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With the growing popularity of wireless sensor networks, network stability has become a key area of current research. Different applications of wireless sensor networks demand stable sensing, coverage, and connectivity throughout their operational periods. In some cases, the death of just a single sensor node might disrupt the stability of the entire network. Therefore, a number of techniques have been proposed to improve the network stability. Clustering is one of the most commonly used techniques in this regard. Most clustering techniques assume the presence of high power sensor nodes called relay nodes and implicitly assume that these relay nodes serve as cluster heads in the network. This assumption may lead to faulty network behavior when any of the relay nodes becomes unavailable to its followers. Moreover, relay node based clustering techniques do not address the heterogeneity of sensor nodes in terms of their residual energies, which frequently occur during the operation of a network. To address these two issues, we present a novel clustering technique, Dynamic Clustering with Relay Nodes (DCRN), by considering the heterogeneity in residual battery capacity and by removing the assumption that relay nodes always serve as cluster-heads. We use an essence of the underlying mechanism of LEACH (Low-Energy Adaptive Clustering Hierarchy), which is one of the most popular clustering solutions for wireless sensor networks. In our work, we present four heuristics to increase network stability periods in terms of the time elapsed before the death of the first node in the network. Based on the proposed heuristics, we devise an algorithm for DCRN and formulate a mathematical model for its long-term rate of energy consumption. Further, we calculate the optimal percentage of relay nodes from our mathematical model. Finally, we verify the efficiency of DCRN and correctness of the mathematical model by exhaustive simulation results. Our simulation results reveal that DCRN enhances the network stability period by a significant margin in comparison to LEACH and its best-known variant.

With the emergence of highly dense fabrication technology and low production costs, Wireless Sensor Networks (WSNs) prove to be useful in a myriad of diversified applications. In a typical WSN application, sensor nodes are scattered in a region from where they collect data to achieve certain goals. Data collection may be continuous, periodic, or event based. Regardless of the data collection technique used, WSNs must operate in a stable manner. This stability is especially important in applications such as security monitoring and motion tracking. The death of just one sensor node may disrupt coverage or connectivity, and thus may reduce network stability in such applications. Therefore, it is important to ensure that all the deployed sensor nodes in WSNs be active during the operational lifetime to ensure uninterrupted service. However, sensor nodes are generally equipped with batteries that have a limited capacity. Therefore, each sensor node must efficiently use its available energy in order to improve the lifetime of the WSN. Different techniques have been proposed to ensure efficient usage of the available energy in a sensor node. Clustering is one of the most well known techniques used to ensure the efficient usage of the battery reserve of the sensor nodes.

Clustering techniques group deployed sensor nodes into clusters. One sensor node in a cluster is solely responsible for communicating with the base station. This sensor node is called the cluster head, and the remaining sensor nodes in the cluster are called followers. Here, the followers collect data and send the collected data to their corresponding cluster heads. Afterwards, the cluster heads aggregate their own collected data with the data received from their followers and send the aggregated data to a base station or sink to accomplish a specific goal. Generally, cluster heads are physically closer to their follower nodes compared to the sink/base-station. Therefore, it takes less energy to transmit data to the cluster head instead of the sink, which allows the sensor nodes to conserve energy and live longer.

There are different clustering techniques in use for wireless ad-hoc networks. However, those techniques cannot be directly used in WSNs because of the fact that WSNs have stricter energy constraints than ad-hoc networks. Therefore, several clustering techniques are proposed in the literature to specifically focus on the constraints of WSNs. We can categorize these clustering techniques in two groups—static and dynamic. Dynamic clustering techniques are more useful for WSNs because they can exploit the dynamic variation in residual energies of the sensor nodes, and thus can maximize network lifetime to a great extent. Different dynamic clustering techniques consider the lifetime of sensor networks in different ways in their optimization processes.

In some early research, network lifetime of a WSN is considered as the time required by the last sensor node to die. Some other research considers the lifetime of a WSN in a different way—the time required by half of the sensor nodes to die. However, network lifetime is frequently defined as n-of-n lifetime [

· Failure of relay nodes causes faulty behavior in the network, and thus the approach makes the network less fault tolerant.

· After a significant amount of service time, the residual energies of relay nodes may become comparable to other live nodes in the network. In this case, the fixed assignment of cluster headships to relay nodes forces them to die even quicker than other live nodes, resulting in faulty behavior and lower network stability.

Therefore, we consider all sensor nodes as candidates for attaining cluster headships. This consideration enables us to avoid the limitations raised from the assumption of previous work.

We use LEACH [

Based on our work, we make the following set of contributions in this paper:

· We propose a novel distributed and dynamic algorithm to cluster a wireless sensor network with relay nodes to improve its network stability in terms of death of the first sensor node. The algorithm is based on four carefully devised heuristics.

· We adopt a mathematical model for our algorithm. The model can effectively find out the optimal percentage of relay nodes. We verify the optimality of our proposed model by comparing its suggested optimal percentage of relay nodes to that suggested by the model of LEACH.

· We demonstrate effectiveness of our proposed algorithm using extensive simulations to compare it with LEACH and the best variant of LEACH. Our simulation results show that our proposed algorithm can achieve up to 188% and 112% improvements in time before the death of the first node in comparison to LEACH and the best variant. Besides, our algorithm also improves the time before the death of half of the nodes by 23% and 15% in comparison to LEACH and the best variant.

We organize the rest of this paper as follows: We point out some related work in the following section. Then, we briefly describe the underlying approach of our technique along with its variants in Section 3. We present our proposed clustering technique with four heuristics and a complete mathematical model after that section. In Section 5, we evaluate the stability period of DCRN by simulation results. In the last two sections, we conclude the paper by discussing our future directions.

Several techniques have already been proposed to improve network lifetime in WSN. Clustering is one of the widely accepted techniques among them. Clustering is also used in wireless ad-hoc networks and mobile ad-hoc networks. Several clustering techniques have already been introduced for partitioning nodes in these areas. Some of the early clustering techniques are—Hierarchical Clustering [

PEGASIS [

In [

PEACH [

Optimal energy aware clustering [

ACE [

HEED [

PADCP [

LEACH [

There are a variety of diversified techniques that maximize network lifetime other than clustering. Lifetime is defined in various ways in these techniques. In [

SPINDS [

LEACH is a self-organizing and adaptive clustering protocol [

In LEACH, the lifetime of the network is divided into some discrete and disjoint time intervals. Each interval is again divided into subintervals or rounds as shown in

In the advertisement phase, each node independently decides whether to become a cluster head or not. In the cluster set-up phase, the clusters are organized based on the decisions made in the advertisement phase. Then a steady-state phase follows. In this phase, the followers, i.e., the sensor nodes except cluster heads, will send data to the corresponding cluster head. The cluster heads accumulate and compress the received data with their own data. Cluster heads send the compressed data to the base station. In order to minimize cluster establishment overhead, the duration of the steady-state phase must be longer than that of the cluster set-up phase.

At the very beginning of the advertisement phase, each node decides whether it wants to become a cluster head for the current round. This decision is based on the suggested percentage of cluster heads for the network, which is set a priori. This decision also depends on the number of times the node has already been a cluster head. This

decision is made by a node n choosing a random number between 0 and 1. If the number is less than a threshold T(n), the node decides to become a cluster head. The threshold is calculated as follows:

where P = the percentage of nodes that can become cluster heads (e.g., P = 0.05);

1/P = the number of subintervals in an interval;

r = the current subinterval;

G = the set of nodes that have not been cluster heads yet in the current interval.

Using this threshold, a node can be a cluster head in any one of 1/P subintervals in an interval. At the first subinterval of an interval (r = 0), each node has a probability P to become a cluster head. The nodes that are cluster heads in the first subinterval cannot be cluster heads in the next (1/P – 1) subintervals of the same interval. Thus, the probability of attaining cluster headships by the remaining nodes increases. After the completion of 1/P subintervals, a new interval will start, and all the nodes are again eligible to become cluster heads.

Each node, which has chosen itself as a cluster head in the current subinterval, broadcasts an advertisement message to the rest of the nodes. The non-cluster-head nodes will choose the cluster to which it will belong in this subinterval. This decision is based on the received signal strength of the advertised message. Assuming symmetric propagation channels, the cluster head whose advertisements have been heard with the largest signal strength will be selected by a non-cluster-head sensor node as its cluster head. In the case of a tie, a cluster head is chosen randomly.

There are some mathematical models available on LEACH. In [_{opt} as:

where N is the total number of sensor nodes, M is the dimension of the sensor area, d_{BS} is the distance between cluster head and the base station, and are the amplifier energies.

In [_{opt}, as:

In [_{opt} as:

where λ is the intensity of homogeneous spatial Poisson process that indicates the sensor node density, E_{elec} is the electronic energy required for coding, modulation, filtering etc., and E_{DA} is the energy required for data aggregation.

However, the lifetime of a sensor node is directly the inverse of its long run rate or expected rate of energy consumption. Therefore, in order to elongate network lifetime, the long run rate of energy consumption must be given more importance than other metrics (e.g., energy required to transmit one frame [

In [

P = the desired percentage of cluster heads;

S = the number of subintervals in an interval, therefore s = 1/P;

P_{h} = the probability of becoming cluster head of a follower node at the start of any subinterval;

= the probability of becoming cluster head of a cluster head node at the start of a subinterval in the next interval;

Φ_{0} = the probability of becoming cluster head of a sensor node at the start of any subinterval;

T(n) = the currently considered threshold value;

N = the total number of sensor nodes in the network;

a × b = the area of the rectangular coverage area.

According to the Renewal Reward Theorem, the rate of reward will be:

where R is the reward and X is the cycle length. The model proposed in [

The model considers different state transition diagrams for a sensor node between two states while changing the subinterval in an interval and between two states while changing the subinterval as well as the interval to compute E(X).

Using these state transition diagrams, the probability of becoming a cluster head, Φ_{0}, at the start of any subinterval is calculated as follows:

where P_{h} = P + (1 – P)^{N}. Here, P contributes to the probability of becoming cluster head by choosing a random number less than the threshold and (1 – P)^{N} is the probability of becoming a one-member cluster head in the case of choosing a random number not less than the threshold with no candidate cluster head found in the network.

After a number of steps, the long run rate of energy consumption is calculated as:

where,

Here, E_{elec} is the energy required per bit to run the circuitry in transmitter or receiver, E_{DA} is the energy required for data aggregation, is the energy constant for the radio transmission of a follower node, is the energy constant for the radio transmission of a cluster head node, k is the number of bits in a message, λ is the path loss exponent, d_{BS} is the distance between the cluster head and the base station, and p_{a} is the percentage of the circular area (centered at a follower and with a radius equal to the distance to a cluster head) falls within the sensor area. As we cannot get any closed form for the derivative of Equation (3), we can get the optimal percentage of the cluster heads by plotting the value of the long run rate of energy consumption from the equation.

This algorithm introduced a fairly simple strategy which is more efficient than the direct transmission and the minimum-transmission-energy (MTE) protocol that chooses the route to minimize the transmitter’s energy. However, it has some limitations:

· LEACH always wants to achieve an even distribution of energy consumption, which might not be rational. Residual energies in different nodes do not remain the same after a significant amount of time of operation. Nodes with higher residual energy should get preference to be elected as cluster heads. Otherwise, longer network stability cannot be ensured.

· LEACH assigns all sensor nodes equal probability to become cluster heads. However, if a sensor node with very low residual energy is chosen to be cluster head, then it may quickly run out of energy. Therefore, there must be some sort of constraint to discount the sensor nodes having very low residual energy during the choice of cluster heads to prolong their lifetime. There is no such constraint in LEACH.

A number of variants have already been proposed for LEACH to overcome its limitations. Some of them are briefly summarized in the following section.

SEP [

SEP assumes m fractions of the nodes are advanced nodes, which have α times energy than that of the normal nodes. As a result, it assumes n(1 + αm) number of virtual normal nodes in the network. It extends the number of subintervals from 1/P to (1+ αm)/P in an interval. The objective of this extension is to elect a normal node once and an advanced node (1 + α) times as the cluster head in an interval. The probability equation to become a cluster head has been modified. In fact, two different equations are used for the normal and the advanced nodes. The weighted election probabilities for the normal and the advanced nodes are p_{nrm} and p_{adv} respectively. Their equations are as follows:

and

where p_{opt} is the optimal probability of a node to become a cluster head. It also uses two different equations for the threshold. One for the normal nodes called T(s_{nrm}) and the other for the advanced nodes called T(s_{adv}). T(s_{nrm}) and T(s_{adv}) are calculated as follows:

where G′ is the set of normal nodes that have not become cluster heads yet within the last 1/p_{nrm} subintervals and G″ is the set of advanced nodes that have not become cluster heads yet within the last 1/p_{adv} subintervals in an interval.

SEP does not make any attempt to enhance the network stability. Besides, in SEP, the percentage of cluster heads is optimized based on the energy consumption in an interval. However, this value should be optimized on the basis of the long run rate or expected rate of energy consumption for achieving the higher network stability period. Finally, this work introduced the heterogeneity to LEACH in terms of two levels of residual energy. However, during the life cycle of the network the different levels of the residual energies may exist which will not be covered by only two types.

Deterministic Cluster Head Selection (DCHS) [

where is the current energy and is the initial energy of the node. The other parameters have the same definitions as of LEACH.

After a significant amount of time of operation, the residual energies of the sensors generally become very low, and then this threshold value will be very small. This can result in a situation where all the live sensors become one-member cluster head. In this case, the energy consumption rate will be very high. To break this stuck condition, another modified equation of the threshold value has been proposed as:

where r_{s} is the number of consecutive rounds in which a node has not been a cluster head.

DCHS uses a random value for the percentage of cluster heads like LEACH. Therefore, it does not consider the optimal value of this parameter. Besides, it does not suggest any optimum value for r_{s}. Finally, it did not attempt any improvement to enhance the network stability.

There are some other variants of LEACH in addition to these approaches. LEACH-C [

This algorithm minimizes the total sum of squared distances between all the non-cluster-head nodes and the corresponding closest cluster head node. Thus, it minimizes the amount of energy required to transmit data to the cluster head nodes by the non-cluster-head nodes. However, the base station selects the cluster heads based on their positions and the average residual energy in the network. Therefore, like LEACH, the individual residual energy in each sensor node has little impact on the cluster head selection process in LEACH-C. This centralized algorithm also suffers from non-scalability. Besides, incorporating a GPS receiver or similar device in the sensor nodes increases sensor node cost. Finally, it did not attempt any improvement to enhance the network stability.

Another variant of LEACH, Adaptive Cluster Head Selection [

In summary, none of the research mentioned in this section makes any attempt to improve the network stability. Moreover, none of them investigate the applicability of relay nodes in a WSN during the clustering of sensor nodes the network. Therefore, in the next section, we propose a novel technique, Dynamic Clustering with Relay Nodes (DCRN), to improve network stability using relay nodes.

In this section, we propose a new algorithm to cluster sensor nodes in a network to improve network stability in terms of the death of the first sensor node. We follow the underlying approach of LEACH. In LEACH, each sensor node is given equal chance to get the cluster headship and thus its lifetime depends solely on its own residual energy. Therefore, a sensor node with low residual energy dies within a short period. However, there may be some other sensor nodes alive after its death. If that sensor node with low residual energy could exploit the residual energies of other high-energy live sensor nodes, then it would live longer. Therefore, we should choose cluster heads according to the residual energies to increase the stability period of a WSN. Moreover, deployment of relay nodes ensures the availability of high-energy sensor nodes in the network. Therefore, we should ensure the high probability of becoming cluster heads for these nodes. Besides, if no cluster head is found by a non-cluster head node, then the nearest relay node should be chosen as its default cluster head to avoid the situation of being a one-member cluster head. Finally, sensor nodes with very low residual energies should be excluded from choices of probable cluster heads to maximize their lifetimes. We present four heuristics to achieve these goals. We illustrate our complete clustering algorithm DCRN in detail after describing these heuristics. We also adapt the mathematical model derived in [

We propose four heuristics for DCRN in this subsection. The first two heuristics basically attempt to choose cluster heads according to relative residual energies of sensor nodes. The third heuristic attempts to avoid one member cluster headship for those sensor nodes that cannot discover any cluster heads in the network. The fourth heuristic provides a safeguard for low energy sensor nodes from becoming cluster heads. We describe these heuris tics as follows:

Heuristic 1: Energy consumption of a cluster head node is higher than that of a follower node. Therefore, sensor nodes with higher residual energy should be elected as cluster heads. In the original LEACH algorithm, network lifetime is divided into disjoint and discrete intervals that are again divided into some subintervals. If a LEACH node becomes a cluster head in a subinterval, it cannot become a cluster head again in any of the subsequent subintervals of the same interval. However, if a sensor node with higher residual energy can attain cluster headship again in other subintervals of the same interval, then a sensor node with lower residual energy can escape from being a cluster head. In that case, the lifetime of this lower energy sensor node will increase by indirect utilization of residual energy of the higher energy sensor node. For this reason, we make the subintervals completely memory-less and eliminate the use of the separate set of nodes that have not been cluster heads yet in the current interval. With this modification, the probability of becoming a cluster head of a sensor node in a subinterval does not depend on its status in the previous subintervals. This heuristic provides a fair increase in the network stability period. We have to consider more proportionate use of the residual energies to obtain further enhancement of the network stability period. Therefore, we adopt our next heuristic to ensure more proportionate use of the residual energies.

Heuristic 2: We can expect a higher stability period of a sensor network if we increase the probability of sensor nodes with higher residual energies becoming cluster heads. We should consider the relative residual energy of a sensor node to determine whether it is with higher residual energy or not. For this reason, we judge the relative residual energy of a sensor node while selecting it as a cluster head.

DCHS [

where E_{current} is the current energy of a sensor node and is the maximum initial energy all over the network. E_{current} is measured at the beginning of each interval, whereas is measured prior to the deployment. This equation is used by all sensor nodes in the network. Therefore, if the residual energy of a relay node becomes lower than that of a normal sensor node, then the probability of the relay node becoming the cluster head will also be lower than that of the normal node. This phenomenon ensures a longer time interval before the death of the first node in network.

The modified threshold value may become very small after a long duration of operation. The small value may inhibit sensor nodes from becoming cluster heads. DCHS proposes another modification in equation of the threshold to adapt this situation. However, we do not need any further modification as our next heuristic will take care of this situation.

Heuristic 3: A sensor node becomes a follower when it picks a random number greater than its threshold, T(n), and it finds a candidate cluster head node. Therefore, the node will become a cluster head even by choosing a random number greater than its threshold if it does not find any candidate cluster head. It may result in two situations—if there is no candidate cluster head in the network or the node cannot successfully receive any of the advertisement messages from the candidate cluster heads. In either case, attaining the cluster headship by the node significantly increases its energy consumption rate. Therefore, we propose a heuristic to impose the choice of the nearest relay node as cluster head in case no candidate cluster head is found. This heuristic completely eliminates the situation where all nodes in the network become one-member cluster heads, and thus it attains significant improvement in the overall energy consumption rate of the network.

Heuristic 4: The stochastic nature of attaining cluster headship imposes a nonzero probability of becoming a cluster head to all live nodes. However, a node with low residual energy will quickly run out of energy if it attains cluster headship. Therefore, we propose our last heuristic to utilize a minimum threshold value on the residual energy to make a node eligible for attaining the cluster headship. A node with residual energy less than the threshold is completely ignored during the cluster head selection. This heuristic guarantees an elongated lifetime for nodes with low residual energy and thus increases the stability period of the network.

Now, we present a complete algorithm for DCRN exploiting all these heuristics in the next subsection.

In the DCRN algorithm, we divide the lifetime of the network into some discrete and disjoint equal length intervals in DCRN. Here, each sensor node operates in these intervals. The intervals are maintained using clock synchronization [55-58]. Each interval has three consecutive phases—advertisement, cluster-setup, and steadystate phase. The DCRN algorithm, depicted in

1) Advertisement Phase: Each relay node broadcasts a RELAY_EXPOSUE message in this phase. Besides, all the sensor nodes independently decide whether or not to become cluster heads. To make this decision, each node computes the threshold, T(n) using Equation 4. Then, it picks a random number and compares the random number with the threshold. If the random number is less than the threshold, then it becomes a cluster head and broadcasts the HEAD_EXPOSURE message.

2) Cluster Set-Up Phase: Each non-cluster-head sensor node independently attempts to choose its cluster head in this phase. It may encounter two cases during the attempt:

· CASE 1: It receives one or more copies of HEAD_ EXPOSURE messages from other candidate cluster head nodes. In this case, the sensor node attempts to become a follower of the nearest candidate cluster head node and sends a FOLLOWER_ACCEPTANCE message to that node. Here, the sensor node chooses the cluster head node with maximum signal strength as the nearest cluster head [

· CASE 2: It does not experience any arrival of the HEAD_EXPOSURE message from other sensor nodes. In this case, the sensor node tries to be a follower of the nearest live relay node. If it does not find any live relay node, it becomes a one member cluster head.

3) Steady-State Phase: In this phase, the followers send data to their corresponding cluster heads. The cluster heads accumulate, aggregate, and compress the received data with its own data. Cluster heads send the aggregated and compressed data to the base station. The duration of steady-state phase is significantly longer than the summation of the durations of the advertisement and cluster set-up phases in order to minimize the cluster establishment overhead.

The difference between the underlying mode of operations of LEACH and DCRN arises because of the new heuristics. The last two heuristics make changes only in the threshold value (T(n)). This change merely affects the probability of becoming a cluster head of a follower node at the start of any subinterval (P_{h}). Otherwise, there is no impact of these two heuristics on Equation (3), which is the latest mathematical formulation of LEACH.

On the other hand, Heuristic 1 of our new clustering algorithm makes the subinterval completely memory-less. For this heuristic, the first state transition diagram of _{0} is formulated from the weighted combination of the two state transition diagrams of _{0} needs to be changed in the mathematical model of DCRN. With the introduction of Heuristic 1, any sensor node can become a cluster head irrespective of its status in the previous sub interval. Therefore, the probability of becoming a cluster head of a follower node at the start of any subinterval (P_{h}) will no longer differ from the probability of becoming a cluster head of a sensor node at the start of any subinterval (Φ_{0}). As a result, we get a new formulation of Φ_{0} as Φ_{0} = P_{h} in the changed mathematiccal model of DCRN.

On the other hand, the third heuristic imposes followership to a node that would be a candidate for becoming a one-member cluster head according to LEACH, if it receives any RELAY_EXPOSURE message. Therefore, this heuristic lowers the probability of becoming a cluster head by reducing the probability of becoming a onemember cluster head. We formulate the new probability of becoming a cluster head, Φ_{0} as:

where is the probability that at least one relay node is live, N_{relay} is the total number of relay nodes, and γ is the probability of successful transmission that reflects environmental effects as well as interference in the network.

With this change, we can use Equation (3) as the mathematical model of DCRN. We compare this mathematiccal model with that of LEACH by simulation results in the next section. We also analyze the efficiency of DCRN in that section.

We conduct our simulation runs on a randomly deployed wireless sensor network. Our simulation program is written in Visual C++. In this section, we first describe our network settings along with various parameters used in the energy rate calculation. Then, we compare the mathematical models of DCRN with that of LEACH. Finally, we evaluate network stability in DCRN with that of LEACH and the best variant of LEACH, DCHS.

We use the network settings as shown in

· The total number of sensor nodes in the network is 100.

· The dimensions of the sensor area are 200 × 200, having the base station at (600, 100). We consider all dimensions and distance values in terms of meters.

· The sensor nodes are uniformly distributed over the sensor area. However, relay nodes are placed to ensure equal coverage for all of them. In our experiment, the optimal number of relay nodes is four, which is found by the mathematical model in Section 5.2. These nodes are placed at (50, 50), (50, 100), (100, 50), and (100, 100).

· Each sensor node is initially equipped with a battery of 1 - 5 Joules. Sensor nodes have initial energy uniformly distributed over this range and the distribution is shown in

We use the following parameters [

· The amount of energy per bit to run sensor node circuitry, E_{elec}, is 5 × 10^{–8};

· The value of energy constant, Є_{amp}, for radio transmission, is 1 × 10^{–10};

· The number of data packets generated during each subinterval by a sensor node is normally distributed in the range of [0, 50], with a mean value of 25. We applied the Box-Muller transformation [

· Each data unit contains 8 bits of data;

· The probability that a message successfully arrives at its destination is 90% (i.e., γ = 0.9).

We first plot the long run rate of energy consumption versus the percentage of cluster heads from the mathematical model of LEACH in _{0}) using Equation (2), and it must not exceed 1. Here, if the percentage of cluster heads (P) exceeds 0.61, then the value will exceed 1. In order to avoid this, we plot the graph against the percentage of cluster heads up to 0.61. According to the graph:

· The energy consumption rate initially decreases very sharply with the increase of the percentage of cluster heads.

· There is an optimal point for which the energy consumption rate is the lowest. After this point, the energy consumption rate increases with the increase of the percentage of cluster heads. In our simulation result, the optimal point for LEACH is (0.045, 0.0005912). The optimal point is explicitly shown in

We also plot the long run rate of energy consumption versus the percentage of heads from the mathematical model of DCRN in _{0} using Equation (5) rather than Equation (2). Here, we plot the graph against the percentage of cluster heads up to 1 as the value of Φ_{0} remains within 1 for these values.

The graph in

· This point indicates that the optimal percentage of cluster heads is 3.5%. It also indicates the optimal percentage of relay nodes as we deploy the relay nodes to primarily act as cluster heads.

· The optimal point provides a 32% lower long run rate of energy consumption than the optimal point for LEACH found in

Next, we evaluate the network stability of DCRN against that of LAECH and its best variant. [

performance comparison.

We utilize an optimal percentage of cluster heads for all the algorithms under evaluation. We have already found the optimal percentages of cluster heads as 0.045 and 0.035 for LEACH and DCRN respectively from the mathematical models. The optimal percentage of cluster heads for the LEACH variant is empirically found as 0.05 in [

1) Data rate of a sensor node;

2) Position of the base station; and 3) Initial energy of the relay nodes.

In the evaluation process, we measure network lifetime in terms of duration before death of the first sensor node as well as duration before death of half of the sensor nodes. The former one is generally termed as First Node Dies (FND) and the later one is termed as Half of the Nodes Die (HND). We plot 15 points of FND and HND for each of the three metrics. We take the average of 100 simulation passes for each of those points. For the first point in each case, we use similar sensor node placements to those already described in Section 5.1. We vary the corresponding metric by a certain constant value to obtain each subsequent point. We present our findings obtained from the evaluation below.

In the initial network settings, the number of data packets generated by a sensor node in a subinterval is symmetrically and normally distributed in the range of 0 to 50, with the mean of 25. We conduct 15 simulation runs varying this range. We change the upper limit of the range from 50 packets with a step of 5 packets in each simulation run. We plot the values of network stability periods in terms of First Node Dies (FND) in

In the initial network settings, the base station is located at (300, 100). Therefore, the distance of the base station from the center of the network area is 200 meters. We conduct 15 simulation runs varying this distance. We change the position of the base station in the first dimension from 300 meters with a step of 20 meters in each simulation run. We plot the values of network stability

periods in terms of FND in

In the initial network settings, the initial energy of relay nodes is 15 Joule. We conduct 15 simulation runs varying this initial energy. We change the initial energy from 15 Joules with a step of 1 Joule in each simulation run. We plot the values of network stability periods in terms of FND in

10(b). There is a significant improvement in FND for DCRN over LEACH and its variant in

These values clearly indicate that DCRN provides significantly higher time before the death of the first node in comparison to LEACH and its variants irrespective of the data rate of the sensor node, the position of the base station, or the initial energy of relay nodes. The perform-

ance of DCRN becomes even better with an increase of initial energy in relay nodes. Moreover, in all cases, DCRN also provides moderate improvement of HND over LEACH and its variant.

We propose DCRN for sensor nodes with similar transmission and sensing ranges. In our future work, we will attempt to enhance DCRN for sensor nodes with varying transmission and sensing ranges. In addition, we will also attempt to enhance DCRN for multi radio sensor nodes, which are now emerging in recent research [

Clustering techniques have the inherent potential to effectively balance the energy consumption throughout a wireless sensor network to improve the stability of the network. Deployment of relay nodes can enhance the potential to a great extent if they are considered in a suitable way during the clustering mechanism. In this paper, we propose a novel dynamic, self-organizing, and adaptive technique DCRN to cluster sensor nodes in WSN with relay nodes to exploit the potential. We do not make any assumption in DCRN such that only relay nodes attain cluster headships, whereas such assumption is enforced in the previous work. Therefore, DCRN does not suffer from any of the limitations of lower fault tolerance or reduced network stability due to the assumption.

To devise DCRN, we use four heuristics with proper justifications. We present a complete mathematical formulation for DCRN exploiting that of LEACH. We present the improvement achieved in DCRN using the mathematical model. Besides, we evaluate the stability period of DCRN with that of LEACH and its best variant through simulation results. The results suggest that DCRN achieves significant improvement in network stability under different circumstances.