Lifetime maximization by partitioning approach in wireless sensor networks
 Mohammed Zaki Hasan^{1}Email author,
 Hussain AlRizzo^{2} and
 Melih Günay^{1}
https://doi.org/10.1186/s1363801608031
© The Author(s) 2017
Received: 3 May 2016
Accepted: 19 December 2016
Published: 13 January 2017
Abstract
Lifetime is a key parameter in the design of routing protocols in energyconstrained wireless sensor networks (WSNs). Conventional singlepath routing schemes may not be optimal in maximizing network lifetime. In this paper, we present a new routing algorithm based on the optimal number of hops to partition the path from the source to the sink. The algorithm is based on energy consumption constrained routing method. The mathematical model uses mixedinteger programming (MIP), based on the Lagrangian relaxation (LR) method, to define critical parameters that control the adaptive hopbyhop switching. LINGO is used to investigate the performance tradeoffs between energy efficiency and quality of service (QoS). Simulation results revealed that our algorithm significantly improves the lifetime by 46.91, 73.00, and 80.00% as compared to the wellknown node density control, upperbound, and WSN optimization of network lifetime algorithms, respectively.
Keywords
1 Introduction
Wireless sensor networks (WSNs) consist of selforganized sensor nodes, which suffer from limited power, computational capabilities, and bandwidth [1]. A WSN is a promising technology that offers a good solution for the design and development of realtime applications using traditional networking paradigms [2], in addition to new types of networks, such as the Internet of Things (IoTs) which has been a core research topic since the beginning of this century [3]. Each sensor node is equipped with a battery, a microcontroller, memory, and a transceiver, whereas the sink node collects data for processing and decisionmaking [4]. The sensor node monitors, collects, and sends information to an allocated area [5]. This means that sensor nodes should operate in a limited energy budget to provide support for applications with an affordable cost [6]. However, batteries possess a finite energy capacity, and this limitation has generated significant interest concerned with the use of many aspects of WSNs to increase battery life by selecting optimal paths with effective power management to maximize operational lifetime [7].
Energy efficiency analysis is notoriously difficult due to the network lifetime that depends on several factors, including network architecture, routing protocols, data collection initiation, lifetime definition, channel characteristics, and power consumption [8–10]. To address these limitations, WSNs offer various types of routing protocols, such as singlehop or multihop to facilitate a route to the sink [11]. These routing protocols have been proposed to address energy efficiency in realtime applications [12]. Some of these algorithms aim at maximizing network utilization and QoS, while several routing and specific path selection mechanisms have been proposed to meet dynamic network topology and applications with specific QoS guarantees [13, 14]. Hence, network parameters such as node density, initial energy in sensor nodes, and data rate could be selected as metrics for the path selection mechanism to achieve the desired network lifetime [15].
Most existing routing protocols select the minimum energy singlepath, whereby each source node transmits data to the sink via the shortest path [16]. The optimal singlepath is selected based on metrics such as the gradient of information, distance to the sink or the node residual energy level [16]. Although the singlepath approach is flexible, simple, and scalable, path breakage due to node failure requires initiation of a new route discovery process which increases energy consumption [17] and leads to an early termination of the network and partition [13]. Therefore, singlepath routing cannot meet the requirements of realtime applications [18]. Several routing protocols that use multipaths have been proposed based on either load balancing or network reliability [19–21]. Load balancing can be achieved by balancing energy utilization among the multipaths to improve network lifetime [22]. Data transmission relies mostly on the optimal path or number of hops. The alternative paths are used only when the nodes on the primary route fail [23].
In this paper, we present a mathematical model for an energyconsumption constrained multipath routing determination mechanism. The aim behind partitioning the multipath is to achieve higher reliability for a given total lifetime in the WSN, i.e., at each moment, every sensor node should have spent the same amount of energy for transmitting and receiving each data packet until delivered by the sink. We highlight the novelties of our proposed algorithm by comparing the results against the node density control [24], the upper bounds of lifetime algorithms [25], and network lifetime optimization in [26, 27]. Most authors derived upper and lower bounds of the network lifetime considering the event detection as spatial behavior of data flow in the network [8]. Furthermore, the optimum length of hop and optimal number of hops in the selected path minimize the total energy consumed for the data transmission. They also eliminate the assumption of source concentrated on a point and assume that the source is distributed over an area.
The node density control algorithm [24] proposes a model to minimize energy consumption that depends on the distribution model of the sensor nodes in the network to explore the relationship of lifetime and the sensor density distribution manner in the events area. However, all of the nodes should use the same transmission range, which causes exhaustion of the energy of the nodes. The model analyzes the network lifetime by deriving the optimum transmission ranges of the nodes.
Studies on the upper bound for the lifetime of data gathering have been reported for various WSNs routing protocols. In [25], a strategy is proposed for collaborative information in routing protocol. This strategy constructs a realistic network topology to simulate the gathering and processing of information to investigate the optimal lifetime for some levels of deployment control. In this specific topology, there are several different multipaths that data packets originating at a specific source node can use to send to the sink node. Therefore, these multipaths also include paths with which the node does not necessarily communicate directly through singlehop. Instead, the node can transmit data packets directly to another node, which is two hops or a multihop away, by spending more energy. Thus, the total number of paths from the source node to the sink grows exponentially as the number of sensor nodes in the network topology increases. However, the implementation of such a strategy is difficult because it is necessary to determine the exact locations of all nodes in the network topology and then to coordinate all the nodes so that different collaborative strategies are sustained over different periods.
The authors in [26] developed a generalized power consumption model to address the optimization of network lifetime and resource allocation for wireless video sensor networks (WVSNs). The authors formulated an algorithm which jointly considered video coding rate, aggregate power consumption, and link rate allocation to maximize the network lifetime. The approach in [27] combines power consumption, video compression power, and network coding power under multipath rate allocation constraints. Unlike lifetime optimization in [26], the authors proposed a solution to provide convex lifetime optimization. Meanwhile, in [28] the authors added a new routing metric to optimize the network lifetime by exploiting the cooperative diversity and jointly considering routing and power allocation schemes. The authors developed flow augmentation algorithm to formulate the objective function under specified constraints to reduce the complexity of nondeterministic polynomial (NP) maximization problem. A collaborative protocol has been proposed in [29] which leads to an increase network lifetime. The authors in [29] extended the work reported in [25] by taking into account the network topology and the effects of aggregation of data streams to permit derivation of bounds for networks with arbitrarily complex capabilities.
As far as analytical studies addressing sensor constraints such as computational capabilities, limited battery power, and less memory in multihop transmission are concerned, the authors in [4] proposed a theoretical data collection transmission scheme from source node to a mobile sink. Currently, research focuses on developing algorithms for network route reconstruction in a multiple sink to minimize energy consumption and to increase network lifetime [30]. Fortunately, this leads to energy balancing through network restructuring and optimizes the network lifetime since the number of disconnected sensor nodes is also reduced. However, the authors in [30] utilized the advantage of having multiple sinks. Indeed, multiple sinks ensure shorter hops to reduce the hop distance [12]. The authors in [31] proposed a 3D gridplanned deployment for heterogeneous WSNs to maintain a prolonged lifetime of reliable WSNs. The problem is mathematically modeled as a mixedinteger linear program (MILP) optimization with the objective of maximizing the network lifetime by reducing energy consumption, while maintaining certain levels of fault tolerance and cost efficiency.
In this paper, we present an approach for multipath routing algorithm that partitions the path from the source to the sink to considerably increase the node lifetime. The proposed algorithm distributes the routing messages to underutilized partitioning multipath and less load to overcommitted paths. It should be noted that our proposed scheme is easy to implement and does not require exact knowledge of the node positions. We present simulations for two scenarios through swapping the role of detecting the event from singlesource to multisource node to enhance the overall network lifetime. Simulation results revealed that our algorithm provides a higher node energy efficiency than the protocols reported in. The rest of the paper is organized as follows. The proposed protocol in multipath data routing scheme is described in Section 2. The performance evaluation of the scheme as well as comparisons against existing protocols are presented in Section 3. Conclusions are given in Section 5.
2 Partitioning algorithm for multipath routing protocol
A magnified portion of the path shows that some sensor nodes are not aligned along the selected path. Thus, this suggests the use of the concept of integer optimization to partition the nodes that are not aligned along the path. Partitioning is performed using the projection of sensors positions onto the path, to determine how close the packet is to the sink. The mathematical model uses mixedinteger programming (MIP) to develop the lower and upper bounds of network parameters using the cutoff method [32].
Critical parameters that control adaptive switching of a hopbyhop QoS routing protocols are illustrated in Fig. 1. The criteria for each objective function as related to the decision constraints are used to determine the cutoff of the optimal number of hops, from which the path from the source to the sink is selected [33]. The main goal is to determine the optimal path that satisfies all QoS requirements for an efficient routing protocol over a multihop route.
MIP defines a critical parameter to solve NPtime problems [32]. The method is motivated by the need to find a plan to increase the capacity of multiservice internet protocol networks [34], and it has been developed over recent years to account for new technologies and mechanisms that enable QoS parameters with different constraints to be satisfied, as well as to guarantee the optimal resource allocation for the task [35, 36].
2.1 Problem formulation
The existing link between two sensor nodes is defined as e=(s _{ ı },s _{ ı+1}) from node s _{ ı } to node s _{ ı+1}, where ı=1,…,n. Each link e∈E is characterized by two integer values: energy consumption and delay. A decision variable x _{ ȷ } is defined as a variable with the value of 1 when two sensor nodes are connected, 0 otherwise. The source node consumes E _{tx} amount of energy to transmit pbit of information with transmission range over a characteristic distance denoted by d with a specified number of hops hop towards the sink. Each intermediate nodes consumes the E _{rx} of the amount energy of the receiving information which is the signal propagated with ε _{ fs } d ^{ α } for a singlepath model and ε _{mp} d ^{ α } for a multipath model for the transmission amplifier, where α is the loss exponent of the signal.
Parameters of the problem
Parameter  Definition 

n  Number of sensor nodes 
link  Set of links in the network 
e _{ ı }  Initial energy in each node 
x _{ ȷ }  Decision variable 
e _{ ı }  Initial energy in each node 
E _{elec}  Overhead energy due to the sensing, receiving and processing 
E _{sense}  Energy cost of sensing 
E _{comp}  Energy cost of computation 
E _{tx}  Amount of energy to transmit pbit information 
E _{rx}  Amount of energy to receive pbit information 
ε _{fs}  Loss coefficient related to pbit transmission propagated over singlepath model 
ε _{mp}  Loss coefficient related to pbit transmission propagated over multipath model 
ξ _{3} and ξ _{4}  Constants coefficients of sensing operation 
t  Lifetime achieved by the sensor node 
d  Distance of sensor nodes to the next hop 
hop  Number of hops for the selected path 
a _{linkȷ }  Partition link indicator that lies on the selected path 
α  Path exponent 
r  Rate in \(\frac {\text {bits}}{\text {sec}}\) of each of the p streams 
P(n)  Total number of events can be detected by a network 
E _{ ı }  Initial energy of each node 
M  Average number of events occurring per unit of time such as day 
t _{ ı }  Energy consumed to sense the event 
Energy_{ ı }  Amount of energy required to report the event from the source node 
p  Number of partitioning intermediate node 
FoV  Field of view of the sensor node in the network 
λ  Total average arrival rate of vehicle 
β  Probability of packet transmission 
2.2 Energy consumption optimization
where r is the rate in \(\frac {\text {bits}}{\text {sec}}\) of each of the p streams and ξ _{4} is a constant.
The definition of all the radio parameters
Parameter  Definition  Unit 

E _{elec} and E _{sense}  Energy dissipation rate to run the radio  50 nJ/bit 
ε _{fs}  Singlepath model for the transmitter amplifier  10 pJ/bitm ^{2} 
ε _{mp}  Multipath model for the transmitter amplifier  0.0013 pJ/bitm ^{2} 
p  Data length  2000 bits 
α  Path loss exponent for free space environment  4 
Theorem 1
where N _{min} is the minimum number of nodes that ensures both network coverage and connectivity, and N ^{∗}=T∗M which is given by number of events M occurring per unit of time T.
where μ is defined as a vector of LR ∀ȷ where ȷ=1,2,3,…,m

(a) Control parameters for the network layer such as the partitioning path selection and the node’s lifetime of the selected path.

(b) The optimization goal is to minimize energy consumption.

(c) Constraint for the physical layer include limited energy.
The proposed model uses the subgradient method reported in [32] to find the global optimal solution of the objective function defined in Eq. 14 assuming that the subgradient of the objective function can be computed. This approach solve the optimization problem with fixed LR. Since, LR is the most attractive method among the few solution methods in optimization that cut across the domains of integer programming.
Algorithm 1 describes the steps of the Lagrangian method applied to find a closedform optimal solution for the constrained optimization. The idea is to relax the explicit constraints by bringing them into the objective function defined by Eq. 11 with the associated Lagrange multiplier μ. Using the LR, the proposed algorithm can choose the optimal μ for a given twonode pair. The constrained optimal path problem can be solved with respect to the modified objective functions of energy consumption (Eq. 11) and delay. We have added the delay constraint along with the energy consumption constraint in Eq. 12 to calculate upper bounds for the objective function in Eq. 14, since adding many constraints can lead to very good formulation. Even though, there is an integer solution to the linear relaxation of the expanded formulation that is also feasible for the linear relaxation of the objective function. This occurs by adaptation of the gradient method in which gradients are replaced by subgradient method by giving an initial value for Lagrangian multiplier by dualizing the constraints with objective function Eq. 14 for the power consumption and Eq. 12 for the delay.
LR enables the development of lower bound constraints for both energy consumption (Eqs. 10 and 12) and delay constraint parameters on the optimal length of a constrained optimal path. These lower bounds are valuable when a specific path from the source to the sink is generated by solving the subproblems of partitioning link quality.
2.3 Energy efficiency metric
Partitioning is performed through projecting the positions of the sensors onto the route to determine how close the packet is to the sink. The sensor nodes are uniformly distributed, and each knows the location and link quality of its neighbors. The lifetime of a sensor node depends basically on two factors: how much energy it consumes over time and how much energy is available for its use. Therefore, to clarify these factors, energy efficiency is defined as the number of data packets delivered from the percentage of alive source or intermediate nodes to sink with optimal spanning over the lifetime of the sensor node in the network [43]. Following this definition, the predominant amount of energy is consumed by a sensor node during sensing, communication, and data processing activities as illustrated in Fig. 3. Indeed, we show that all parameters such as coverage, connectivity, and node availability can be detrimental to lifetime considerations. From this definition, the key results with respect to this definition are established in the following theorem [43].
Theorem 2
For fixed network sizes, the operational lifetime of a wireless sensor network decreases in the order of \(\frac {1}{\sqrt {n}}\) as the number of nodes n grows.
Theorem 3
For fixed node densities, the operational lifetime of a wireless sensor network decreases in the order of \(\frac {1}{n}\).
Let ε be a real number that satisfies 0<ε<1, we define the operational lifetime of a WSN as follows [43]:
Definition 1
The operational lifetime of a network is the expected time after which at least 100(1−ε ^{2}) data transmissions fail.
The understanding of the asymptotic behavior of lifetimes is essential to determining of sensor network whether or not a sensor network can function till the end of its operation.
3 Performance evaluation

(a) The source and sink nodes are placed inside the wireless sensor area.

(b) The senor network architecture is considered homogeneous.

(c) Each sensor node has a connectivity that is associated with two positive QoS constraints in terms of energy consumption and average delivery delay.

(d) The total number of vehicles on the highway is very high.

(e) A single vehicle uses a certain percentage of the highway resources depending on the type of highway.

(f) The decision to enter the highway is made independently by each driver.

(g) Each sensor node is assumed to be aware of its geographic location with a transmission range of approximately 12.00 m, since a few sensor nodes are assumed to be video sensor nodes that have more constraints, such as the limitation on the sensing coverage and the FoV.
where u(τ) denotes the unit step function.
The key to find the optimal number of multipath hops from the source to the sink can be exposed to the convex envelope or the convex hull of the objective function by adding an objective cut. Hence, the evolution of energy consumption and endtoend delay can be generated for the multipath that adopts the objective functions for the power consumption and delay level cutoff at an optimal number of hops corresponding to a feasible solution. This evolution corresponds to feasible solution to the constraints that are added together to produce values of the upper bounds. It is observed that the LR μ permits developing lower and upper bounds on the optimal length of a constrained optimal multipath [33].
After completing the discovery phase and constructing all multiple paths, the mechanism starts to select a set from the constructed paths to transfer the data packet. The selection phase of partitioning the multipath is based on the routing metric in order to minimize energy consumption of the selected paths. However, the selection is based on the definition of critical parameters to control the adaptive switching of hopbyhop until the sink node.
The path selection is based on the critical parameters to control the adaptive hopbyhop switching routing. It is also based on partitioning paths where the sensor nodes are distributed into partitioning. All sensor nodes inside the partitioning area can communicate with other each. Therefore, all sensor nodes in each partitioning area have equal link quality, i.e., transmission range. The exact partitioning of multipath is used to optimize performance metrics such as energy balance and network lifetime. After constructing all multipath and data is received in the source node, the source selects an optimal path to its next most preferred neighbor by partitioning.
Simulation parameters
Parameter  Value 

E _{elec}  50 nJ/bit 
ε _{fs}  10 pJ/bitm ^{2} 
ε _{mp}  0.0013 pJ/bitm ^{2} 
Topology structure  Square (1000×1000 m), sensor node uniformly distributed 
Total number of sensor nodes  50 sensor nodes 
Message payload  64 B 
Data length p  2000 b 
λ  Total average of packet arrival rate 
P _{ x }(τ)  Poisson arrival of seeing a number of vehicles in a specified time period 
τ  Specified time period 
Transmission range  12.00 m 
α  4 
D  150 m 
There are several factors that affect the lifetime of the energylimited systems. These factors are network topology, detecting the event, and number of sources. However, our routing algorithm also provides results on the impact of these factors along with their variations (time/place) on the energy efficiency of the WSN when solving the optimization problem.
Usually, the lifetime of the wireless sensor node increases when the packet travels along many partition routes that are estimated with an efficient link quality [37]. Therefore, an increasing lifetime demonstrates that each sensor node unaligned on the partition route enters a sleep state once because in/ongoing transmission is lacking; each sensor node aligned on the partition route enters the wakeup state.
where M is the average number of events occurring per unit of time.
The node density control algorithm, on other hand, defines lifetime reporting as a noncumulative function that depends on the total number of events detected by a network composed of n nodes. The node density explores the relationship between energy efficiency and the number of nodes. These nodes are dense nodes that are uniformly deployed to optimize the lifetime.
where Energy_{ ı } is the amount of energy required to report the event from the source node along the multiple hops to the sink node calculated by Eq. 2, t _{ ı } is the energy consumed to sense the event, and M is the average number of events detected during a day [28]. A total of n nodes are deployed along a highway to monitor the traffic vehicles captured in the camera’s FoV. Processing is performed for a motion differentiating between consecutive frames. The location of the vehicle is defined as a complete trajectory of the vehicle centroid moving in the FoV. The duration of the detection is the time interval that a vehicle spends in the camera’s FoV [33]. A vehicle is detected when no centroid is overlooked during the event duration; otherwise, the event is overlooked [44].
The focus of the simulation is to evaluate various energy consumption optimization algorithms in [26, 27] by adjusting the number of hops from the source to the sink. For a given optimal number of hops, the optimal energy in the sensor network is obtained by jointly considering the data rate and transmit power. Since nodes in the network are deployed among different partitioning selected paths, therefore, the node lifetime is proportional to the initial energy spent to detect the event at each node in a unit of time. The lifetime decreases as the time to detect the vehicles is prolonged because of the inefficiency of the regular deployment of nodes. This deployment is performed to monitor the events and to route the report of the occurring events. The proposed algorithm designs an efficient regular partitioning topology by performing link measurements when the nodes are awake for event transmission or reception with minimal extra energy expenditure.
Figure 13 shows the energy efficiency of the WSN when the optimization problem by is solved for energy consumption. The lifetime of the sensor node increases when the packet travels along a number of partition routes estimated with an efficient link quality that connects the sensor nodes in the network domain. This increasing lifetime demonstrates that each sensor node unaligned on the partition route enters a sleep state once because of the lack of ongoing transmission; by contrast, each sensor node aligned on the partition mode enters a wakeup state.
It is important to address the average cost of routing algorithm in order to understand its computational complexity. The cost of routing algorithm depends on searching the optimal selected path from the source to the sink. Usually, this depends on the number of effective fitness functions required to evaluate the objective functions in Eqs. 11 and 14 in terms of power consumption and delay, respectively, which encompass to the desired search space of the optimization problem. The percentage of complexity is defined as the optimal value of μ required to obtain the optimal selected partitioning path as referred to in (Eqs. 11 and 14) with Z _{ d }=Z. The average value of \(\phantom {\dot {i}\!}(Z_{d^{*}}*100)\) divided by the average value of Z (Eq. 11), where \(\phantom {\dot {i}\!}Z_{d^{*}}\) denotes the number of actual lower and upper bounds obtained from the determination of cutoff [54, 55].
This lifetime is proportional to the ratio of the total number of nodes n that deploys an average number of detecting event M in a unit of time multiplied by the number of partitioning intermediate nodes denoted as p. This effect is understandable due to the increasing number of partitioning intermediate nodes. In other words, the more nodes involved to detect the event, the more energy is consumed in a unit of time, thus, decreasing the node lifetime.
This can be attributed to the fact that the network lifetime is proportional to the ratio \(\frac {P(n)}{pM}\). With the same number of partitioning nodes, the network lifetime decreases when more than one source detects the event as seen in Fig. 16. Therefore, more nodes need to be involved for event detection, and more energy is consumed by the network in a unit of time. Hence, the lifetime decreases when the number of sources increases. Similar to the singlesource scenario in Fig. 15, the lifetime decreases when the number of initially partitioning nodes in the network increases. The proportionality of the network lifetime (Eq. 25) reasonably justifies this explanation. Therefore, our asymptotic analysis has been clarified based on Theorem 2 and Theorem 3 which state that for a fixed network density, network lifetime decreases in the order of \(\left (\frac {1}{P(n)}\right)\) as the number of initially partitioning deployed nodes n which aligned in select partitioning path. This analysis also shows that the network lifetime for multisource nodes scenario is shorter than the case of singlesource scenario by a certain factor, i.e., energy consumption depends on the hop distance that the data has been transmitted which supports our definition of network lifetime for both scenarios. More precisely, Fig. 14 for singlesource scenario illustrates a better performance of the proposed routing protocol than multisource scenario. Unfortunately, it seems that most existing multipath routing protocols are inherently illsuited to perfom efficiently with multisources scenarios where the data needs to report to the sink as shown in Fig. 16. Indeed, to achieve multihop communication with the multisources scenario, more multipath routing have been constructed and selected independently (e.g., by flooding a control message from each source node and having each node remember the reverse selected paths to the sink) for data routing. However, by reducing the number of paths exploited, the amount of redundant information flowing in the network decreases and less nodes are involved in routing the messages. This increases the network lifetime and reduces the contention on wireless medium and packet collision which ultimately increases the reliability of multipath routing.
4 Discussion
Multipath routing in WSNs provides efficient strategies to increase the bandwidth, improving the load balancing, reliability, and faulttolerance. It also provides path resilience. The generation of multiple paths is compared between the singlesource as shown in Fig. 15 and multiplesource scenarios as seen in Fig. 16. The transmission along multiple paths from multiple sources may interfere with the transmission along another path even when the discovered paths are nodedisjoint. Therefore, the interference may limit the achievable network performance and lead to multiple paths, affecting data packet transmission. The multiplesource scenario is usually referred to as route coupling [56]. In other words, multiple paths are located physically close enough to interfere with each other during selection and data transmission. However, the route coupling caused by interference among multiple paths may affect the performance of multipath routing protocols. It may even lead to worse results as depicted in Fig. 17 than routing over the singlesource scenario shown in Fig. 14.
This observation demonstrates that reducing of the number of partitioning multipath decreases the amount of redundant information flowing in the network. Moreover, reducing the number of node density involved in routing messages increases the lifetime and decreases the contention medium and packet collisions. As a result, the reliability of communication is increased. Consequently, most realtime multimedia sensor networks are based on a manytoone paradigm. Therefore, a multipath routing approach is geared towards the efficient and reliable partitioning of transmission between the source and the sink to optimize the QoS parameters in the multihop sensor networks. In order to conserve energy, the multipath routing protocol must ensure that the traffic path selection along the selected path does not interfere with each other. In most cases, this situation is difficult to achieve.
5 Conclusions
Lifetime maximization is considered a key challenge for energyconstrained WSNs. In this paper, we propose a mathematical model for an energyconstrained routing algorithm. The proposed routing is based on the determination of cutoff for the optimal number of hops to partition the path from the source to the sink. We presented a LR method to maximize lifetime by defining critical parameters to control the adaptive hopbyhop switching. Our results significantly improve the lifetime compared with the three wellknown algorithms. The first comparison was done with node density control algorithm, and our proposed algorithm has improved the lifetime by 46.91%. The second algorithm was the upper bounds of the lifetime protocol, and our algorithm has improved lifetime by 73.00 and 80.00% for WSNs optimization of network lifetime algorithms.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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