Data transmission optimization algorithm for network utility maximization in wireless sensor networks

Abstract

We provide a joint scheme for rate control, scheduling, routing, and power control protocol for wireless sensor networks based on compressive sensing. Using a network utility maximization formulation, we present cross-layer optimization solutions using Lagrangian multipliers in the transport, network, media access control, and physical layers. Inspired by compressive sensing, we focus on the construction of utility functions based on the constraints of the link capacity, rate, routing, and power to decrease the computational cost, accelerate the convergence rate, and degrade the error ratio. The optimization solutions are developed by solving the optimization model of network utility maximization. We prove the effectiveness by the theory analysis at the stability of the transmission rate and error ratio. Finally, simulation results demonstrate the performance in terms of stability of the error ratio of compressive sensing, energy consumption, and transmission delay in wireless sensor networks.

Keywords

Wireless sensor networks cross layer optimization network utility maximization compressive sensing

Introduction

Wireless sensor networks (WSNs) have sensed the physical world and played an important role in the Internet of Things. Data transmission is one of the most significant properties of WSNs. Sensor nodes are responsible for receiving and relaying data simultaneously. Because of the relevance of the data, it is not necessary to transmit all the data to the sink, as that expends too much energy and reduces transmission efficiency. A substantial amount of research has concentrated on methods to improve data transmission effectiveness, such as cross-layer designs^1–4 and network utility maximization (NUM).^5–8

Khalek et al.,¹ Cammarano et al.,² and Chen et al.⁴ proposed the cross-layer transmission schemes and derived joint algorithm for media access control (MAC), scheduling, and routing through optimization theory,³ and comprehensively studied the state of the art for wireless communication at the application, transport, network, MAC, and physical layers.

Yang et al.,⁵ Zhou et al.,⁶ He et al.,⁷ and Chen et al.⁸ formulated optimization solutions by NUM in WSNs with the constraints of specific network topologies. Ideal schemes were obtained at the expense of sacrificing network utility and rigorous network topologies,⁷ which showed that each node cannot transmit and receive data simultaneously. Furthermore, no excessive consumption was guaranteed, and the effective saving of energy consumption was not solved.

Dimension reduction of data has aroused more and more concerns to network optimization,⁹ which proves that the full-view area coverage can replace the full-view point coverage which leads to a significant dimension reduction for the full-view area coverage problem. To economize energy, Tapparello et al.¹⁰ proposed a joint compression–transmission algorithm for energy-harvesting multi-hop networks, where both compression and transmission activities were implemented using the energy available in the energy buffer. Compressive sensing (CS) is rapidly being developed in WSNs for data compression with a low complexity and low energy consumption. The features of CS for data transmission in WSNs have been recently researched in various studies.^11–16 The protocols decrease transmission delay and reduce the transmission power for less transmitted data.

In order to achieve network optimization,^1–8 improved data transmission strategies based on maximum utility or cross-layer ideas without full consideration of the sparse feature of the original data and the efficiency of the data transmission were used. In general, the sensor node is responsible for harvesting energy from the environment, sampling data, and transmitting to the sink. It is worth mentioning that data compression will greatly reduce the amount of data transmitted after sampling.^9,10,17 However, this approach only reduced the amount of data transmitted, and the sampling frequency was not reduced. Despite data compression,^11–16 the cross-layer collaboration has not been fully considered. In this article, we improve the bottleneck constraints of the aforementioned algorithms. The optimal transmission policy will be implemented through NUM combined with CS in transport, network, MAC, and physical layers, respectively. The policy greatly reduces the data transmission by reducing the sampling frequency, which fundamentally constrains data process. Our major contributions are summarized as follows:

Motivated by data transmission applications, we propose a rate control for the transport layer, scheduling for the network layer, routing for the MAC layer, and a power control algorithm for the physical layer. The proposed scheme reduces the transmission flow through CS, which can significantly reduce energy consumption. In addition, the algorithm implements a trade-off between transmission rate and energy consumption based on Lagrangian optimization and dynamic programming. Logical routing and appropriate link capacity allocation are presented, which effectively relieve network congestion.

We demonstrate the effectiveness of the scheme through Lyapunov theory. Through theoretical analysis, we find that the transmission rate is stochastically stable, which should be attributed to the compression technique and optimization method. Furthermore, we also validate that the rate–distortion ratio is less than 4/N (where N is the number of nodes in the WSNs).

We conduct extensive simulations to validate the effectiveness of the proposed algorithm. Simulation results demonstrate that the scheme can significantly reduce transmission delay and communication costs.

The article is organized as follows: section “Preliminaries and model statement” describes the preliminaries and model statement; section “Problem formulations” presents the problem formulation; section “Joint control algorithms” proposes the joint rate control, scheduling, routing, and power control algorithm; section “Performance analysis” provides the analysis of the performance of the proposed algorithm; section “Simulation results” presents the simulation results; and section “Conclusion” presents the conclusions.

Preliminaries and model statement

CS theory

The unknown original signal $v (t) \in ℜ^{n}$ , where n is the vector length, is projected into a low-dimensional space through a sensing matrix, $Φ \in ℜ^{m \times n}$ (m << n), that is, $y (t) = Φ v (t)$ , where a new data vector $y (t) \in ℜ^{m}$ is obtained and transmitted, and processed in the networks. However, the signal is generally not sparse, and there is an invertible basis matrix $Ψ \in ℜ^{n \times n}$ , $y (t) = Φ v (t) = Φ Ψ θ (t)$ .

Since the lengths of $y (t)$ and $v (t)$ satisfy m << n, the gathered data are less than the original data by the sensing node. Thus, an exact reconstruction is difficult in this situation at the sink node. The more rigorous condition of restricted isometric property (RIP) should be considered.¹⁸ A matrix $Φ$ obeys RIP with parameters $(k, δ)$ for $δ \in (0, 1)$ if

(1 - δ) ‖ v (t) ‖_{2}^{2} \leq ‖ Φ v (t) ‖_{2}^{2} \leq (1 + δ) ‖ v (t) ‖_{2}^{2}

(1)

The original signal should be reconstructed when the measurement vector is transmitted to the sink node. The reconstruction of $v (t)$ from $y (t)$ is straightforward using the $l_{1}$ -norm optimization¹⁹ as shown in equation (2)

min ‖ θ (t) ‖_{1} s . t . y = Φ Ψ θ (t)

(2)

Model statement

In this study, we consider a WSN with N nodes that are randomly distributed in the region. One sensor node can receive sensory data from other $N - 1$ sensors and transmit the data to the sink node through a multi-hop message relay. The WSN model can be denoted as the directed graph $G = (V, L)$ , where V is the set including N nodes, each of them transmitting and receiving data simultaneously. Let L denote the directed link set, where $(i, j) \in L$ denotes that node i transmits data to node j. For simplification, $(i, j)$ can be represented as link l.

Special relationships exist between the transmission rate and transmission vector. Suppose the transmission rate $x_{i} (t)$ is the linear function of the transmission vector $y (t)$ and the temporal entropy $H_{i}^{j} (t)$

x_{i} (t) = \frac{V_{i}^{j} (y (t), H_{i}^{j} (t))}{T}

(3)

where $V_{i}^{j} (y)$ , the linear function of $y (t)$ , is the transmission aggregate at time t from node i to j. T denotes cycle duty. The temporal entropy $H_{i}^{j} (t)$ at time t is calculated as follows²⁰

H_{i}^{j} (t) = \sum_{(i, j) \in S} \Pr_{i}^{j} (t) \ln \frac{1}{\Pr_{i}^{j} (t)}

(4)

where $\Pr_{i}^{j} (t) = \Pr {W_{i} (t) = j}$ denotes the transmission probability at time t from node i to j. Let $W_{i} (t)$ be the next transmission node from node i at time t. For simplicity, $H_{i} (t) = \sum_{j = 1}^{N - 1} H_{i}^{j} (t)$ . The temporal entropy is smaller, and there is less unknown information, which is better for a network environment.

Problem formulations

This section presents a cross-layer algorithm based on CS. The architecture of the proposed cross-layer framework is shown in Figure 1.

Figure 1.

Architecture of the proposed cross-layer framework.

Rate and capacity constraints

Let $T (i)$ be the set of nodes to which node i can directly transmit data and $ϒ (i)$ be the set of nodes from which node i can directly receive data. Let $r_{ji} (t)$ denote the rate from node j to node i at time t (receiving rate at node i) and $r_{ij} (t)$ denote the rate from node i to node j at time t (transmitting rate at node i). The rate conservation law implies that for rate $x_{i} (t)$ and node i

x_{i} (t) + \sum_{j \in ϒ (i)} r_{ji} (t) = \sum_{j \in T (i)} r_{ij} (t), (i, j) \in L

(5)

which states that the sum of the created rate itself and the receiving rate is equal to the transmission rate for node i. Because the rate is associated with the link capacity, we have the following constraint

x_{i} (t) + \sum_{j \in ϒ (i)} r_{ji} (t) \leq f_{i}^{l} (t)

(6)

where $f_{i}^{l} (t)$ is the link capacity associated to node i of link l at time t.

Power and routing constraints

The energy model in this study considered transmitting, receiving, and sensing energy consumption, which accounts for the bulk of all energy consumption. Let $E_{ij}^{tr} (t), E_{ij}^{re} (t), E_{i}^{sn} (t), and E_{i} (t)$ , respectively, denote the transmitting, receiving, sensing energy consumption, and the total amount of energy consumption for node i at time t. The total amount of energy consumption can be presented as follows

E_{i} (t) = \sum_{j \in T (i)} E_{ij}^{tr} (t) + \sum_{j \in ϒ (i)} E_{ij}^{re} (t) + E_{i}^{sn} (t)

(7)

The main energy limitation of WSNs is the use of battery supplies; therefore, we provide an upper bound of energy consumption, that is

\sum_{i = 1}^{N} E_{i} (t) \leq \bar{E} (t)

(8)

where $\bar{E} (t)$ is the upper bound of energy consumption at time t in the network.

Channel allocation supports multiple transmissions and reduces interference.^21,22 Let $P_{r}^{I} (H_{i}^{j} (t))$ denote the occupancy probability of link $(i, j)$ . If the entropy is smaller, the channel performance is better. Therefore, $P_{r}^{I} (H_{i}^{j} (t))$ can be formulated as

P_{r}^{I} (H_{i}^{j} (t)) = e^{- H_{i}^{j} (t)}

(9)

Cross-layer optimization model

Utility function U(·) reflects the “utility” when the impact factors play a role in maximizing utility.²³ In this study, we design the U(·) for the function of rate $x_{i} (t)$ , link capacity $f_{i}^{l} (t)$ , routing $I_{l} (t)$ , and power $p_{i} (t)$ at time t. We assume that U(·) is differentiable and non-decreasing. In addition, all variables are functions of entropy $H_{i}^{j} (t)$ , that is, $x_{i} (t) \Leftrightarrow x_{i} (H_{i}^{j} (t))$ .

Combining equations (5)–(9) associated with each node in the network, a fair standard utility function for node i is given by

max U (x_{i} (t), f_{i}^{l} (t), I_{i}^{l} (t), E_{i} (t))

(10)

s . t . {\begin{matrix} (5) - (9) \\ x_{i} (t) \geq 0, f_{i}^{l} (t) \geq 0, I_{i}^{l} (t) \geq 0, E_{i} (t) \geq 0 \end{matrix}

Constraint (5) is the rate constraint for node i at time t; constraint (6) expresses the relationship between rate and link capacity; constraints (7) and (8) denote the power constraint; and constraint (9) states the probability of routing selection, which is inversely proportional to entropy $H_{i}^{j} (t)$ .

The NUM challenge of dynamic-routing WSNs with the transmission rate, routing, link, and battery capacity constraints is how to maximize the network utility (10) under constraints (5)–(9)

max \sum_{i = 1}^{N} U (x_{i} (t), f_{i}^{l} (t), I_{i}^{l} (t), E_{i} (t))

(11)

s . t . {\begin{matrix} (5) - (9) \\ x_{i} (t) \geq 0, f_{i}^{l} (t) \geq 0, I_{i}^{l} (t) \geq 0, E_{i} (t) \geq 0 \end{matrix}

The solution for maximizing the utility function involves the joint design of all nodes and links to achieve reliable and secure communication for all users while at the same time maintaining a stable stochastic transmission rate.

Joint control algorithms

In order to achieve an efficient trade-off between energy consumption and quality of service, we present a cross-layer algorithm for physical, MAC, network, and transport layers. The proposed WSN algorithm is derived from the solution of the NUM problem.^24,25 We use the Lagrangian multiplier method to solve the cross-layer optimization problem and jointly achieve optimal rate control, scheduling, routing, and power control.

Lagrangian multiplier decomposition

The cross-layer optimization model (9) is approximately convex⁷ if the entropy satisfies $e^{- H_{i}^{j} (t)} \approx 1 - H_{i}^{j} (t)$ . We form the Lagrangian problem by relaxing constraints (5)−(9) as follows

\begin{matrix} L (x_{i} (t), f_{i}^{l} (t), I_{i}^{l} (t), E_{i} (t), λ_{i}, α_{i}, β_{i}, γ_{i}) \\ = \sum_{i = 1}^{N} U (x_{i} (t), f_{i}^{l} (t), I_{i}^{l} (t), E_{i} (t)) \\ + λ_{i} (t) (x_{i} (t) + \sum_{j \in ϒ (i)} r_{ji} (t) - \sum_{j \in T (i)} r_{ij} (t)) \\ + α_{i} (t) (f_{i}^{l} (t) - x_{i} (t) - \sum_{j \in ϒ (i)} r_{ji} (t)) \\ + β_{i} (t) (P_{r}^{I} (H_{i}^{j} (t)) - e^{- H_{i} (t)}) \\ + γ_{i} (t) (\bar{E} (t) - \sum_{i = 1}^{N} E_{i} (t)) \end{matrix}

(12)

where $λ_{i} (t), α_{i} (t), β_{i} (t), and γ_{i} (t)$ are Lagrangian multipliers with the conservation constraints. Because L(·) is only piecewise differentiable, we use the distributed sub-gradient method. The implementation details are as follows.

Rate control

Our goal is rate maximization under conditions where link capacity, routing, and power allocation are relatively satisfied

P_{1} max x_{i} (t)

s . t . (5) - (9)

Taking the derivative of L(·) with $x_{i} (t)$ and $λ_{i} (t)$ , we obtain the solution as follows

\frac{\partial L}{\partial x_{i}} (t) = \frac{\partial \sum_{i = 1}^{N} U_{i}}{\partial x_{i}} (t) + λ_{i} (t) - α_{i} (t)

(13)

\frac{\partial L}{\partial λ_{i}} (t) = x_{i} (t) + \sum_{j \in ϒ (i)} r_{ji} (t) - \sum_{j \in T (i)} r_{ij} (t)

(14)

Since $P_{1}$ is the approximately convex problem, it can be solved by

x_{i} (t + 1) = {[x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t)]}^{+}

(15)

λ_{i} (t + 1) = {[λ_{i} (t) - ε \frac{\partial L}{\partial λ_{i}} (t)]}^{+}

(16)

where ${[x]}^{+} = \max (0, x)$ and $ε$ is the step size.

Scheduling

$P_{2}$ is also approximately convex for the maximization link capacity $f_{i}^{l} (t)$ , and the implementation approach is similar to $P_{1}$

P_{2} max f_{i}^{l} (t)

s . t . (5) - (9)

Taking the derivative of L(·) with $f_{i}^{l} (t)$ and $α_{i} (t)$ , the solutions are obtained, respectively, as

\frac{\partial L}{\partial f_{i}^{l}} (t) = \frac{\partial \sum_{i = 1}^{N} U_{i}}{\partial f_{i}^{l}} (t) + α_{i} (t)

(17)

\frac{\partial L}{\partial α_{i}} (t) = f_{i}^{l} (t) - x_{i} (t) - \sum_{j \in ϒ (i)} r_{ji} (t)

(18)

The link capacity and Lagrangian multiplier can be updated, respectively, as follows

f_{i}^{l} (t + 1) = {[f_{i}^{l} (t) - ε \frac{\partial L}{\partial f_{i}^{l}} (t)]}^{+}

(19)

α_{i} (t + 1) = {[α_{i} (t) - ε \frac{\partial L}{\partial α_{i}} (t)]}^{+}

(20)

Link capacity is allocated according to the aggregate flow and transmission rate. Transmitting with unnecessarily high link capacity reduces the lifetime of the network and leads to excessive interference. Therefore, distribution according to one’s needs is the best allocation principle for link capacity.

Power control

The common transmission power used by each node should be large enough so that the bit error rate (BER) is within the toleration span; however, the sensor node relies on limited battery power. The trade-off between power and transmission efficiency is one of the most important issues in WSNs

P_{3} min E_{i} (t)

s . t . (5) - (9)

Taking the derivative of L(·) with $E_{i} (t)$ (21) and $γ_{i} (t)$ (22)

\frac{\partial L}{\partial E_{i}} (t) = \frac{\partial \sum_{i = 1}^{N} U_{i}}{\partial E_{i}} (t) - γ_{i} (t)

(21)

\frac{\partial L}{\partial γ_{i}} (t) = \bar{E} (t) - \sum_{i = 1}^{N} E_{i} (t)

(22)

Using the same technique to obtain $p_{i} (t)$ and $γ_{i} (t)$ , it can be shown that

E_{i} (t + 1) = {[E_{i} (t) - ε \frac{\partial L}{\partial E_{i}} (t)]}^{+}

(23)

γ_{i} (t + 1) = {[γ_{i} (t) - ε \frac{\partial L}{\partial γ_{i}} (t)]}^{+}

(24)

Routing

Routing access strategy should be considered for solving channel competition resulting in network congestion. In this study, we give the maximal probability link access at the next hop. Thus, we have

I_{i}^{l} (t) = \arg max_{(i, j)} (P_{r}^{I} (H_{i}^{j} (t)))

(25)

Note that routing access $I_{i}^{l} (t)$ is updated in real time through dynamic network environment, which achieves routing access with least congestion ratio.

Remark 1

The proposed algorithm accounts for the need to allocate rate, power, link capacity, and routing also for node i. It can be proven that this algorithm has optimal properties as summarized in Theorems 1 and 2. We omit a formal statement of this result here since it is a straightforward extension of Theorems 1 and 2.

Performance analysis

In this section, we validate the effectiveness of the proposed algorithm for rate stability and transmission accuracy. Obviously, $x_{i} (t) - x_{i}^{*}$ and $\frac{\partial L}{\partial x_{i}} (t)$ are opposite, in addition, L(·) is convex and $x_{i} (t)$ is bounded. Thus, we assume the following relationship between $x_{i} (t) - x_{i}^{*}$ and $\frac{\partial L}{\partial x_{i}} (t)$ .

Assumption 1

For transmission rate $x_{i} (t)$ , if $x_{i}^{*}$ denotes the desired rate at node i, $x_{i} (t)$ and $x_{i}^{*}$ satisfy $x_{i} (t) - x_{i}^{*} \leq \frac{δ}{2} \frac{\partial L}{\partial x_{i}} (t)$ .

Stability analysis

Theorem 1

Let $x_{i}^{*}$ be the desired rate at node i, under Assumption 1, the transmission rate (11) is stochastically stable.

Proof

In the proposed algorithm, the sensor node can transmit and receive data simultaneously. To achieve a balance between transmission and reception, the created rate should be adjusted according to entropy $H_{i}^{j} (t)$ . Define the Lyapunov function as $V (t) = \sum_{i = 1}^{N} {(x_{i} (t) - x_{i}^{*})}^{2}$ .

Based on formula (11), the sub-gradient vector is

\begin{matrix} Δ V (t + 1) = V (t + 1) - V (t) \\ = \sum_{i = 1}^{N} ({[x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t)]}^{+} - x_{i} (t)) \\ ({[x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t)]}^{+} + x_{i} (t) - 2 x_{i}^{*}) \end{matrix}

(26)

Next, we discuss the positivity and negativity of $Δ V (t + 1)$ from two aspects.

1. If $x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t) > 0$ , we have

\begin{matrix} Δ V (t + 1) = \sum_{i = 1}^{N} (x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t) - x_{i} (t)) \\ = (x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t) + x_{i} (t) - 2 x_{i}^{*}) \\ = \sum_{i = 1}^{N} (2 ε (x_{i} (t) - x_{i}^{*}) \frac{\partial L}{\partial x_{i}} (t) + {(ε \frac{\partial L}{\partial x_{i}} (t))}^{2}) \end{matrix}

(27)

According to Assumption 1, the inequality of $Δ V (t + 1) < 0$ is obtained. Hence, the transmission rate (15) is stochastically stable.

2. If $x_{i} (t) + ε \frac{\partial L}{\partial x_{i}} (t) < 0$ , then $x_{i} (t + 1) = 0$ , thus we obtain

\begin{array}{l} Δ V (t + 1) = \sum_{i = 1}^{N} (- x_{i}^{2} (t) + 2 x_{i}^{*} x_{i} (t)) \\ \leq \sum_{i = 1}^{N} (x_{i}^{2} (t) + ε x_{i} (t) \frac{\partial L}{\partial x_{i}} (t)) \end{array}

(28)

Obviously, we achieve the solution of Theorem 1.

Remark 2

In fact, bounded link capacity and energy consumption efficiently facilitate rate stability. The stable transmission rate (11) guarantees for effective data transmission, which makes the compression maximization for the initial data and sampling frequency minimization. This can significantly optimize data transmission environment of WSNs.

Transmission accuracy

The transmitted data are incomplete because the data are compressed before transmission, but they still contain effective information. CS is efficient for reducing energy consumption; however, the transmitted data may not be completely accurate. Let $ρ_{i} = \frac{{(x_{i} (t) - x_{i}^{*})}^{2}}{{(x_{i} (t))}^{2}}$ denote the rate–distortion ratio. Considering the utility function

\begin{array}{l} U (x_{i} (t), f_{i}^{l} (t), I_{i}^{l} (t), E_{i} (t)) = \frac{1}{N ε} x_{i}^{2} (t) + x_{i} (t) (α_{i} (t) - λ_{i} (t)) \\ + {(f_{i}^{l} (t))}^{2} + {(I_{i}^{l} (t))}^{2} + {(E_{i} (t))}^{2} \end{array}

(29)

we discuss the transmission accuracy based on CS and rate control (15).

Theorem 2

Considering the utility function (25) under Assumption 1, the rate–distortion ratio is satisfied by $ρ_{i} = \frac{{(x_{i} (t) - x_{i}^{*})}^{2}}{{(x_{i} (t))}^{2}} \leq \frac{4}{N}$ .

Proof

When $\frac{\partial L}{\partial x_{i}} (t) \leq 0$ , the rate–distortion ratio is

ρ_{i} = \frac{{(x_{i} (t) - x_{i}^{*})}^{2}}{{(x_{i} (t))}^{2}} \leq \frac{{(\frac{ε}{2} \frac{\partial L}{\partial x_{i}} (t))}^{2}}{{(x_{i}^{*})}^{2}} \leq \frac{\frac{ε^{2}}{4} Σ_{x_{i} (t) = x_{i}^{*}}^{2}}{{(x_{i}^{*})}^{2}} = \frac{1}{N}

(30)

where $Σ = \frac{\partial \sum_{i = 1}^{N} U_{i}}{\partial x_{i}} (t) + λ_{i} (t) - α_{i} (t)$ . If $\frac{\partial L}{\partial x_{i}} (t) > 0$ , then

ρ_{i} = \frac{{(x_{i} (t) - x_{i}^{*})}^{2}}{{(x_{i} (t))}^{2}} \leq \frac{{(ε \frac{\partial L}{\partial x_{i}} (t))}^{2}}{{(x_{i}^{*})}^{2}} \leq \frac{4}{N}

(31)

Hence, Theorem 2 is proven.

Remark 3

The fact that (31) implies that the transmission rate (15) satisfies the rate constraints (5) and (6) at each time-slot since transmission vector, after compression, is relatively stable.

Simulation results

In this section, we present our results and analyses of transmission delay, transmission rate, error ratio, and energy consumption to demonstrate the performance of the proposed algorithms (DTOCLD) and to compare them with those of CLC_DD and CLC_OOD.⁶ Parameters used in Zhou et al.⁶ are selected. Considering a 100-node network, the nodes are stochastically distributed. We set the utility function for formula (29), and the timing of each simulation at 3 s. The sensing matrix $Φ$ of size 100 × 200 was generated with independent normally distributed entries $N (0, σ^{2})$ , where $σ^{2} = 1 / 1000$ (expected l₂-norm of each column is unity).

Figure 2 shows the comparison of transmission delays of CLC_DD and CLC_OOD. Despite the larger delay of DTOCLD during the first second as a result of the heavy upload based on CS, it descended quickly in the next second because of the solutions from the Lagrangian method.

Figure 2.

Comparison of transmission delays of CLC_DD, CLC_OOD, and DTOCLD.

Figure 3 shows the transmission rate of DTOCLD compared to CLC_DD and CLC_OOD. DTOCLD has a much larger transmission rate than CLC_DD and CLC_OOD. Data compression can depress the amount of traffic, which effectively improves network conditions and significantly enhances transmission efficiency.

Figure 3.

Comparison of transmission rates of CLC_DD, CLC_OOD, and DTOCLD.

Figure 4 presents the error ratio of DTOCLD. According to Theorem 2, the error ratio is less than 0.04 in the 100-node network, which is in accordance with the results of Figure 4. After 2 s, the error ratio is stable and less than 0.04, which validates the lower influence of loss compression.

Figure 4.

Error ratio of DTOCLD.

In the DTOCLD algorithm, the energy consumption in nodes is low in the key region of Figure 5. This demonstrates that the DTOCLD algorithm consumes less energy, which is attributed to decreased transmission data by the CS theory. This confirms that the DTOCLD algorithm optimizes network utilization. DTOCLD chooses a congestion-avoidance routing that consumes the least energy based on the number of nodes. The CLC_DD and CLC_OOD algorithms consume more energy because of the enormous amount of data transmitted.

Figure 5.

Comparison of energy consumptions of CLC_DD, CLC_OOD, and DTOCLD.

Conclusion

This article presents the implementation of a cross-layer optimization design for data transmission in WSNs consisting of rate control, scheduling, routing, and power control. Unlike the existing cross-layer protocols, we focus on a joint optimization strategy based on CS in physical, MAC, network, and transport layers. Taking into account the relationships among rate, routing, link capacity, and energy allocation, NUM is proposed for efficient data transmission, and we solve the optimal solutions with the Lagrangian multiplier method. The performance of the proposed algorithm, in theory and practice, perfectly achieves the desired solutions.

Footnotes

Academic Editor: Pelegri-Sebastia

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 was supported by the National Natural Science Foundation of China under grant no. 61374097, Fundamental Research Funds for the Central Universities of China (no. 142303013), Program of Science and Technology Research of Hebei University (no. QN2014326), and School Funds Project of Northeastern University at Qinhuangdao (no. XNB2015004).

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