1 Introduction

Over the past few decades, Wireless sensor networks (WSNs) have been used in many of indoor and outdoor applications. Although the size of the Sensor nodes (SNs) is small but these are capable of performing many operations in unattended geographical regions. WSNs contain nodes having limited energy and computing power deployed in a certain region for specific applications. SNs in typical WSN communicate with each other in multi hop manner for information dissemination.

With the rapid advancement of wireless technology, WSNs are now used for various mission critical applications, e.g., weather forecasting, remote healthcare and disaster management military applications. All these applications require efficient mechanisms for coverage and connectivity preservation along with energy saving as energy level of these devices deplete quickly when they perform certain operations. Specifically, the energy of the nodes is wasted to large amount if they are performing no operations in a given time slot. This may happen because nodes did not know when to transmit or receive the data from other SNs in the network [1, 2]. Generally, the data collected by the SNs is transferred din hop by hop manner to the remote locations where Base Station (BS) is deployed. As this involves several round of communication among the nodes in the network so lot of energy is wasted in this mechanism. As energy is a scared resource in WSNs so a novel mechanism is required which has self healing features so that the valuable resource such as energy can be saved.

In literature, clustering is the most common technique which is used for efficient operations to save energy in WSNs [3–10]. The area under investigation is divided in to various clusters depending upon the various factors such as density, remaining energy, or coverage and connectivity. Out of the participating nodes in the network, some of the nodes may be elected as cluster heads (CHs) and rests of them are treated as non CHs. The CHs are responsible for monitoring and data transmission to BS in a particular region. All non CHs nodes transit the gathered information to their respective CH. This process will save lot of energy as all the nodes need not remain in contact with the BS all the time. The respective CH is responsible for collecting and maintaining the data in the region under investigation.

Although clustering has been widely used in WSNs to save the energy as discussed above [3–10], but it also raises many challenges such as selection of CH nodes and coverage and connectivity in the network depending upon various metrics. An efficient coverage mechanism covers all the nodes in the network with minimum overlap and resource consumption. This will minimize the number of CHs in the network and energy consumption of the nodes. Both of these issues are challenging and required a specific approach to solve the same. There are many proposals for efficient clustering with coverage and connectivity in WSNs covering these aspects. Some protocols have used duty cycle which performs the on/off in WSNs nodes to save the energy [2]. But still there is a requirement of efficient optimized clustering mechanism which not only performs the efficient clustering and maintains the coverage and connectivity by consuming less energy but also have features of self healing and learning in it. Keeping in view of the above, in the paper we propose a new Efficient Learning Automata (LA) based Cell Clustering Algorithm (ELACCA).

It has been found in literature that LA can be used to solve wide variety of applications [11–16]. It is an optimization technique that uses the concept of learning based upon the input parameters and produces an output, i.e., it is an adaptive learning technique with a decision-making machine which has the capability to improve by learning from its environment so that it can choose the optimal action. The automaton is assumed to be deployed at each of the CH for capturing the information from the environment and then adaptively selects the operation to be performed. For each action taken by the automaton, it receives reward or penalty and based upon these inputs from the environment, automaton decides its next action to be taken. The automaton deployed at each of SN collaborate with each other also o share the valuable information with each other. After finite number of steps, the automaton converges to particular value which is taken as the solution to a particular problem.

The rest of the paper is organized as follows. Section 2 describes the related work. Section 3 describes the background of learning automata and basic terminology used. Section 4 describes the system model and problem definition. The proposed LA based cell clustering algorithm is described in Sect. 5. Section 6 presents the simulation environment with results and discussions. Finally, Sect. 7 concludes the paper and provides the future scope.

2 Related Work

As WSNs have been used in many of our day to day life applications, lot of research proposals covering various aspects of the same have been reported in the literature. Fuad et al. [6] have proposed an energy efficient adaptive algorithm for selection of CH based upon the residual energy of the nodes. Extensive simulation with respect to various parameters is done in the proposed scheme for validation. Yong et al. [7] have proposed a new clustering algorithm which depends upon the density and size of the network. The proposed scheme is found to be better in comparison to the existing schemes. Khalil et al. [8] have proposed energy aware evolutionary algorithm in WSNs. Authors have formulated a new objective function having parameters such as network lifetime, stability and energy consumption. Jin et al. [9] have explored the coverage and connectivity problems in WSNs. Authors have assumed a uniform distribution of SNs in a circular region and then a probabilistic lower bound is derived to get an idea about the connectivity preservation for the area under investigation and an algorithm is also proposed for the same. Lin et al. [10] have proposed an ant colony algorithm for energy optimization in WSNs. In the proposed scheme, by estimating the remaining energy the selection of next hop is done in probabilistic manner. Moreover, the updates in the values of the pheromone are also done using the global and local values.

Liu et al. [17] have proposed a unique identification for every SN in the network and then provide an analytical model for CH election in distributed manner based upon the remaining energy and node distribution in a particular region of the network. Aioffi et al. [18] have proposed a multi objective network model for energy saving and increase the lifetime of the network. Authors proposed various policies for the minimization of energy consumption and data dissemination latency. Lai et al. [19] have proposed a new mechanism for energy saving by varying the size of the cluster. The reason for varying the size the cluster is that the nodes which are closer to BS consume more energy than the other nodes in the network as they suffer heavy traffic load during data dissemination from other SN to BS. Francesco et al. [20] have proposed a multi objective approach for lossy compression which gives the flexibility to the users to select the parameters depending upon a particular application. The proposed approach is tested with respect to complexity and data compression by collecting data from real environment. Ting et al. [21] have presented an efficient algorithm for increasing the lifetime of WSNs. The proposed scheme consists of exploration and at both local and global levels. The proposed algorithm is executed without any restriction on the cover set of the number of SNs in the network.

Bandyopadhyay et al. [22] have proposed a distributed clustering algorithm for energy conservation in WSNs. By arranging the CHs at different levels, authors proved that with an increase in levels of CHs, more energy can be saved. Wang et al. [23] have proposed residual energy based traffic splitting algorithm. In the proposed scheme, authors have used linear programming to describe a new protocol which distributes the energy among the participating nodes equally to prolong the network lifetime. Alippi et al. [24] have proposed an algorithm for k-level clustering by taking into consideration of node’s residual energy and aggregation degree and provide a detailed simulation of the proposed scheme. Attea et al. [25] have formulated a new fitness function having parameters such as cohesion and separation error for CH election using evolutionary algorithm. The proposed algorithm tries to remove the deficiencies in the earlier algorithms by improving the network lifetime, and preservation of energy. Yu et al. [26] proposed an energy-aware clustering (EADC) routing by introducing the concept of load balancing on CHs. The proposed algorithm uses the competition range for efficient selection of CH and routing in sparse dense regions. Pinaki et al. [27] have proposed security enhanced communication in WSNs using R-M codes. Pan et al. [28] have proposed recommendation of by clustering of tag neighbors. Silas et al. [29] have proposed a novel fault tolerance service selection framework for pervasive computing. Dhurandher et al. [30] have proposed communication mechanism for underwater networks.

All of the above proposals have used various techniques for different levels of clustering to save the energy without any learning and self healing mechanisms at SNs. But in literature, it has been observed that LA based techniques are very useful in finding the optimal solution with in the specified constraints and converge rapidly to solve many engineering applications [11–16]. Esnaashari et al. [11] have proposed a strategy which guides the movements of SNs within the area of the network without any sensor to know its position or its relative distance to other sensors. Same authors [12] have proposed LA based scheduling algorithm to solve the network coverage problem in WSNs. In the proposed scheme, automaton learns the sleeping time of the node by knowing exactly the movement of the target points. Torkestani et al. [13] have described the mobility pattern of the nodes using LA by knowing the mobility pattern which can also be used in the construction of multicast tree. Also they have proposed LA based sampling algorithm to solve the minimum spanning tree problem in stochastic graphs [14]. In the proposed scheme, a set of LA are randomly activated and determine the graph edges used in sampling phases. Using this idea, authors have constructed backbone formation algorithm to find an optimal solution for the minimum connecting dominating set problem [15]. Savvas et al. [16] proposed a new method for visual multimedia content encryption using cellular automata. The scheme was based on the retrieval of the contents based upon type of application.

3 Overview of Learning Automata

Learning automata (LA) is an optimization technique which operates according to some inputs and produce the output. Figure 1 describes the generalized architecture of LA. An automaton takes some input parameters and then performs an action to produce an output. Moreover, the machine has the capability of learning from its environment so that it can choose the desired action from a finite set of allowed actions through repeated steps [11–16]. The initial action chosen is random in nature, but by taking input from the environment in terms of reinforcement signal, after finite number of steps, the solution converges and produces the desired output.

Fig. 1
figure 1

Learning automata

Full size image

Mathematically, LA is defined as \(\langle Q,K,P,\delta , \omega \rangle \), where \(Q=\left\{ {q_{1}, q_{2}, \ldots ,q_{n} } \right\} \) are the finite set of states of LA, \(K=\left\{ {k_{1}, k_{2},\ldots ,k_{n} } \right\} \)are the finite set of actions performed by the LA, \(P=\left\{ {p_{1}, p_{2},\ldots ,p_{n}} \right\} \) are the finite set of response received from the environment, and \(\delta :Q\times P\rightarrow Q\) maps the current state and input from the environment to the next state of the automaton and \(\omega \) is a function which maps the current state and response from the environment to the state of the automaton [11–16, 31, 32].

The environment in which the automaton operates can be defined as a triplet \(\langle X,Y,\rho \rangle \), where \(X=(X_{1}, X_{2},\ldots ,X_{n})\) are finite number of inputs, \(Y=(Y_{1}, Y_{2}, \ldots ,Y_{n} )\) are values of reinforcement signal, and \(\rho =(\rho _{1}, \rho _{2} ,\ldots ,\rho _{n})\) are penalty probability associated with each \(X_{i} ,1\le i\le n\). Based upon each action taken by the automaton its action is rewarded or penalized and its action probability vector is updated as follows:

The formula for probability update is as follows [11–16]:

$$\begin{aligned} p_j (n+1)&= (1-a)p_{j} (n),\quad j\ne i,\quad Y=0 \nonumber \\ p_j (n+1)&= ap_j (n),\quad j=i,\quad Y=0 \nonumber \\ p_j (n+1)&= p_j (n),\quad Y=1, \end{aligned}$$
(1)

where \(a\) is the parameter.

4 System Model

4.1 Network Model

Figure 2 shows the network model used in the proposed scheme. All SNs are uniformly distributed in geographical region and send the data after collecting at regular interval to BS via CHs. The area under investigation is divided in to various cells and each cell has cluster and sensor nodes deployed. The SNs periodically sense the data in the respective regions and send it to CHs of that region. Each CH in the respective region sends the collected information to the nearest BSs for further processing. Each CH communicates with the other CH for information sharing. Let \(S=\{S_{1}, S_{2}, S_{3}, \ldots ,S_{n} \}\) and \(CH=\{CH_{1}, CH_{2}, \ldots ,CH_{n}\}\) are the n number of SNs and CH nodes deployed in the region under investigation. We have assumed that finite number of nodes are divided into different CHs so that a CH in a particular region is overall responsible for all the activities such as data transmission, message transmission etc. Initially all the nodes are assumed to have same energy value but as they perform any operation their energy level decreases.

Fig. 2
figure 2

Network model in the proposed scheme

Full size image

To preserve the energy at various levels, we have defined an upper threshold on the value of SNs in a particular CH. This value is called as the cutoff value and is calculated by defining a function. We assume that \(value(f^{cut})\) is the upper threshold on the number of nodes in a particular CH in a network field of area [0, L]\(^{2}\) where L is the length of the cell. The cutoff for the number of nodes in a particular CH consists of parameters such as remaining energy and angular position of nodes from the base line which defines the location of the node. Also, the defined cutoff is assumed to have success or failure and accordingly its probability is also defined. This probability is calculated by varying the angular position of the nodes and remaining energy level after operating any specific operation. In this way by applying the concept of cutoff on the sensor nodes, we can consider them as heterogeneous in terms of different resources they have.

4.2 Problem Definition

Definition

A cutoff is a function that defines the upper threshold on the number of nodes in a particular CH and can be defined as follows:

$$\begin{aligned} f^{cut}:K\rightarrow N^{CH} \end{aligned}$$
(2)

where \(K=\sum _{i=1}^{n} {S_{i}}\) where \(S_{i}\) are the SNs in a particular region and \(value(f^{cut})\) is the value of the cut off nodes in a particular CH. The values in the variable \(value(f^{cut})\) can be set according to the application to application. The cut off defined on the number of SNs can be successful it may be a failure. The success or failure is controlled by the blocking probability of the cutoff, i.e., the number of nodes in a particular CH should be optimized so that a particular CH should not be overloaded or under loaded. The success or failure is counted by the fact that some of nodes are not in the range of the BS or CH due to location of the same in a particular geographical region. We have taken this factor as the SNs may be deployed in dense or sparse region of any geographical location. Hence keeping in view of the same, following terms are defined as follows:

Define the probability of failure \((P^{F})\) and success \((P^{S})\) of cutoff on number of nodes in a cell as:

$$\begin{aligned} P^{F}&= B^{i}\sum _{j=0}^{i-1} {(1-B^{j})} \end{aligned}$$
(3)
$$\begin{aligned} P^{S}&= \sum _{i=0}^{n} {(1-B^{i})} \end{aligned}$$
(4)

\(B^{i}\) is the blocking probability of the cutoff in the region. It is defined as the ratio of upper threshold on the number of cutoff to the total number of SNs in the network.

So the combined probability of the cutoff can be defined as follows:

$$\begin{aligned} P^{com}=LP^{S}+\sum _{j=1}^L {jP^{F}} =\text{ L }\sum _{i=0}^n {(1-B^{i})} +B^{i}\sum _{j=0}^{i-1} {(1-B^{j})} j \end{aligned}$$
(5)

The cutoff on number of SNs is an iterative process in which after finite number of steps, the solution would converge to a particular value. The process of cutoff has some probability of success or failure in each step as defined above. Let \(z\) are the number of steps to perform the final cutoff which includes the count of success or failure in a particular time interval.

So, cumulative \((tot)\) probability of cutoff can be defined as follows:

$$\begin{aligned} tot=\sum _{z=1}^\infty {zP^{com}} p_s (1-p_s )^{z-1} \end{aligned}$$
(6)

Let \(D(S)\) and \(\overline{D(S)}\) is the amount of data disseminated before and after the cutoff operation on number of SN in particular CH. The cumulative data disseminated can be calculated in modified form as defined in [33]:

$$\begin{aligned} {\overline{D(S)}} =\max (D(S),D(S_{1} ))P^{S}+\min (D(S),D(S_{1} ))P^{F} \end{aligned}$$
(7)

Equation (7) gives detailed description about the data dissemination in between the two SNs in the network before and after cutoff operation. While selecting the maximum value, probability of success of cutoff is multiplied but probability of failure during cutoff is multiplied for selection of minimum value.

Now, using Eqs. (2)–(7), we have defined a combined objective function \(\mu ^{obj}\) for the given problem as follows:

$$\begin{aligned} \mu ^{obj}=\min \{\overline{D(S)}\times {tot}\} \end{aligned}$$
(8)

5 Proposed Approach

In this section, we would explain various steps of the proposed scheme in detail. Basically we have assumed that each automaton is deployed at each of CH which is responsible for overall monitoring and control of the SNs in the respective regions. The operations controlled by the automaton are classified in the following steps and accordingly automaton updates its action probability vector.

5.1 Maintenance of the Connectivity in the Cell by the Automaton

Each automaton controls the connectivity ratio of the SNs for respective cells and facilitates the various operations. The connectivity ratio is the probabilistic measure of number of SNs in a particular cell to the Lot of energy is preserved by the way the particular operation is performed by the automaton. For each action taken by the automaton, its action probability vector is updated according to the input provided by the environment which is defined in the section above. The main concern in WSNs is to preserve the connectivity of the SNs. If few of the nodes are not connected to the network then communication between the SNs can not be performed and these nodes are considered to be dead nodes. The nodes in the cell are monitored by the respective CH which maintain the connectivity both in the intra and inter domain. The automaton takes input as the structure of the cell and then calculates the probability of connectivity of the nodes with respect to the position of the nodes. The position of each node in the cell is estimated by variation of the angle of inclination from the base of the cell. The area under investigation is divided in to cells with size of each cell is assumed to be as rhombus which is a variation of the assumption made in Hybrid Energy Efficient Distributed (HEED) Clustering protocol [37].

Assumptions of our protocol are similar to those for HEED [37]. The main difference between HEED [37] and the proposed scheme is the size of the cell. In the proposed scheme we have taken a rhombus shape of the cell, area ABCD is the cell structure for the sample calculation in which we have taken \(\theta = 60^{\circ }\).

In worst case the distances between the CHs which are on the edge of cell are 2.7 \(R_{c}\) [33]. From Fig. 3, in triangle ADE,

$$\begin{aligned} \sin 60&= \frac{DE}{AD},\text{ where } AD=\left( 2+\frac{1}{\sqrt{2}}\right) R_{c} \cong 2.7R_{c} \end{aligned}$$
(9)
$$\begin{aligned} \text{ DE }&= \text{ AD }\sin 60^{\circ },\quad \mathrm{DE}=\left( {2+\frac{1}{\sqrt{2}}} \right) \times \frac{\sqrt{3}}{2}R_{c} \cong 2.44R_{c} \end{aligned}$$
(10)

Therefore the area of rhombus ABCD can be easily calculated,

$$\begin{aligned} \text{ Area }=\text{ AB }\times \text{ DE }=2.7R_{c} \times 2.44R_{c} \cong 6.588R_{c}^{2} \end{aligned}$$
(11)

In Fig. 3, area \(\text{ P }_{1}\mathrm{P}_{2}\mathrm{P}_{3}\mathrm{P}_{4}\) covers two cells assuming that the two CHs are located at the \(\text{ P }_{1}\mathrm{P}_{3}\). So even if this is the worst case that the two CHs are located at the corners of the cell, for multi hop clustering these two CHs should be connected and the distance between these two CHs will be \(\text{ R }_\mathrm{t}\). [33]. If ‘d’ be the perpendicular distance from \(\text{ P }_{2}\) to \(\text{ P }_{1} \mathrm{P}_{4}\) than we can calculate d:

$$\begin{aligned} \sin \,60=\frac{d}{P_{1} P_{2}}\text{ where } \text{ P }_{1} \text{ P }_{2} =2\times 2. 7\times R_{c} \cong 5.4R_{c} \end{aligned}$$
(12)

\(\text{ d } = 4.68\, R_{c}\) further if the distance from the \(\text{ P }_{1}\) to the perpendicular from \(\text{ P }_{2}\) is ‘r’ then by simple calculation it is found that \(\text{ r } = 2.7\, R_{c}\). Once these values are calculated by the automaton for respective cell then it is able to preserve the connectivity of the nodes located at different positions of the cell.

Fig. 3
figure 3

a Structure of cell, b intra cluster communication

Full size image

5.2 Calculation of Distance Between CHs by the Automaton

The distance between the two CHs and average number of CHs in the whole geographical region is computed next by the automaton. It keeps track of the nodes and CHs association at different time intervals with respect to the various conditions of connectivity. The distance between different CHs and number of CHs connected are required to be computed for complete operations which are calculated as follows:

  1. (i)

    Calculation for \(R_t \): From Fig. 2, in triangle \(\text{ P }_{1}\mathrm{P}_{3}\mathrm{P}_{5}\)

    $$\begin{aligned} \text{ P }_{3} \mathrm{P}_{5}&= \text{ d } ={4}.{68}R_{c} \nonumber \\ \text{ P }_{1} \mathrm{P}_{5}&= \text{ P }_{1} \mathrm{P}_{4} + \text{ P }_4 \mathrm{P}_5 = 2.7 R_{c} +2.7R_{c} =5.4R_{c} \nonumber \\ R_t&= \sqrt{{\left( {P_{1} P_5 } \right) ^{2}+\left( {P_3 P_5 } \right) ^{2}}}\cong 7.15R_{c} \end{aligned}$$
    (13)

    From this result it is clear that CHs are connected if \(R_t \ge 7.15R_{c}\) when \(\theta =60^{\circ }\)

  2. (ii)

    Calculation of no. of CHs in network field: For \(\theta =60^{\circ }\), we calculated the area of cell \(6.588R_{c} ^{2}\). Hence numbers of CHs in the network field are

    $$\begin{aligned} N^{CH}=\frac{L^{2}}{6.588R_{c} ^{2}}\cong 0.152\frac{L^{2}}{R_{c} ^{2}} \end{aligned}$$
    (14)

Equations (13) and (14) are computed at each of the CH by respective automaton for calculation of the above defined parameters. Table 1 below computes the distance between the CHs and average number of CHs for different values of \(\theta \). Table 1 is maintained by the respective automaton at each of the CH to update these two values [33]. For each value of \(\theta \), the different conditions for maintaining the connectivity among the SNs and CHs are maintained at respective CH, i.e., by variation of \(\theta \).

Table 1 Variation of \(R_t\) in terms of \(R_{c}\) and CH selected [33]
Full size table

Theorem

The cutoff on the number of nodes in the cluster head is \(\sqrt{K}\left( {\frac{L^{2}}{R_{c} ^{2}}+1} \right) \).

Proof

Initially we have assumed the degree of the cluster head node as \(m\). The new node leave or join the network and can migrate from one CH to another. In this case the size of CHs change and their degree also vary. As the degree of the CH reaches a threshold value \((thr)\) then all the nodes calculates the following values and decide their participation degree \(S^{part}\) in respective CH as follows:

$$\begin{aligned} S^{part}=\left\langle {\begin{array}{l} \left\{ {N^{CH}} \left( {\frac{P_m }{P}} \right) \right\} ,\quad m\le thr\\ \frac{value(f^{cut})-S(f^{cut})}{S},\quad m>thr \\ \end{array}} \right. \end{aligned}$$
(15)

At any instant the position of the SNs is defined by it location L that consists of horizontal and vertical components with its degree of participation as defined above. As defined in the above equations, degree of participation consisting of cutoff value and number of CHs within and above the defined threshold value. So the combined attributes can be defined by the following equations:

$$\begin{aligned} x((t+\Delta t),S^{part})&= x(t)+x\Delta t+S^{part} \nonumber \\ y((t+\Delta t),S^{part})&= y(t)+y\Delta t+S^{part} \nonumber \\ f^{cut}((t+\Delta t),S^{part})&= f^{cut}(t)+f^{cut}(\Delta t)+S^{part} \end{aligned}$$
(16)

Subtracting the x and y co-ordinates, we get the exact position along with the participation of the nodes in the respective cluster as follows

$$\begin{aligned} L&= x((t+\Delta t),S^{part})-y((t+\Delta t),S^{part}) \end{aligned}$$
(17)
$$\begin{aligned} L^{2}&= (x((t+\Delta t),S^{part})-y((t+\Delta t),S^{part}))^{2} \end{aligned}$$
(18)

Equation (18) gives the location with respect to the participation degree and cutoff bound in the respective CH for SNs.

Dividing the value by square of the radius value \(R_{c}\) and multiplying by K, we get the value of cutoff in the respective CH. Hence the proof of the theorem. \(\square \)

5.3 Update of Action Probability Vector According to the Participation Degree

The main part of LA is how the action probability vector is updated. The action probability vector contains the values that needs to be updated corresponding to each action of the automaton. The environment provides a feedback corresponding to each action of the automaton. The feedback may be positive or negative and accordingly the values in the action probability vectors are updated. The participation degree of each node in a particular CH is computed by the automaton. The participation degree of the node is computed by calculating the number of packets successfully sent to the total number of packets sent with respect to the geographical region of the nodes in a particular time interval in the given network. The location is introduced in the participation degree as some of the nodes may have poor connectivity in a particular region. Moreover, the time is also a crucial factor as the information may be sent in strict deadline as defined above in Eqs. (15)–(18).

For each operation, if environment provides reward to LA then participation degree of SNs is get incremented as follows: \(S^{part}=S^{part}+p_j (n)\). Similarly if environment provides penalty to LA then its participation degree of SNs is get decremented as follows: \(S^{part}\) \(=S^{part}-p_j (n)\). The probability factor is changed only in the case of reward function else it remains same as we have considered the reward inactive scheme in the current proposal. Now based upon these values, the probability of success or failure and combined probability of cutoff on the number of nodes in a particular cell are computed using Eqs. (3)–(6). The cut off values are used to decide the connectivity of the nodes in a cell as illustrated above. Table 1 below provides a complete detailed overview of the various parameters variation in the proposed scheme, e.g., with an increase in the values of \(\theta , \,R_{t}\) in terms of \(R_{c}\) is decreased and number of CHs for the selected network is also decreased.

The proposed algorithm consists of two steps: Cluster head maintenance and connectivity preservation. Cluster head maintenance consists of CH election by application of cutoff at different levels and by computing the node participation ratio in the cell. The connectivity preservation phase consists of maintaining the connectivity of the nodes with respect to their position, e.g., some of the nodes are in the middle of the cell and some of them are on the edge of the cell. In the proposed algorithm, the automaton is activated at various levels of clustering hierarchy. Initially the node having maximum energy is selected as CH and automaton is activated at this node (lines 1–3). Once the automaton is activated at respective CH then it performs its action by computing the participation degree and also compute the probability of selection of CH for the next round (lines 4–8). The probability of success or failure is computed next (line 9). If LA receives an award or penalty the participation ratio of SNs is incremented or decremented (lines 10–15). Based upon these values, the cumulative probability of cutoff is computed using Eq. (6) (line 16). First level cutoff for SNs is applied and if participation ratio of the node is less than the value of cutoff then this node is made CH for the next level else the second level cutoff is applied to select the node for CH (lines 17–22). To preserve the connectivity of the nodes in the network, cell radius is computed using Eqs. (9), (10) and (11) and distance between them is computed if they are on the edge of the cell (lines 23–25). The angular position of the nodes from the base position is computed to find the location of the same using Eqs. (16), (17) and (18) (lines 26–32). The value of \(R_t\) is computed in terms of \(R_{c}\) (lines 33–35). In the final step, the action probability of LA is computed based upon the input received from the environment and accordingly the action of LA is performed (lines 36–39). The reward and penalty received by the environment is assumed as proposed in [11–16, 31, 32]. The ratio of number of times an action is rewarded to the total number of times an action is taken is computed and maximum value of this ratio is taken for maximum benefits (lines 40–43). The complete pseudo code for the proposed algorithm is described as follows:

figure c
figure d

6 Simulation Environment with Results and Discussion

The simulation of the proposed scheme is performed on ns-2 [34] simulator with the parameters as defined below in Table 2. As discussed above automaton are deployed at each of the selected CH which controls all the operations a particular SN performs in the given geographical region. We have compared the performance of the proposed scheme with LEACH [35], EECF [36] and HEED [37]. Various parameters such as cluster size, number of nodes alive, energy consumption are evaluated by varying the number of nodes in the network, with number of rounds for selection of CH are evaluated in the proposed scheme and existing well known schemes. The detailed values of the parameters are listed in the Table 2 below.

Table 2 Simulation parameters
Full size table

6.1 Impact of the Proposed Scheme on Cluster Size

Figure 4 shows the performance of the proposed scheme on cluster size with varying the number of rounds. As shown in the figure below, with an increase in number of rounds for CH selection, the size of the CH is decreased but the proposed scheme contains more nodes than the other existing schemes. It shows the effectiveness of the proposed scheme because with an increase in number of rounds the connectivity of the nodes is higher than the other existing schemes. The reason for the improved performance of the proposed scheme is due to the fact that LA adaptively selects the number of nodes in the CH based upon the participation ratio of the nodes in the cell. Based upon the participation ratio of the nodes, first level and second level cutoff are applied to the number of nodes in the cell. As illustrated in Fig. 4a, b, with an increase in number of rounds for selection of CHs the size of the cluster is decreased most in the proposed scheme compared to other schemes as cutoff is being applied at different levels based upon the participation ratio of the SNs.

Fig. 4
figure 4

Proposed scheme with cluster size having a 200 nodes b 400 nodes

Full size image

6.2 Impact of the Proposed Scheme on Number of Nodes Alive

Figure 5 shows the impact of the proposed scheme on number of nodes alive after finite number of rounds. Each node in the network has some initial energy but the level of energy decreases as the nodes perform a particular operation and may become zero after finite number of rounds. In such a case the node are considered to be dead and does not perform any action. As shown in the figure, the number of nodes alive in the proposed scheme is higher than the other schemes even high number of rounds. This is primarily due to the reason that in the proposed scheme, LA performs the cutoff operation on the upper bound on the number of nodes in each CH. This will save a lot of energy and increases the life of the nodes in the network. The selection and CH in the other schemes are random and consequently the energy levels of the nodes decrease to considerable amount. Moreover using the defined participation ratio of the nodes in the network the coverage and connectivity is preserved and consequently cutoff at different levels in the network is applied. As LA is deployed on the CH so computational complexity is handled by respective LA and not by the individual node. Hence lot of energy of the nodes is saved using the proposed methodology as compared with the other approaches of its category.

Fig. 5
figure 5

Proposed scheme with nodes alive. a 200 nodes, b 400 nodes

Full size image

6.3 Impact of the Proposed Scheme on Total Energy Consumed

Figure 6 shows the impact of the proposed scheme on total energy consumed with number of rounds. With an increase in number of rounds, the total energy consumption increases in all the schemes for both the cases (a) and (b) of 200 and 400 nodes. But compared to other schemes, the energy consumption is less in the proposed scheme in comparison to other schemes. This is primarily due to the reason that the LA learns from the environment and adapt to the situation quickly and select the best node for maintaining the coverage and connectivity among all the participating nodes. Moreover, LA performs the cutoff operation as defined above with respect to the number of nodes in the CH which further reduces the energy consumption in the proposed scheme compared to other schemes. The advantage of using the cutoff is that only nodes satisfying particular criteria are allowed to participate in CH election which saves lot of energy as compared to other approaches of its category.

Fig. 6
figure 6

Proposed scheme on total energy consumed. a 200 nodes, b 400 nodes

Full size image

6.4 Impact of Cutoff Value on Energy Consumed

Figure 7 shows the impact of cutoff value on energy consumption with varying number of rounds. As seen from the figures, with an increase in the cutoff value in a particular CH, the average energy consumption is also increased. This is primarily due to the fact that with an increase in number of rounds and cutoff, more and more nodes will belong to a particular CH and to maintain all such nodes in the network, lot of energy is consumed which is also reflected in the Fig. 7 also. As shown in the Fig. 7, with an increase in the value of cutoff nodes from 10 to 50, the average energy consumption is increased by 40 %. Moreover, some of the nodes are not located in geographical regions where coverage and connectivity is preserved for them, so in the proposed approach estimates about their location with connectivity to the CH is computed and then accordingly major decisions such as selection of CH, coverage and connectivity about the nodes is made.

Fig. 7
figure 7

Variation of energy consumption with cut off value

Full size image

6.5 Impact of Variation of \(\theta \) on Number of CHs

Figure 8 shows the impact of variation of values of \(\theta \) on number of CHs. As we can see with an increase in the value of \(\theta \), the number of CHs is decreased to large extent. Moreover, with an increase in the values of cutoff nodes in the cell, the number of CHs is also decreased. The primary reason for the same is due to the fact that with an increase in the value of more number of nodes are competing to become CH as with smaller values of \(\theta \), every node in the cell has chance to become the CH. Moreover, as the value of cutoff is increased more stringent conditions are applied for selection of CH and hence there is a decrease in the number of CHs as can be seen from Fig. 8 below.

Fig. 8
figure 8

Variation of \(\theta \) with number of CHs

Full size image

7 Conclusions and Future Scope

Clustering with coverage and connectivity in wireless sensor networks (WSNs) are the two key issues which are backbone of many applications in WSNs. But some of the nodes in WSNs may not be able to make contact with the CHs or BS and hence loose communication with them. So an intelligent approach is required in which complexity of the proposed solution is less. To tackle these issues in WSNs, in this paper we propose a new LA based clustering with coverage and connectivity scheme for WSNs. The area under investigation is divided into various cells having CHs and SNs. A cutoff value is provided which sets the bound on the number of nodes in a particular CH depending upon the participation ratio of SNs. In the proposed scheme, cell shape has been changed to rhombus which is different from the earlier approaches in which it has been assumed as square. The angular displacement from the base of the cell to other points is taken and rotated to give different values of angle. The coverage of maximum number of points from the base of the cell is taken into account. Moreover, Efficient Learning automata based Cell Clustering Algorithm (ELACCA) is proposed. The performance of the proposed scheme is evaluated using various metrics. The results obtained show that the proposed scheme is better than the existing schemes with respect to these metric. In future, we would like to extend this work for secure communication between CHs and SNs.