A reaction-diffusion mechanisms for node scheduling design of wireless sensor networking systems

Abstract

The wireless sensor networking (WSN) systems have been used for several applications from target tracking to environment monitoring. Recently, there has been increased interest in the design of WSN protocols for delay-sensitive applications, such as military surveillance, health monitoring, and infrastructure security. In this paper, we apply gene regulatory networks (GRN) principles to the WSN system and analyze a reaction-diffusion mechanisms for decentralized node scheduling design. Using control theory, we verify that the proposed scheme guarantees system stability. Also, we provide conditions of system parameters to ensure system convergence and stability. Simulation results indicate that the proposed scheme achieves superior performance with energy balancing as well as desirable delay compared with other well-known schemes.

Keywords

Reaction-diffusion mechanism decentralized node scheduling wireless sensor networks

1. Introduction

Wireless sensor networks (WSNs) are generally comprised of a large number of tiny sensor nodes that perform network processing of the acquired data and then forward the monitored data to the sink via multi-hop paths. The sensor nodes are typically deployed to cover a geographical space and are usually battery powered. Recently, delay-sensitive applications, such as, emergency and rescue applications, require application-specific functionalities and performance guarantees. Also, large scale WSNs demand a high level of self-organization so that each entity of a system can make decisions based on local interactions with its neighbors [8].

Many researchers have been engaged in developing biologically-inspired design paradigms to address the scalability and adaptability issues for wireless networks while guaranteeing system robustness with individual simplicity [4]. Concepts and principles derived from biological systems, such as immune system [1], insect colonies [6], activator-inhibitor systems [7], and cellular signaling systems [17], have been applied to WSN system design. Also, epidemic-based communication models provide an effective means of transferring data in wireless networks, in which infection transmission process corresponds to message passing among nodes [3]. Among theses approaches, Gene Regulatory Networks (GRNs), a representation of genes/proteins and the interactions between them, have been proposed for robust network design [13]. The analogy between GRN and WSN is evident; like sensor nodes, genes perform major functions, i.e., sensing, actuating, and signaling. In their sensing phase, genes sense the levels of proteins in the cells through signals mediated by other interacting genes and environmental variables, to determine their own gene expression levels. Then, in the actuating phase, each gene produces activator or inhibitor proteins to regulate the expression level of other genes in the network. In the signaling phase, genes interact with other genes to regulate protein levels in the cell. In [18], the authors determined the locations of wireless sensor nodes using GRN topology to achieve robustness and resilience to node failures. In [5], they proposed a GRN model to solve the issue of optimal coverage in WSNs. In specific, a non-linear differential equation model of GRN was used to identify the minimum number of sensors required for maximum coverage in WSNs. In [14], they used the non-linear differential equation based model to emulate the evolution process of genes in GRNs for network self-configuration. In [15], they proposed an embedded controller that offers every sensor the ability to regulate its sampling rate based on its data and its neighbors’ behavior and shared information. In [20], they showed how a GRN inspired controller can be used to configure submarines robots. In [9], they proposed the use of the attractor theory in GRN to achieve a fault-tolerant WSN routing.

However, these GRN-inspired approaches have not been theoretically analyzed in terms of system stability. Stability issues were discovered only during simulations due to the analytical complexity of these systems. Therefore, they could not provide parameter conditions which would ensure the stability of the networking systems that use bio-inspired algorithms. In [2], we proposed GRN-inspired self-organizing control scheme for WSNs. Based on the existing works, we provide insights into the performance of the proposed scheme by deriving the steady states. Also, a theoretical analysis of the system’s stability is provided with parameter conditions to ensure the system convergence to the desired states.

2. GRN model

Gene Regulatory Networks (GRNs) are models of genes and gene interactions at the expression level. Each GRN is a collection of DNA segments in a cell which interacts with each other indirectly through their RNA, protein product, and other chemicals in the cell, thereby governing the rates at which genes in the network are transcribed into mRNA. Gene regulation is a general name for a number of sequential processes, the most well known and understood being transcription and translation, which controls the level of a gene’s expression, and ultimately results in a specific quantity of a target protein. The organism is thus able to coordinate the actions of multiple cells without requiring a central controller. Proteins are produced by genes in the cell and genes control the further production or suppression of proteins in a time dynamic manner. This mechanism, which regulates cell behavior through protein production, is referred to as a GRN.

Research efforts in recent years have focused on computational modeling of signal transduction and developmental genetic networks in a variety of ways. For a multi-cell organism, it is necessary to consider external factors, transcription factors diffused from other cells, and interactions between cells into the GRN model. In [16], the authors suggested a generalized GRN model that considered diffusion of transcription factors among the cells: $\begin{array}{rcl} (1) & \frac{d x_{i j}}{d t} = - γ_{i} x_{i j} + ϕ [\sum_{l = 1}^{n_{g}} W^{j l} x_{i l} + θ_{j}] + D_{j} \nabla^{2} x_{j i}, \end{array}$ where $x_{i j}$ denotes the concentration of the jth gene product (protein) in the ith cell. The first term on the right-hand side of (1) represents the degradation of the protein at a rate of $γ_{i}$ , the second term specifies the production of protein $x_{i j}$ , and the last term describes the diffusion component at diffusion rate $D_{j}$ . ϕ is an activation function for the protein production, which is usually defined as a sigmoid function: $\begin{array}{rcl} (2) & ϕ (z) = \frac{1}{1 + exp (- μ z)} . \end{array}$

GRN systems possess some attractive properties, like robustness and resilience to failures. Thus, the evolution process of genes in GRNs inspires solutions to many problems in WSNs, such as maximizing network coverage, topology control, routing, duty-cycle adaptation, and clustering [2,5,9,10,12,14–16,18–20]. In particular, a GRN inspired real-time controller for a group of underwater robots has been proposed [20]. Then, a genetic algorithm was applied to evolve the controller for a simple clustering task. In [10], the authors proposed a distributed GRN-based algorithm for a multi-robot system to organize themselves autonomously into predefined shapes. The movement dynamics of each robot is described by a GRN model, where the concentration of two proteins of type G represents the x and y positions of the robots, respectively, and that of the proteins of type P represent the internal state of the robot: $\begin{array}{l} (3) & \frac{d g_{i, x}}{d t} = - a z_{i, x} + m p_{i, x} \\ (4) & \frac{d g_{i, y}}{d t} = - a z_{i, y} + m p_{i, y} \\ (5) & \frac{d p_{i, x}}{d t} = - c p_{i, x} + k f (z_{i, x}) + b D_{i, x} \\ (6) & \frac{d p_{i, y}}{d t} = - c p_{i, y} + k f (z_{i, y}) + b D_{i, y}, \end{array}$ where $i = 1, 2, \dots, n$ and n is the total number of robots (cells) in the system. $g_{i, x}$ and $g_{i, y}$ are the x and y position of the ith robot, respectively, which corresponds to the concentration of two proteins of type G. $p_{i, x}$ and $p_{i, y}$ are the concentration of two proteins of type P, which denotes the internal state of the ith robot along the x and y coordinates, respectively. $D_{i, x}$ and $D_{i, y}$ are the sum of the distances between the ith robot and its neighbors, i.e., the sum of the concentration of protein type G diffused from neighboring cells. When a robot detects its neighbor, it will receive the protein emitted from that neighbor so that it can keep itself away from the neighbor to avoid collision. In [11], the location and velocity of the robots were described by the protein concentration of a few genes whose expression is influenced by each other.

3. Proposed algorithm

3.1. Model

We briefly introduce the GRN-inspired model for node scheduling in [2]. The notations used in the paper are as follows:

$g_{i}$ , the expression level of the gene (mRNA) of sensor node i.

$p_{i}$ , the concentration of sensor nodes i’s proteins produced by the gene.

$N_{i}$ , the set of neighboring nodes of a node i.

$N_{i}$ , the cardinality of $N_{i}$ .

$E_{i}$ , the consumed energy level, defined by the ratio of consumed energy to the initial energy of node i.

${\overline{E}}_{i}$ , the ratio of consumed energy level to the neighbors’ average.

$d_{i}$ , the delay time between the source node i and the sink node.

$d_{r}$ , the delay requirement of an application.

${\overline{D}}_{i}$ , the difference between the measured delay and delay requirement.

$f_{g}^{h}$ , the inhibition function.

$f_{p}^{a}$ , the activation function.

The dynamics of the proposed GRN model is defined by the following equations: $\begin{array}{rcl} (7) & \frac{d g_{i}}{d t} & = & - a g_{i} + η f_{g}^{h} ({\overline{E}}_{i} p_{i}) \\ (8) & \frac{d p_{i}}{d t} & = & - c p_{i} + κ f_{p}^{a} ({\overline{D}}_{i} g_{i}) + b \sum_{\forall j \in N_{i}} (g_{i} - g_{j}), \end{array}$ where a and c are the decay rates of mRNA and protein concentration, respectively. η and κ are the synthesis rates of mRNA and protein concentration, respectively. Functions $f_{g}^{h}$ and $f_{p}^{a}$ are defined as follows: $\begin{array}{l} (9) & f_{g}^{h} (x) = e^{- x} \\ (10) & f_{p}^{a} (x) = \frac{1 - e^{- x}}{1 + e^{- x}} . \end{array}$ The gene expression g of (7) represents the local environmental state of a node in WSNs. In specific, the expression level of the gene of a node is determined by its consumed energy level and protein concentration. When the consumed energy level of a node is larger than the average over its neighbors or the protein concentration increases, the level of gene becomes smaller, and vice versa. Considering (8), the concentration of protein is determined by the gene expression level and external factors, i.e., the delay requirement of an application and transcription factors diffused from other nodes. When the measured delay is larger than the delay requirement or the expression level of gene is larger than the average over all the neighbors, the protein concentration increases, and vice versa. The last term of (8) specifies the sum of the concentration of g diffused from neighboring nodes, which aims to keep the energy consumption of nodes in balance. The diffusion term in the regulatory model simulates intercellular signaling in a multi-cellular system.

When each sensor node determines its protein concentration, it generates a random value ω following the uniform distribution within [0, 1]. Each node independently generates a random value. If the protein concentration is less than ω, then the node goes to sleep. On the other hand, if the protein concentration is greater than ω, the node becomes active by turning on its sensing circuitry.

3.2. Theoretical analysis

The discrete-time formula of the model updates take place every control period τ. Thus, the time is divided into $[k, k + 1)$ , $k = 0, 1, \dots$ , with time duration equal to τ. Then, (7)–(8) can be formulated as follows: $\begin{array}{l} (11) & g_{i} (k + 1) = (1 - a) g_{i} (k) + η f_{g}^{h} ({\overline{E}}_{i} p_{i} (k)) \\ (12) & p_{i} (k + 1) = (1 - c) p_{i} (k) + κ f_{p}^{a} ({\overline{D}}_{i} g_{i} (k)) + b \sum_{j = 1}^{N_{i}} (g_{i} (k) - g_{j} (k)) . \end{array}$

Let $g_{i s}$ and $p_{i s}$ be the steady state values of $g_{i}$ and $p_{i}$ , respectively. From (11)–(12), the steady states of $g_{i s}$ and $p_{i s}$ are derived as follows: $\begin{array}{l} (13) & g_{i s} = \frac{η}{a} e^{- {\overline{E}}_{i s} p_{i s}} = \frac{1}{N_{i}} \sum_{j = 1}^{N_{i}} g_{i s} \\ (14) & p_{i s} = \frac{κ}{c} (\frac{1 - e^{- g_{i s} {\overline{D}}_{i s}}}{1 + e^{- g_{i s} {\overline{D}}_{i s}}}), \end{array}$ where ${\overline{E}}_{i s}$ and ${\overline{D}}_{i s}$ are the steady states of ${\overline{E}}_{i}$ and ${\overline{D}}_{i}$ , respectively. For $\forall i \in N$ , since $0 < p_{i s} < 1$ , it results that $0 < κ / c < 1$ . In order to ensure the energy balancing among nodes, the following lemma is provided.

Lemma 1.
For each node, the steady state value of the gene expression is given by $\begin{array}{rcl} (15) & g_{i s} = g_{j s}, \forall i \neq j . \end{array}$
Proof.
Let $φ_{i j} = g_{i s} - g_{j s}$ and ${\bar{g}}_{i s} = \frac{1}{N_{i}} \sum_{j \in N_{i}} g_{j s}$ . From (12) and (13), $\sum_{j = 1}^{N_{i}} φ_{i j} = 0$ which means that $\begin{array}{rcl} (16) & g_{i s} - {\bar{g}}_{i s} = \frac{1}{N_{i}} \sum_{j \in N_{i}} φ_{i j} . \end{array}$ Then, (16) means that $\begin{array}{rcl} (17) & g_{i s} = {\bar{g}}_{i s}, \forall i . \end{array}$ That is, there is a zero fixed point $φ_{i j} = 0, \forall i \neq j$ . Consequently, the following steady states of the system for all i and j are derived: $\begin{array}{l} g_{i s} = {\bar{g}}_{i s} = g_{j s} \\ (18) & p_{i s} = p_{j s} \\ {\overline{E}}_{i s} = {\overline{E}}_{j s} . \end{array}$ □

From these equations, the proposed scheme balances the energy consumption among nodes by converging the gene expression level to the average one for all the neighbor nodes and this makes the energy consumption level balanced globally among all nodes.

Next, I prove the system’s stability and provide a guide of parameter selections. For simplicity, I neglect the dynamics of the protein diffusion in the proof of the system convergence. Also, I consider that ${\overline{E}}_{i}$ and ${\overline{D}}_{i}$ are arbitrary constants, such as ${\overline{E}}_{i} = {\overline{D}}_{i} = 1$ . However, the results of this paper can be generalized to the time-dependent variables of ${\overline{E}}_{i}$ , ${\overline{D}}_{i}$ . Then, (11)–(12) can be rewritten as follows: $\begin{array}{l} (19) & g_{i} (k + 1) = (1 - a) g_{i} (k) + η f_{g}^{h} (p_{i} (k)) \\ (20) & p_{i} (k + 1) = (1 - c) p_{i} (k) + κ f_{p}^{a} (g_{i} (k)) . \end{array}$
Theorem 1.
The proposed system converges to the steady states defined by ( 18 ) provided that $\begin{matrix} (21) & \begin{matrix} 0 < a, η, c, κ < 1, \\ 0 < κ < c, \\ 0 < κ < \sqrt{a (1 - a)}, \\ C_{1} + C_{2} < c (2 - c), \end{matrix} \end{matrix}$ where $\begin{array}{l} (22) & C_{1} = (ε^{2} + \frac{{(1 - a)}^{2}}{a} + 1) η^{2} \\ C_{2} = \frac{{((1 - a) η ε - (1 - c) κ)}^{2}}{a - (a^{2} + κ^{2})} . \end{array}$ Therefore, the energy consumption level of each node is balanced across the networks by converging its energy consumption level to the average level over its neighbors.
Proof.
Since $f_{g}^{h}$ and $f_{p}^{a}$ are the inhibition and activation function, respectively, the following conclusions are derived:
$f_{g}^{h} (x) = e^{- x} ⩽ 1 - ε x$ ,

$| f_{p}^{a} (x) | = | (1 - e^{- x}) / (1 + e^{- x}) | ⩽ | x |$ and $| f_{p}^{a} (x) | = | x |$ only if $x = 0$ ,
where $ε = 0.63$ .

For the system stability analysis, I use Lyapunov Theory to claim that the system defined by (11)–(12) will be convergent if I can find a Lyapunov function $V (g_{i} (k), p_{i} (k))$ that satisfies the following conditions:
$V (g_{i} (k), p_{i} (k))$ is positive definite;

$V (g_{i} (k + 1), p_{i} (k + 1)) - V (g_{i} (k), p_{i} (k))$ is negative definite;
I consider the Lyapunov function in the following form: $\begin{array}{rcl} (23) & V (g_{i} (k), p_{i} (k)) = g_{i}^{2} (k) + p_{i}^{2} (k) . \end{array}$ Here, I remove the subscript i since every node shares the same dynamics. I denote $V (k) = V (g (k), p (k))$ and obviously $V (k) ⩾ 0$ . Let $Δ V (k) = V (g (k + 1), p (k + 1)) - V (g (k), p (k))$ . Then, I can get $\begin{array}{l} Δ V (k) = & {((1 - a) g + η f_{g}^{h} (p))}^{2} - g^{2} + {((1 - c) p + κ g)}^{2} - p^{2} \\ ⩽ & {((1 - a) g + η (1 - ε p))}^{2} - g^{2} + {({(1 - c)}^{2} + κ g)}^{2} - p^{2} \\ = & (- a + a^{2} + κ^{2}) g^{2} - a g^{2} - 2 (1 - a) ε η p g \\ (24) & + 2 (1 - c) κ p g + 2 (1 - a) η g + (- 2 c + c^{2}) p^{2} + η^{2} - 2 ε η^{2} p + ε^{2} η^{2} p^{2} . \end{array}$ Let $C_{2} = - B^{2} / A$ , where $A = (- a + a^{2} + κ^{2})$ , $B = (1 - a) ε η - (1 - c) κ$ . Then, (24) can be rewritten as follows: $\begin{array}{l} Δ V (k) ⩽ & A {(g - \frac{B}{A} p)}^{2} - a {(g - \frac{(1 - a)}{a} η)}^{2} \\ - \frac{B^{2}}{A} p^{2} + \frac{{(1 - a)}^{2}}{a} η^{2} + (- 2 c + c^{2}) p^{2} + η^{2} - 2 ε η^{2} p + ε^{2} η^{2} p^{2} \\ = & A {(g - \frac{B}{A} p)}^{2} - a {(g - \frac{(1 - a)}{a} η)}^{2} \\ + p^{2} (- 2 c + c^{2} - \frac{B^{2}}{A} + ε^{2} η^{2}) + η^{2} - 2 ε η^{2} p + \frac{{(1 - a)}^{2}}{a} η^{2} \\ ⩽ & A {(g - \frac{B}{A} p)}^{2} - a {(g - \frac{(1 - a)}{a} η)}^{2} - 2 c + c^{2} - \frac{B^{2}}{A} + (ε^{2} + \frac{{(1 - a)}^{2}}{a} + 1) η^{2} . \end{array}$ So, if the following stability conditions are satisfied, I can ensure $Δ V (k) ⩽ 0$ .
$0 < a, η, c, κ < 1$ .

$0 < κ < c$ .

$A = - a + a^{2} + κ^{2} < 0 \Rightarrow κ < \sqrt{a (1 - a)}$ .

$- \frac{B^{2}}{A} + (ε^{2} + \frac{{(1 - a)}^{2}}{a} + 1) η^{2} < c (2 - c) \Rightarrow \frac{{((1 - a) η ε - (1 - c) κ)}^{2}}{a - (a^{2} + κ^{2})} + (ε^{2} + \frac{{(1 - a)}^{2}}{a} + 1) η^{2} < c (2 - c)$ .
By Lyapunov theorem, the equilibrium point is asymptotically stable if there is a continuous positive definite function $V (k)$ , that is, $V (0) = 0$ and $V (k) > 0$ for $k \neq 0$ , so that $Δ V (k)$ is negative definite. □

Based on the above analysis, both conditions of the Lyapunov function V have been satisfied, thus the system is stable and will converge to the steady states if the system parameters are selected according to the derived stability conditions. Therefore, our proposed GRN model automatically drives the WSN system performance to the desired application requirement while guaranteeing energy balancing among sensor nodes.
4. Simulation results

4.1. Configuration

In order to evaluate the performance of our proposed algorithm, a simulation environment using the MATLAB simulator is developed. The simulation area is 100 m × 100 m, where the entire network is divided into equally shaped grids, and the sensor nodes are deployed uniformly. Source nodes generate packets in a Poisson distribution with an average packet arrival rate of one packet per second. Each packet is 100 bytes and the controller time slot duration (τ) is one second. The channel capacity is set to 200 kbps. According to the derived stability conditions of (21) and (22), I set $a = η = c = 0.1$ and $κ = 0.05$ .

4.2. Results

Fig. 1.

Time behavior of the proposed algorithm with two neighbor nodes for each node: (a) g and p with $d_{r} = 10 s$ , (b) $\overline{E}$ with $d_{r} = 10 s$ , (c) average delay with $d_{r} = 10 s$ , (d) g and p with $d_{r} = 50 s$ , (e) $\overline{E}$ with $d_{r} = 50 s$ , and (f) average delay with $d_{r} = 50 s$ .

The simulation results show that indicate how the system behaves over time with the proposed algorithm. Figure 1 shows the variation of g, p, average delay, and energy consumption. The average delay is the time between when the source nodes send packets and the sink node in the network receives the packets, averaged over all source nodes. The delay requirement is set to two different values, 10 s and 50 s, respectively. The number of neighbor nodes of each node is set to two ( $N_{i} = 2, \forall i$ ). The values of g and p of node i are denoted as $g_{i}$ and $p_{i}$ , respectively. In order to show the performance of energy balancing, the initial power consumption of each node is set differently; the initial energy consumption of nodes 1, 2, and 3 are set to 0 mW, 10 mW, and 20 mW, respectively.

Figure 1(a), (b), (c) shows the results for $d_{r} = 10 s$ . From Fig. 1(a), during the first 450 s, $g_{3}$ is lower than $g_{1}$ and $g_{2}$ . The lower g leads to a smaller p, i.e., a lower probability of achieving an active state. Since node 3 starts with lower initial battery power compared with node 1 and node 2, node 3 is considered less fit to be active compared to its neighbors, which leads to the smallest p among them, thereby going into sleep mode more often in order to save energy. Meanwhile, node 1 has relatively good local environmental quality in terms of consumed energy, resulting in the highest g and p among nodes, i.e., the highest probability of being active. After 450 s, g and p of all nodes become identical and converge to almost the same value. This means that the probabilities of becoming active are almost identical for all nodes, resulting in energy balancing among nodes. Figure 1(b) shows the ratio of consumed energy level to the neighbors’ average, ${\overline{E}}_{i} (i = 1, 2, 3)$ . The values of ${\overline{E}}_{1}$ , ${\overline{E}}_{2}$ , ${\overline{E}}_{3}$ become almost the same after 450 s in spite of the difference in initial power among nodes. It means $E_{i} = \sum_{\forall j \in N_{i}} E_{j} / N_{i}$ , for all i. In this way, when an imbalance in the energy consumption level is detected, each node adapts its probability of being active and balances the energy consumption among the neighboring nodes by controlling both the gene expression and protein concentration. Figure 1(c) shows the average delay and our proposed scheme keeps the average delay around the desired delay requirement, 10 s, for the entire simulation time.

Figure 1(d), (e), (f) shows the system behavior with $d_{r} = 50 s$ . In Fig. 1(d), g and p become close to one and zero, respectively, during the first 100 s. This is due to the rather loose delay requirement compared to $d_{r} = 10 s$ . The looser delay requirement causes a lower probability to be active state and sensor nodes go into sleep mode more often in order to save energy. Accordingly, as shown in Fig. 1(e), the gradient of $\overline{E}$ of each node is rather smaller than the result of Fig. 1(b). After 450 s, the energy consumption ratio of each node is stabilized to the same value of one. Also, Fig. 1(f) shows that the average delay lies around the desired delay requirement, $d_{r} = 50 s$ , after 200 s. From Fig.1, the proposed scheme successfully achieves energy balancing among nodes and keeps the average delay to the desired delay requirement by controlling the gene expression and protein concentration. Also, the time behavior of the proposed scheme verifies that our proposed scheme achieves both energy balancing and delay guarantee in spite of the changes in the number of neighbor nodes and the differences in initial power. These results are consistent with the theoretical analysis in Section 3.2.

Fig. 2.

Average performance varying delay requirements: (a) gene expression (G), protein concentration (P), and (b) active node ratio.

The average performance under the variable delay requirements is evaluated. The delay requirement is varied within the range of [3 s, 50 s]. For each delay requirement, 1000 simulations are used to obtain averages of gene expression (G), protein concentration (P), active node ratio, delay and consumed energy level (E), so each point in the graphs represents an average of 1000 executions. To show the effectiveness of the proposed scheme, the proposed algorithm is compared with two existing schemes, QoS-guaranteeing Duty Cycle Control (Q-DCC) [21] and GRN-based Optimal Coverage Control (G-OCC) [5]. Q-DCC proposes a feedback controller which controls the duty cycle to guarantee an end-to-end communication delay while achieving the energy efficiency for WSNs. To do this, Q-DCC decomposes the end-to-end delay requirement problem into a set of single-hop delay requirement problems. The duty cycle of each node is determined based on the single-hop delay requirement and the actual packet delay, measured using time stamps. G-OCC introduces the GRN as a computing paradigm and demonstrates its effectiveness for sensor coverage. G-OCC assigns values of ON or OFF to each sensor node to attain the maximum possible coverage while simultaneously keeping the number of active sensors as low as possible to reduce power consumption. The sensor nodes turn ON when it acquires high values of its corresponding gene expression levels. A threshold is selected a priori, and any sensor with a higher value in its corresponding gene expression level can be turned ON.

Figure 2 illustrates the average performance of the proposed scheme in terms of gene expression $(G)$ , protein concentration $(P)$ , and active node ratio for different delay requirements. As the delay requirement becomes looser, the value of G increases while protein concentration P decreases, which increases the number of nodes entering sleep mode instead of active mode, as shown in Fig. 2(b). On contrary, as the delay requirement become stricter, the active node ratio increases with the value of P, leading to more frequent packet transmissions in order to meet the delay requirement.

Fig. 3.

Average performance of the proposed algorithm, Q-DCC [21], and G-OCC [5] varying delay requirements: (a) delay and (b) energy consumption level.

Figure 3 illustrates the average delay and consumed energy levels of the proposed algorithm, Q-DCC, and G-OCC. From Fig. 3(a), the proposed algorithm and Q-DCC successfully control the average delay according to the desired requirements. However, G-OCC keeps the average delay close to zero irrespective of the varying delay requirements. That leads to excessive energy consumption as shown in Fig. 3(b). Q-DCC shows a lower energy consumption level compared with G-OCC, but the energy consumption level is almost constant in spite of different delay requirements. In contrast, our proposed scheme achieves the smallest energy consumption ratio, while meeting delay requirements. Also, as the delay requirement becomes looser, the energy consumption level decreases, resulting in energy savings. This is because as the delay requirement becomes looser, the proposed algorithm makes a greater number of nodes enter sleep mode instead of active mode, resulting in much lower energy consumption compared to G-OCC and Q-DCC.

5. Conclusions

In this paper, the conditions of system parameters of the proposed decentralized node scheduling scheme is presented to ensure system convergence and stability. The coordination problem of on-off cycles of wireless sensor nodes is represented by modifying gene regulation dynamics of multi-cellular mechanisms. From the theoretic analysis, the insights into the performance of the proposed scheme is provided by deriving the steady states of the systems.

Footnotes

Acknowledgement

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2014050424).

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