c_{2} \log n}{λ (n) e (r_{n}) M (i, j) + λ (n) e (r_{m}) \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n} \\ = \frac{c_{1} c_{2} \log n}{λ (n) (M (i, j) e (r_{n}) + \frac{n c_{1} \log n - 4 l^{2} c_{0} c_{1} {(\log n)}^{2}}{n} e (r_{m}))} \end{matrix}

(24)

where $r_{m}$ is the radius of base station cell. The network lifetime is still a decreasing function of the information density. The per-node energy-limited capacity of hybrid ad hoc network is

\begin{array}{l} λ_{e} (n) \leq \frac{c_{1} c_{2} \log n}{e (r_{n}) M (i, j) + e (r_{m}) \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n} \\ = \frac{c_{1} c_{2} \log n}{e (r_{n}) n \frac{12 + 22 L + 12 L^{2} + 2 L^{3}}{3 N^{4}} + e (r_{m}) \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n} \\ = \frac{3 c_{1} c_{2}}{e (r_{n}) c_{0}^{2} (12 + 22 L + 12 L^{2} + 2 L^{3})} \frac{n}{\log n} \end{array}

(25)

Thus there is a watershed for the value of $L$ . When $L = O ((n / \log n)^{1 / 3})$ , the per-node energy-limited capacity of hybrid ad hoc network is

λ_{e} (n) \leq Θ (\frac{n}{L^{3} \log n})

(26)

When $L = Ω ((n / \log n)^{1 / 3})$ , we have

λ_{e} (n) = Θ (1)

(27)

As shown in equation (25), the energy-limited capacity is constrained by the value of $L$ . With the decrease of $L$ , $λ (n)$ is increasing because the offloaded traffic to the base station layer is correspondingly increasing. Besides, comparing the energy-limited capacity of pure ad hoc network and hybrid ad hoc network, it can be discovered that the energy-limited capacity of hybrid ad hoc network is larger than that of pure ad hoc network, especially when $L = Ω ((n / \log n)^{1 / 3})$ . The reason is that in hybrid ad hoc network, the data can be offloaded to the base station layer such that the nodes do not have to relay a massive number of routings.

Network lifetime maximization via energy configuration

To maximize the network lifetime, the time period before a cell exhausts its energy needs to be extended. The number of routings at the center cell is maximum, which means that the traffic load at the center cell is heaviest. Thus more energy needs to be allocated at the center cell of the network.

The maximum network lifetime is the time period that all cells run out of energy simultaneously. According to equations (20) and (24), the network lifetime for pure ad hoc network and hybrid ad hoc network is proportional to $(c_{1} c_{2} \log n) / I_{p} (i, j)$ and $(c_{1} c_{2} \log n) / I_{h} (i, j)$ , respectively, which is the energy capacity of a cell divided by the information density. In order to guarantee the maximum network lifetime, the energy capacity of a cell should be proportional to the information density of the cell.

Assuming the total energy capacity of the network is $c_{3} n$ , for pure ad hoc network, the per-node energy-limited capacity at cell $(i, j)$ is

c_{2} (i, j) = \frac{c_{3} n}{c_{1} \log n} \frac{I_{p} (i, j)}{\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} I_{p} (i, j)}

(28)

For hybrid ad hoc network, the per-node energy-limited capacity at cell $(i, j)$ is

c_{2} (i, j) = \frac{c_{3} n}{c_{1} \log n} \times \frac{e (r_{n}) M (i, j) + e (r_{m}) \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n}{\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} e (r_{n}) M (i, j) + e (r_{m}) \frac{n - 4 L^{2} c_{0} \log n}{n} c_{1} \log n}

(29)

In this way, it can be guaranteed that the nodes with heavy traffic load have more energy. Thus the network life can be maximized, namely, all the nodes in ad hoc network exhaust energy simultaneously. Equation (28) can also be applied to the hybrid ad hoc network, since in hybrid ad hoc network, the cell $(0, 0)$ has the minimum number of routings.

Discussions

Pure ad hoc network

The value of $p (i, j)$ is related with the information density and energy configuration. Figures 5 and 6 illustrate the $p (i, j)$ and the contours of $p (i, j)$ , respectively. In Figures 5 and 6, the number of nodes is $2500$ . It is obvious that with HDP-VDP routing scheme, the number of routings passing through each cell is inhomogeneous. The number of routings at the corners of the region is minimal. On the contrary, the central cell handles the most routings. Hence the information density and energy consumption for each cell are also inhomogeneous. The energy configuration needs to match the information density. Thus the energy capacity of central cell is the largest to improve the network lifetime.

Figure 5.

The probability $p (i, j)$ for pure ad hoc network. The horizontal and vertical ordinates are the indexes of the cells.

Figure 6.

The contours of the probability $p (i, j)$ for pure ad hoc network. The horizontal and vertical ordinates are the indexes of the cells.

Hybrid ad hoc network

It is noted that we have ignored the boundary effect in Lemma 3. With considering the boundary effect, there are 64 cases that need to be considered. Hence, we provide the simulation results to demonstrate the probability that a routing passes through cell $(i, j)$ considering the boundary effect. In Figures 7 and 8, the number of nodes satisfies the condition of $N = 2500$ . In Figure 7, $L = 5$ , while in Figure 8, $L = 20$ . In Figure 7(a), the $p (i, j)$ in Lemma 3 is plotted versus the axis $i$ and $j$ . Figure 7(b) illustrates the contours of Figure 7(a). Notice that the number of routings in the central area of the entire region is maximum. On the contrary, the number of routings on the boundaries of the region is smaller than that in the central area of the entire region. Specifically, the number of routings in the corner of the region is the smallest. Comparing with Figure 5, the probability is decreased, and the information density of hybrid ad hoc network is smaller than that of pure ad hoc network, which means that the traffic load of each node is relieved due to the fact that the base station layer has offloaded data traffic of ad hoc layer.

Figure 7.

The probability $p (i, j)$ and its contours for hybrid ad hoc network with $L = 5$ . The horizontal and vertical ordinates are the indexes of the cells (a) the probability p (i, j) and (b) the contours of the probability p (i, j).

Figure 8.

The probability $p (i, j)$ and its contours for hybrid ad hoc network with $L = 20$ . The horizontal and vertical ordinates are the indexes of the cells (a) the probability p (i, j) and (b) the contours of the probability p (i, j).

Figure 7 illustrates the results with $L = 5$ . Figure 8 shows the results with $L = 20$ . It can be discovered that with the increase of $L$ , the probability $p (i, j)$ is increasing correspondingly because less traffic will be offloaded to the base station layer in this case. Thus with the increase of $L$ , the energy consumption of hybrid ad hoc network is increasing.

Conclusion

In order to characterize the energy consumption of ad hoc network, we propose the concept of information density. First, the entire region is divided into cells. Second, the probability that a routing passes through each cube is derived. The number of routings passing through a cell is obtained and the information density is achieved. Third, with the energy consumption model, the energy configuration to maximize the network lifetime is designed, with which all the nodes in the ad hoc network exhaust energy simultaneously. In order to present the information density of more general wireless network, we derive the information density of hybrid network. The theoretical results show that the information density of hybrid network is more homogeneous than pure ad hoc network. Besides, hybrid ad hoc network consumes less energy than pure ad hoc network because the base station layer will offload traffic from ad hoc layer. The information density proposed in this article shows more details of the information flow in ad hoc networks, which may be applied in the analysis and optimization of ad hoc networks.

Footnotes

Acknowledgements

The authors appreciate editor and anonymous reviewers for their precious time and great effort in handling this paper.

Handling Editor: Nan Cheng

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (No. 61631003, No. 61601055).

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Information density–based energy-limited capacity of ad hoc networks

Abstract

Keywords

Introduction

System model

Distribution of routings and information density

Pure ad hoc network

Lemma 1

Proof

Lemma 2 22

Hybrid ad hoc network

Lemma 3

Proof

Energy-limited capacity

Pure ad hoc network

Hybrid ad hoc network

Network lifetime maximization via energy configuration

Discussions

Pure ad hoc network

Hybrid ad hoc network

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References

Lemma 2²²