1 Introduction

Due to various digital devices connect to cloud computing network as well fog computing network, data is generated in an explosively increasing manner. The development of big data imposes heavy burden to the underlying infrastructure. Because it requires vast computation, storage and communication resources. But the big data not only contains the sensed data, but also contains the process of diffusing program codes. One important application of big data is to diffuse program codes. In fog computing network, in order to enhance the intelligence of edge access devices, a large number of devices are updated, upgraded and reconfigured using wireless software defined network, so that the deployed devices obtain new vitality. Programming codes need spread to devices, the process is an inverse operation of big data collection. The process of program codes diffusion from data collection center to network edge is widely used in practical applications. For example: wireless sensor network (WSN) is a promising platform, and it is widely used in military and civilian domains [1,2,3,4]. In a smart wireless software-define network, many sensor nodes are deployed in the monitored area, sensor nodes sense data from surrounding environment, then the sensed data is transmitted to base station. The base station sends message to devices after the base station receives message, then the devices update and reconfigure software. Compare to traditional industrial automation systems using wired communications, a smart wireless software-define network brings notable advantages including lower cost, higher flexibility, and self-organizing capability, which significantly improves industrial efficiency and productivity [5].

With the rise of software defined network (SDN), the deployed devices can have new function by updating and configuring program codes. Due to the flexibility of software update and configuration, it has attracted the attention of researchers. In such environment, the program codes is usually transmitted to all devices. After the devices receive new program codes, the program codes in devices have been recompleted, then those devices have new functions. Broadcast is a fundamental operation in wireless software defined sensor networks (WSDSNs) [6, 7, 9]. Given a base station, the aim is to propagate data packets to all nodes in a minimum-transmission broadcast (MTB) while incurring minimum latency. This problem, called minimum-transmission broadcast (MTB), has been studied extensively [6,7,8,9]. Many applications such as industrial automation often have very stringent requirements on communication transmission delay [5]. Nevertheless, such environments faces severe challenges on the design of efficient codes diffusion in WSDSNs. First, energy of sensor nodes are limited, sensor nodes usually adopt duty-cycle style, in which a node alternates between dormant and active state [10,11,12]. Due to duty cycle style of nodes, program codes need transmit multiple time to nodes in the network. As a result, the MTB problem in duty-cycled networks (MTB-DC problem) is difficult. Second, the issue of tradeoff between lifetime and delay is a challenge task. Because duty cycle of node should be big to transmit program codes to all nodes in the network quickly, and duty cycle of nodes should be as small as possible to prolong network lifetime. Therefore, how to reduce the network transmission delay while retaining network lifetime is a challenge issue.

Though there are many researches about program code (or other large big data) broadcast in WSDSNs, it is still a place for further study about network lifetime, minimum transmission broadcast and broadcast delay. In this paper, an adjustable duty cycle based fast disseminate (ADCFD) approach is proposed for minimum-transmission broadcast (MTB) in the WSDSNs. The main contributions of the ADCFD approach are as follows:

  1. (1)

    ADCFD approach optimizes broadcast, which can significantly improve broadcast performance while retaining original broadcast protocol. It has been widely used in WSDSNs. All previous schemes optimize scheduling algorithm under the condition of the given duty cycle, thus, the ability for improving network performance is limited. ADCFD scheme adopts different methods that increase duty cycle of nodes far from the sink to reduce the number of transmissions and broadcast delay. Because the number of broadcasts and delay has an inverse relationship with duty-cycle.

In this paper, we not only consider the energy consumption for transmitting program codes, but also consider the energy consumption of the whole network. Thus, the energy left can be full used to improve broadcast performance. According to [14], the energy consumption of nodes near to the sink is far higher than energy consumption of nodes far from the sink in the process of data collection. The data collection operation is regular operation in WSDSNs, updating and broadcasting program code is temporary. Therefore, the energy left of nodes far from the sink can be used to increase duty cycle of nodes far from the base station to improve broadcast performance.

  1. (2)

    The optimization parameters and duty cycle are discussed and calculated for ADCFD scheme in theory. And the performance of the approach is given in this paper in theory. Through simulation experiment, broadcast times, broadcast delay can be reduced and energy-efficient can be enhanced in ADCFD scheme simultaneously. Compare to the previous scheme, the broadcast delay can be reduced by 17.895%–107.527%, the number of broadcast is reduced by 44.776%–118.519%. The ADCFD scheme has better performance while retaining network lifetime.

The rest of this paper is organized as follows: In section 2, the related works are reviewed. The system model is described in section 3. In section 4, a novel ADCFD approach is presented for WSDSNs. The performance analysis for ADCFD approach is provided in section 5. Section 6 are experimental results and comparison. We conclude in section 7.

2 Related work

In the wireless software defined network, in order to enhance the intelligence of access network devices, when software in network devices need to be updated, upgraded, and re-configured, the data control center (the sink) define software, and then transmit program codes with specific function to devices in the network, thus, a large number of software in devices can be updated, upgraded. Thus, the deployed device obtained new vitality. The broadcast in WSDSNs have been formulated and investigated in [13,14,15,16,17,18,19]. It can be divided into the following kinds based on the different applications requirements.

  1. (1)

    The minimum-transmission broadcast (MTB) problem. In those researches, the aim of those schemes is how to reduce the number of times. In previous schemes, considering nodes are in active status all the time, thus, reducing the number of transmission is to find a Minimum Connected Dominating Set (MCDS) of the network [14, 15]. Nodes in the set can cover the whole network. Thus, as long as nodes in the set broadcast program codes once, all nodes in the network can receive program codes. In [17], the authors proved that building a minimum flooding tree is identical to finding an MCDS.

In fact, the aim of reducing transmission times is to reduce energy consumption. There are many researches about reducing energy consumption. Ref. [14] proposed a Heuristic broadcast algorithm BIP (broadcast incremental power). It builds a tree whose root is source node, in the process of constructing tree, one node uncovered is joined the broadcast tree, the energy cost in this way is minimum. Ref. [15] further optimize the BIP algorithm, proposed effective search algorithm r-shrink, it reschedules the non-leaf nodes in the process of building tree to reduce the total energy consumption of the radio.

In order to save energy in WSNs, a node alternates between dormant and active state [7]. In broadcast, the program codes need transmit many times to all nodes in the network. As a result, it becomes a problem in MTB-DC problem that how to select set of forwarding nodes and identify broadcast schedules.

Le Duc et al. [21] proposed Level-Based Approximation Scheme to identify forwarding nodes and their corresponding receivers for all time slots; then constructs a broadcast backbone by connecting these forwarding nodes to the broadcast source.

  1. (2)

    The minimum latency broadcast scheduling (MLBS). In those schemes, they not only reduce energy consumption of nodes, but also reduce the time for transmitting program codes [13, 20].

Zhao D et al. [16] considered MLBS in duty cycled WSNs, and gives two algorithms, BS-1 and BS-2, the maximum broadcast delay in this two algorithms are (∆ − 1)TH and 13TH, respectively. Here, ∆ is the maximum degree of nodes, T denotes the number of time slots in a working cycle, and H is the broadcast latency lower bound obtained from the shortest path algorithm.

Khiati M et al. [22] proposes a Broadcast over Duty-Cycle and LEACH (BOD-LEACH) protocol, takes advantage of the LEACH’s energy-efficient clustering. The new common static and dynamic broadcast periods are included to accelerate broadcasting. The dynamic periods are scheduled following the past arrivals of messages and using a Markov chain model.

3 System model and problem statements

3.1 The network model

We consider that n homogenous sensor nodes and one sink node deployed in a WSDSN. The sink node is regarded as the base stations, s means the base station. The network can be regarded as ℳ = {s, v 1, v 2, v 3, v 4 … v n }. v i is the node ID. Those sensor nodes deployed in the target area such as environment monitor, industrial field, smart field and so on [23,24,25]. The network radius is R, the transmission radius of a sensor node is r. The sensor nodes are powered by battery whose energy are limited, and the energy of the sink is unlimited. Fig. 1 shows the initial connectivity setup of the network or the weighted graph established by a routing protocol. The network structure studied in this paper is given in the Fig. 1, s is the center of data collection. v i is the sensor node in the network.

Fig. 1
figure 1

Network structure diagram

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In consequence of saving energy, it would be better that nodes are set to sleep-wake when nodes don’t need to receive program codes. Each node adopts asynchronous duty-cycle model, the duty cycle starts over between two consecutive periods. Each node has active/sleep two models. Each working cycle T is divided into ∣T∣ time slots with same length. Thus, one working cycles can be expressed as {0, 1, 2, 3,….., ∣T∣ − 1} time slots. It is shown as Fig. 5, considering working cycle of a node is 3, the working cycle is regarded as three time slots, they are 0, 1 and 2. Each node randomly selects active time slots. s is the sink node, the sink node sends program codes to neighbor nodes, then the program codes are transmitted from those neighbor nodes to outside nodes, until program codes reach the network boundary. The number in nodes are active time slots of nodes.

3.2 Energy consumption model

In this paper, nodes can periodically switch on and off according to a normalized duty cycle. The energy consumption of a node contains three aspects. (1) The energy consumption for receiving data packets, (2) the energy consumption for sending data packet, (3) When node v i at lm away from the sink is in the sleep state, the energy consumption of node v i for sensing message is \( {e}_{sen}^i \), the energy consumption for sensing data in different schemes are the same. When node v i at lm away from the sink is in active status, the energy consumption of node v i is \( {e}_r^i \), and the energy consumption of node v i at lm away from the sink is \( {e}_s^i \) when node v i sends program codes to neighbor nodes. Thus, the total energy consumption e i of node v i at lm away from the sink is as follows.

$$ {e}^i={e}_r^i+{e}_{s en}^i+{e}_s^i $$
(1)

Thus, the power consumption of node contains (1) The power for receiving packets \( {\varpi}_r^i \), (2) the power for sending packets \( {\varpi}_s^i \). Thus the energy consumption at node v i at lm away from the sink is as follows.

$$ {e}^i={e}_{s en}^i+{\varpi}_r^i{Q}_r^i+{\varpi}_s^i{Q}_s^i $$
(2)

\( {Q}_r^i \) is the amount of receiving data of node v i at lm away from the sink, \( {Q}_s^i \) is the amount of sending data of node v i at lm away from the sink.

The main parameters of system model adopted in this paper are similar to those parameters in [26,27,28]. The parameters are adopted in Table 1.

Table 1 Main notations and values adopted in this paper
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3.3 The problem statements

Definition 1: Minimization network updated delay. The network updated delay D is refer to the time for transmitting program codes from the base station to all nodes in the network. The aim of this paper is to reduce network updated delay, it can be calculated as follows.

$$ \min (D)= \min \left({\sum}_{i\subseteq route}{d}_i\right) $$
(3)

Definition 2: Minimization broadcast times. Broadcast time ϱ is refer to the number of transmission when program codes are transmitted to the whole network. When nodes transmit program codes to neighbor nodes, the neighbor nodes can receive program codes only when nodes are in active statues. The aim of this paper is to reduce the number of transmission by increasing duty cycle of nodes far from the base station. The minimum broadcast times is expressed as follows.

$$ \min \left(\varrho \right)= \min \left({\sum}_{i\subseteq route}\kern0.2em {\varrho}_i\right) $$
(4)

Definition 3:

Maximization of network lifetime (denoted as ℓ). The fundamental purpose of application is to maximize network lifetime [29,30,31]. Network lifetime can be defined as the time of the node that dies first. For after the first node dies, it may affect the connectivity and coverage of the network severely leading the network cannot play a proper role. Thus, network lifetime in this paper is the time of the node that dies first. The energy consumption of node e i contains the energy consumption \( {e}_r^i \) for receiving data packet, the energy consumption \( {e}_s^i \) for sending data packet and the energy consumption for sensing data \( {e}_{sen}^i \). Where E init represents the initial energy of node v i . Thus, network lifetime is as follows.

$$ \kern0.75em \max \left(\ell \right)={}_{0\le i\le n}{}^{\mathit{\max}}\left({E}_{i nit}/\left({e}_r^i+{e}_{s en}^i+{e}_s^i\right)\right) $$
(5)

Definition 4: Maximization of effective energy utilization rate (denoted as φ). φ refers to the ratio of energy efficiently utilized to the total energy of nodes in the network. The aim in this paper is to make full use of energy left in nodes far from the sink, then maximize energy utilization of the whole network. e i is the energy consumption of node v i , E init represents the initial energy of node v i . Thus, the effective energy utilization rate is as follows.

$$ \max \left(\upvarphi \right)= \max \left(\left({\sum}_{1\le i\le n}{e}^i\right)/\left({\sum}_{1\le i\le n}\kern0.2em {E}_{i nit}\right)\right) $$
(6)

Obviously, the goal of ADCFD is to minimize delay, D, minimize broadcast time, ϱ, maximize the network life, ℓ, and effective energy utilization, φ, which can be summarized as follows.

$$ \left\{\begin{array}{c}\mathit{\min}(D)=\mathit{\min}\left({\sum}_{i\subseteq route}{d}_i\right)\\ {}\mathit{\min}\left(\varrho \right)=\mathit{\min}\left({\sum}_{i\subseteq route}{\varrho}_i\right)\\ {}\kern0.75em \mathit{\max}\left(\ell \right)={}_{0\le i\le n}{}^{\mathit{\max}}\left({E}_{i nit}/\left({e}_r^i+{e}_{s en}^i+{e}_s^i\right)\right)\\ {}\mathit{\max}\left(\varphi \right)=\mathit{\max}\left(\left({\sum}_{1\le i\le n}{e}^i\right)/\left({\sum}_{1\le i\le n}{E}_{i nit}\right)\right)\end{array}\right. $$
(7)

4 Main design of ADCFD

4.1 Research motivation

The research motivation in this paper is based on the following two aspects:

  1. (1)

    When a sensor node in a wireless software-define network senses surrounding environment, the sensed data can be transmitted to the sink. Due to nodes near to the sink can bear a large amount of data, its energy consumption is higher than the energy consumption of nodes far from the sink. The energy consumption is given in Fig. 2. According to some researches, there are 90% energy left when the network is dead [31]. That is to say, the energy left in the network is wasted. The paper mainly researches program code transmission. Program codes are transmitted to all network, then software in devices can be updated. The aim in this paper is to reduce transmission delay. Due to the much energy left in a network, the energy left can be used to increase duty cycle of nodes, which can reduce the transmission delay.

Fig. 2
figure 2

The energy consumption for collecting data in wireless software-define networks

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  1. (2)

    The higher number of transmission and broadcast delay in the previous scheme.

LBAS scheme proposed in Ref. [21] constructs a broadcast backbone for reducing broadcast delay and transmission times. Before program codes start transmitting, the broadcast backbone should be built. The broadcast backbone construction is accomplished by a two-stage traversal on all forwarding nodes. The process of building broadcast backbone is as follows.

For each time slot, a covering set with a minimum number of nodes is identified such that these nodes can cover all nodes active at that time slot. Nodes in such a set are called covering nodes, and nodes covered by these covering nodes are called covered nodes.

  1. (1)

    Finding the minimum covering node set

Considering the broadcast source s is a covering node, but not as a covered node in any case. σ a is the set of nodes with active time slot a. ω a is the minimum covering node set of nodes with active time clot a [21]. For each time slot a ∈ T, if the number of neighbor nodes with active time slot a of node y is largest in σ a , node y can be covering node. This process is executed until all nodes in σ a are covered.

For example, considering the network with 20 nodes in the Fig. 3, the duty-cycle |T| is 3, there are 0, 1 and 2 time slots [21]. For the time slot 0, the neighbor nodes of v 6 are v 3, v 4, v 7, v 12, v 13, there are two nodes whose active time are 0. There are two nodes whose active time are 0 in those neighbor nodes of node v 5. There are one nodes whose active time are 0 in those neighbor nodes of node v 2. The minimum covering node set for time slot 0 is ω 0={v 2 , v 5 , v 6}. Because neighbor nodes of three nodes (v 2 , v 5 , v 6) are include all nodes whose active time is 0. The minimum covering node set 1 is ω 1={v 1 , v 3 , v 7 ,   v 10}, the minimum covering node set 2 is ω 2={v 4 , v 8 , v 9}.

  1. (2)

    Constructing broadcast backbone

Fig. 3
figure 3

Network structure diagram in wireless software-define networks

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The broadcast backbone is built based on the minimum covering node sets. It is can be seen that if all nodes in the minimum covering node sets are connected as a tree in the network, the program codes can be transmitted to all nodes in the network along the broadcast backbone. Because nodes in the minimum covering node sets cover all nodes in the network. Thus, connecting all nodes in the minimum covering node sets to build broadcast backbone to reduce transmission. The principle of this paper is as follows.

In the LBAS scheme, nodes are divided into several levels, nodes near to the sink have upper level, and nodes far from the sink have lower level. As shown in Fig. 4.

  1. (3)

    If a covering node near to the sink has a neighbor node in the sets of covering node, and the covering node covers the neighbor node, the node will connect to his neighbor node. An example is shown in Fig. 4a, u is the parent of node v.

  2. (4)

    If a covering node covers the neighbor node in the covering sets, and the neighbor node covers the covering node [21], the two nodes are in the same level near to the sink. One can connect to other node. An example is shown in Fig. 4b.

  3. (5)

    If one covering node covers the neighbor node in the covering sets, and the covering node and the neighbor node are in the same level [21], the covering node is selected as the parent. An example is shown in Fig. 4c, where node u is selected as the parent of node v.

Fig. 4
figure 4

The way for connecting covering nodes in wireless software-define networks

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Though some nodes are selected for building broadcast backbone, some nodes may not be selected. Such as Fig. 4d, node v forms a covering sub-tree with itself.

Through the following step, a broadcast backbone can be completed. When the source node (the sink) sends program codes to neighbor nodes, and program codes are transmitted along the broadcast backbone. At last, all nodes in the network can receive program codes.

Though LBAS scheme can reduce the number of transmission and broadcast delay to a certain extent, the performance of LBAS scheme is improved only from adjusting the algorithm, the ability of improving performance is limited. The duty cycle in the LBAS scheme is showed in Fig. 5. The working cycle in LBAS scheme is T, each working cycle is divided into |T| time slots. Each sensor node selects a time slot as an active time slot, sensor node in an active time slot can receive data packets. It is shown in Fig. 5, the length of working cycle T is 3, and it contains 0, 1 and 2 time slots. Only when one node is in an active time slot, the node can receive data packets. If node A want to transmits data, the next node B is in sleep time slot, node A must wait for a while, until node B is in active time slot. This process can cause longer transmission delay.

Fig. 5
figure 5

the time slots of the LBAS scheme in wireless software-define networks

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Though program codes can be transmitted to all nodes in the network, the transmission delay is still longer. The reason is that when one node receives the program codes from the sink, the program codes is only transmitted to the nodes when nodes is in the active time slot. Each node randomly selects a time slots as an active time slot, if the active time slots of nodes in the lower level in the broadcast backbone is higher than the active time slots of nodes in the upper level, the program codes are transmitted to the lower level nodes until the next working cycle, it needs longer transmission delay. As showed in the Fig. 6, the active time slot of node u in the upper level is 1, the active time slot of node v in the lower level is 0, node u wait for the next working cycle to transmit the program codes to node v when node u receives program codes. It need longer transmission delay.

Fig. 6
figure 6

the level in wireless software-define networks

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It can be seen that duty cycle of nodes in the LBAS scheme in different area are the same. Each node selects one time slots as an active time slot, which can cause larger transmission delay and higher energy left in the no-hotspots area.

Due to reduce broadcast delay farther, ADCFD scheme adjusts duty cycle of nodes to reduce broadcast delay. In the ADCFD scheme, duty cycle of nodes are adjusted based on energy left of the nodes. The duty cycle of a node in the ADCFD is showed in the Fig. 7. Considering the working cycle T is 3, it is divided into 0, 1 and 2 time slots. Nodes with higher energy consumption can select two or more time slots as active time slots. The different of duty cycle of nodes in the ADCFD scheme from LBAS scheme is that nodes with higher energy left can select many time slots as active time slots. In the above analysis, each node sends the detected data to the sink. Therefore, the energy consumption is imbalanced in the whole network. Nodes far from the sink have much energy left, the duty cycle of those nodes can be increased to reduce transmission delay. It is showed that the longer the duty cycle is, the transmission delay is lower.

Fig. 7
figure 7

the time slots of the ADCFD scheme in wireless software-define networks

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Broadcast performance depends on duty cycle of nodes in the network. Larger duty cycle brings higher monitoring performance. Figure 8 shows the number of broadcast in different duty cycle. The number of broadcast decrease with the increase of duty cycle of nodes. Figure 9 shows the broadcast delay in different duty cycle. It can be seen that the number of broadcast and transmission delay decrease with the increase of duty cycle of nodes. The larger duty cycle of nodes are, and the probability of nodes that receive program codes is higher when one node broadcasts program codes once. Thus, the number of broadcast and transmission delay are smaller.

Fig. 8
figure 8

The number of broadcast with different duty cycle

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Fig. 9
figure 9

Broadcast delay with different duty cycle

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Figure 10 shows that the remaining energy of nodes is used for increasing duty cycle in different regions. It can be seen from Fig. 2 that duty cycle of non-hotspot nodes are higher than duty cycle of hotspot nodes. According to the above analysis, there are much energy left in the non-hotspots area, the energy left can be used to increase duty cycle of nodes. The adjustment duty cycle of nodes in different areas is given in Fig. 10. In the previous scheme, the duty cycle of nodes in different areas are the same. But the duty cycle of nodes far from the sink in the ADCFD scheme are greater than that of previous scheme. Adjusting the nodal duty cycle dynamically can achieve better and more comprehensive broadcast performances. For this reason, an adjustable duty cycle based fast disseminate (ADCFD) approach is proposed to improve program codes diffusion while retaining network lifetime.

Fig. 10
figure 10

duty cycle for different distances

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4.2 The calculation of energy consumption of the network

This section gives the amount of data loaded in different area, then the energy consumption in different area for collecting data is shown.

Inference 1. Considering the probability of producing packet of one node is λ, the node v i at l ,  l = hr + x (0 < x < r) away from the sink receives the amount of data \( {Q}_r^i \), it is as follows.

$$ {Q}_r^i=\left( z+\frac{z\left( z+1\right) r}{2 l}\right)\lambda $$
(8)

Proof: It can be seen from Eq. (8), one node generates a data packet in a collection cycle. If the probability is λ. If one node produces a data packet, the number of data packet at node v i is \( z+\frac{z\left( z+1\right) r}{2 l} \), thus, the amount of data loaded by the nodes at l away from the sink is as follows.

$$ {Q}_r^i=\left( z+\frac{z\left( z+1\right) r}{2 l}\right)\ \lambda $$

Inference 2. Considering the probability of generating packet of one node is λ, the node v i at l ,  l = hr + x (0 < x < r) away from the sink sends the amount of data \( {Q}_s^i \), it is as follows.

$$ {Q}_s^i=\left(\left( z+1\right)+\frac{z\left( z+1\right) r}{2 l}\right)\lambda $$
(9)

Proof: It can be seen from Eq. (8), when one node receives the data packet, it will generate one data packet with probability λ by itself, then those data packets are transmitted to the next node, the amount of data transmitted by the nodes at l away from the sink are as follows.

$$ {Q_s^i= Q}_r^i\lambda =\left(\left( z+1\right)+\frac{z\left( z+1\right) r}{2 l}\right)\lambda $$

The amount of data of nodes varies from area to area, hotspot nodes have to receive and forward large amount of data packets. Considering nodes at l m away from sink have \( {Q}_z^i \) packets to be transmitted in a communication cycle τ c , and \( {Q}_r^i \) packets to be received.

e i denotes the total energy consumption of node v i at l m away from sink in data working cycle τ c . According to Refs. [32, 33], there are three states in the energy consumption e i: (i) transmission, (ii) reception, and (iii) sensing. e i can be computed as follows.

$$ {e}^i={e}_{s en}^i+{\varpi}_r^i{\mathrm{Q}}_r^i+{\varpi}_s^i{\mathrm{Q}}_s^i $$
(10)

\( {\varpi}_r^i \) is the power used for receiving a packet. \( {\varpi}_s^i \) is the power for transmitting a packet. It can be calculated as follows.

$$ {\varpi}_s^i={\varsigma}_s{\Theta}_d+\left(\frac{\tau_a{\tau}_c}{4\left({\Theta}_p+{\Theta}_a\right)}+\frac{1}{2}\right)\left({\varsigma}_s{\Theta}_p+{\varsigma}_r{\Theta}_a\right) $$
(11)

ς s Θ d is the energy consumption for sending a data packet. ς s is the transmission power consumption, ς r is the reception power consumption, Θ d is the packet duration, Θ a is the ACK window duration, Θ p is the preamble duration. τ a is the active period radio, τ c is the working cycle.

\( {\varpi}_r^i \) can be calculated as follows.

$$ {\varpi}_r^i={\varsigma}_r{\Theta}_d+\left({\varsigma}_r{\Theta}_p+{\varsigma}_s{\Theta}_a\right) $$
(12)

ς r Θ d is the energy consumption for receiving a packet and ς r Θ p  + ς s Θ a is for the reception of the preamble and the transmission of the returned message. Eq. (2) calculates the total energy consumption.

The energy consumption for receiving and sending data packet in previous scheme are given in Fig. 11 and Fig. 12, respectively. In the previous scheme, the duty cycle of all nodes are the same. The energy consumption for receiving and sending data packet is higher, and the energy consumption of nodes is unbalanced. The energy consumption for receiving and sending data packet in the previous scheme are given in Fig. 13. It can be seen that the energy consumption of nodes near to the sink is higher than the energy consumption of node far from the sink. There are enough energy left in the non-hotspots to increase duty cycle to reduce the number of transmission and transmission delay.

Fig. 11
figure 11

Energy consumption of receiving data

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Fig. 12
figure 12

Energy consumption of sending data

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Fig. 13
figure 13

The total energy consumption

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Theorem 1. Considering the initial energy of each node is E init . The remaining energy \( {E}_{left}^i \) of data operation of node v i at l m away from the sink in a data collection period τ c is as follows.

$$ {E}_{left}^i={E}_{i nit}-{e}_{s en}^i-{\varpi}_r^i{\mathrm{Q}}_r^i-{\varpi}_s^i{\mathrm{Q}}_s^i $$
(13)

Proof. The energy consumption of node i at l m away from the sink is \( {e}_r^i+{\varpi}_r^i{Q}_r^i+{\varpi}_s^i{Q}_z^i \), the initial energy of each node is E init . Thus the remaining energy of data operation of node v i is as follows.

$$ {E}_{\mathrm{left}}^i={E}_{\mathrm{init}}-{e}_{s en}^i-{\varpi}_r^i{Q}_r^i-{\varpi}_s^i{Q}_s^i $$

4.3 Adjustable duty cycle calculation

The main difference from ADCFD scheme to previous scheme is the different duty cycles of nodes. Therefore, the calculating method of sensing and communicating duty cycle in different regions of the network is given in this section.

Theorem 2. Considering the number of sending and receiving packets in hotspots are \( {Q}_s^{max} \) and \( {Q}_r^{max} \), respectively. The residual energy is only used for increasing τ a , the active period of node v i at l m away from sink is \( {\tau}_a^i \), and the active period of nodes nearest to the sink is \( {\tau}_a^{hotspots} \). \( {\tau}_a^i \) is calculated as follows.

$$ {\tau}_a^i=\frac{\tau_a^{hotspots}{Q}_s^{max}}{Q_s^i}-2\vartheta \frac{2\alpha \left({Q}_r^i-{Q}_r^{max}\right)-2{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)+\beta \left({Q}_s^i-{Q}_s^{max}\right)}{\beta {\tau}_c} $$
$$ \mathrm{Where}\ \alpha ={\varsigma}_r{\Theta}_d+\left({\varsigma}_r{\Theta}_p+{\varsigma}_s{\Theta}_a\right),\beta ={\varsigma}_s{\Theta}_p+{\varsigma}_r{\Theta}_a,\vartheta =\left({\Theta}_p+{\Theta}_a\right)\frac{\tau_c}{{\mathrm{Q}}_s^i} $$
(14)

Proof. The scheme proposed in this paper increases active period to reduce the number of transmission and transmission delay while retaining network lifetime. Nodes in hotspots area consume largest energy according to Refs. [34, 35], the number of sending and receiving packets in hotspots are \( {Q}_s^{max} \) and \( {Q}_r^{max} \), respectively. In order to ensure the network lifetime, the maximization energy consumption of nodes far from the sink can equal to the energy consumption of nodes in hotspots area. It is as follows.

$$ {\varpi_s^i=\varsigma}_s{\varTheta}_d+\left(\frac{\tau_a{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right)\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right),{\varpi}_r^i={\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right) $$
$$ {e}_{s en}^i+{\varpi}_r^i{Q}_r^i+{\varpi}_s^i{Q}_s^i={e}_{s en}^i+{\varpi}_r^{max}{Q}_r^{max}+{\varpi}_s^{max}{Q}_s^{max} $$
$$ \Rightarrow {\varpi}_r^i{Q}_r^i-{\varpi}_r^{max}{Q}_r^{max}={\varpi}_s^{max}{Q}_s^{max}-{\varpi}_s^i{Q}_s^i $$
$$ \Rightarrow \left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right){Q}_r^i-\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right){Q}_r^{max}=\left({\varsigma}_s{\varTheta}_d+\left(\frac{\tau_a^{hotspots}{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right)\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right)\right)\ {Q}_s^{max}-\left({\varsigma}_s{\varTheta}_d+\left(\frac{\tau_a^i{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right)\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right)\right){Q}_s^i $$
$$ \Rightarrow \left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)={\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)+\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right)\left(\left(\frac{\tau_a^{hotspots}{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right){Q}_s^{max}-\left(\frac{\tau_a^i{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right){Q}_s^i\right) $$
$$ \Rightarrow \frac{\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)-{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)}{\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right)}=\left(\left(\frac{\tau_a^{hotspots}{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right){Q}_s^{max}-\left(\frac{\tau_a^i{\tau}_c}{4\left({\varTheta}_p+{\varTheta}_a\right)}+\frac{1}{2}\right){Q}_s^i\right) $$
$$ \Rightarrow \frac{\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)-{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)}{\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right)}\ast \left(4\left({\varTheta}_p+{\varTheta}_a\right)\right)=\left(\left({\tau}_a^{hotspots}{\tau}_c+2{\varTheta}_p+2{\varTheta}_a\right){Q}_s^{max}-\left({\tau}_a^i{\tau}_c+2{\varTheta}_p+2{\varTheta}_a\right){Q}_s^i\right) $$
$$ \Rightarrow \frac{\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)-{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)}{\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right){\tau}_c}\ast \left(4\left({\varTheta}_p+{\varTheta}_a\right)\right)+\left(2{\varTheta}_p+2{\varTheta}_a\right)\left({Q}_s^i-{Q}_s^{max}\right)=\left({\tau}_a^{hotspots}{\tau}_c{Q}_s^{max}-{\tau}_a^i{\tau}_c{Q}_s^i\right) $$
$$ \Rightarrow \left(\frac{\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)-{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)}{\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right){\tau}_c}\ast \left(4\left({\varTheta}_p+{\varTheta}_a\right)\right)+\left(2{\varTheta}_p+2{\varTheta}_a\right)\left({Q}_s^i-{Q}_s^{max}\right)\right){\tau}_c=\left({\tau}_a^{hotspots}{Q}_s^{max}-{\tau}_a^i{Q}_s^i\right) $$
$$ \Rightarrow {\tau}_a^{hotspots}{Q}_s^{max}-\left(\frac{\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)-{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)}{\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right){\tau}_c}\ast \left(4\left({\varTheta}_p+{\varTheta}_a\right)\right)+\left(2{\varTheta}_p+2{\varTheta}_a\right)\left({Q}_s^i-{Q}_s^{max}\right)\right){\tau}_c={\tau}_a^i{Q}_s^i $$
$$ \Rightarrow {\tau}_a^i=\frac{\tau_a^{hotspots}{Q}_s^{max}}{Q_s^i}-\left(\frac{\left({\varsigma}_r{\varTheta}_d+\left({\varsigma}_r{\varTheta}_p+{\varsigma}_s{\varTheta}_a\right)\right)\left({Q}_r^i-{Q}_r^{max}\right)-{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)}{\left({\varsigma}_s{\varTheta}_p+{\varsigma}_r{\varTheta}_a\right){\tau}_c}\ast \left(4\left({\varTheta}_p+{\varTheta}_a\right)\right)+\left(2{\varTheta}_p+2{\varTheta}_a\right)\left({Q}_s^i-{Q}_s^{max}\right)\right)\frac{\tau_c}{Q_s^i} $$
$$ \Rightarrow {\tau}_a^i=\frac{\tau_a^{hotspots}{Q}_s^{max}}{Q_s^i}-2\vartheta \frac{2\alpha \left({Q}_r^i-{Q}_r^{max}\right)-2{\varsigma}_s{\varTheta}_d\left({Q}_s^{max}-{Q}_s^i\right)+\beta \left({Q}_s^i-{Q}_s^{max}\right)}{\beta {\tau}_c} $$

Where α = ς r Θ d  + (ς r Θ p  + ς s Θ a ), β = ς s Θ p  + ς r Θ a , \( \vartheta =\left({\varTheta}_p+{\varTheta}_a\right)\frac{\tau_c}{Q_s^i} \).

Figure 14 shows the adjustment duty cycle of nodes in the ADCFD scheme. In the ADCFD scheme, the duty cycle of nodes far from the sink is increased depended on the energy left of nodes. It can be seen that the duty cycle of nodes far from the sink is higher, it up to 1, and the duty cycle of nodes near to the sink is lower.

Fig. 14
figure 14

The adjustment duty cycle in the ADCFD scheme

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4.4 Scheduling method of ADCFD scheme

The duty cycle of nodes are adaptively adjusted by the ADCFD scheme which use feedback control method, the aim of this paper is to reduce transmission delay while retaining network lifetime [36].

In the process of data collection, when one node v i sends a data packet, the maximum energy consumption \( {e}_{max}^i \), added in the header of the data packet, is set 0. Each node v k receives a data packet, then compare the energy consumption e k to the maximum energy \( {e}_{max}^i \) in the header of data packe, if \( {e}_{max}^i<{e}_k \), then.

$$ {e}_{max}^i={e}_k $$

When the sink receives the data packet, the maximum energy consumption is also received by the sink. After all nodes send a data packet in one data collection cycle, the sink calculates the maximum energy consumption e max .

In the process of program codes diffusion, when the sink sends program codes, the maximum energy consumption e max is added in the header of program codes. Each node v k receives program codes, then compare the energy consumption e k to the maximum energy \( {e}_{max}^i \) in the header of data packet, if e max  < e k , then.

$$ {e}_{max}={e}_k $$

The program codes are sent to all nodes in the network. After all nodes in the network receive program codes, the ACK message contains maximum energy consumption e max is returned to the sink. Then the sink sends adjustment message to adjust duty cycle of nodes according to the energy consumption of nodes and the maximum energy consumption.

In the process of adjusting duty cycle, the sink sends adjustment message contains maximum energy consumption e max to nodes for adjusting duty cycle of nodes. When one node receives the adjustment message, the duty cycle of the node is adjusted according to the Eq. (15) based on the energy consumption of the node and the maximum energy consumption. The process continues until reaches balanced status.

When sensor nodes send data to the sink, the message about energy is also transmitted to the sink, the energy consumption of nodes near to the sink can be obtained. If the energy consumption of nodes far from the sink is less than the energy consumption of nodes near to the sink, the duty cycle of nodes far from the sink can be adjusted using feedback message. But if the energy consumption of nodes far from the sink is bigger than the energy consumption of nodes near to the sink, the duty cycle of nodes far from the sink can be decreased using feedback message. The duty cycle of nodes can be adjusted better through several feedback control.

figure e
figure f

In the above analysis, it can be known that nodes with maximum energy consumption are in the area near to the sink, the area called hotspots area. For the given duty cycle of nodes in the hotspots area, the duty cycle of nodes can be adjusted according to algorithm 1. At the first, if duty cycle of nodes in the hotspots area are big, the transmission delay and network lifetime can’t be satisfied the required value. In order to prolong network lifetime, Algorithm 2 is given. It can be seen that in order to prolong network lifetime, the maximum energy consumption must be reduced, the method is to reduce the duty cycle of nodes in the hotspots area is given in the algorithm 2. The duty cycle can be reduced Δτ, that is as follows.

$$ {\tau}_{min}={\tau}_{min}-\Delta \tau $$

Then adjusting duty cycle of other nodes in the network, until reaches balanced status.

figure g

5 Performance analysis in theory

5.1 Analysis of transmission times

Theorem 3. The number of nodes with k active time slot is N k . Considering node v i at l m away from sink, \( {\Lambda}_k^i \) is the set of neighbor nodes with k active time slot of node v i . \( \mid {\Lambda}_k^i\mid \) is the number of nodes in the set \( {\Lambda}_k^i \). \( {\mathrm{Y}}_k=\left\{{v}_{i_1},{v}_{i_2},{v}_{i_3},\dots {v}_{i_n}\right\} \) is the set of nodes, the \( \left|{\Lambda}_k^{i_1}\right|>\left|{\Lambda}_k^{i_2}\right|>\left|{\Lambda}_k^{i_3}\right|>\left|{\Lambda}_k^{i_4}\right|> \)… \( {\Lambda}_k^{i_n} \) and ∣Γ k ∣ = N k , where the \( {\Gamma}_k=\left\{{\Lambda}_k^{i_1},{\Lambda}_k^{i_2},{\Lambda}_k^{i_3},\dots {\Lambda}_k^{i_n}\right\}. \) Considering the number of transmission in the network is Ξ, it can be calculated as follows.

$$ \Xi ={\sum}_{k=0}^{\left| T\right|-1}\left|{\Upsilon}_k\right| $$
(15)

Proof. It can be seen that in order to reduce the number of transmission, the broadcast backbone is built. The first step is to find the minimum covering node sets. The method is that u k is the set of nodes with active time slot k. c k is the minimum covering node set. For each time slot k ∈ T, one node y is selected as a covering code for time slot k if y covers the largest number of uncovered nodes in u k . This process of selection is executed, until all nodes in u k are covered. That is to say, for active time slot k, the minimum covering node with active time slot k is Υ k . The total number of covering node is Ω = {Υ1, Υ2, Υ3 … Υ|T|}.

In the ADCFD scheme, the broadcast backbone is built based on the minimum covering node sets. For one node in the Ω, the node can be selected as a transmission node to transmit data. If all nodes in Ω transmit program codes once, all nodes in the network will receive program codes. Thus, the number of transmission is the number of nodes in the Ω. Thus, the number of transmission times is as follows.

$$ \Xi =\sum_{k=0}^{\left| T\right|-1}\mid {\mathrm{Y}}_k\mid $$

5.2 Analysis of broadcast delay

Theorem 4. The number of nodes with k active time slot is N k . Considering node v i at l m away from sink, \( {\Lambda}_k^i \) is the set of neighbor nodes with k active time slot of node v i . \( {\tau}_{\epsilon}^i \) is the maximum active time slots of nodes in the set \( {\Lambda}_k^i \), For example, if the active time slots of node A is 0, and the active time slots of node B is 1, the \( {\tau}_{\epsilon}^i \) is 1. \( \mid {\Lambda}_k^i\mid \) is the number of nodes in the set \( {\Lambda}_k^i \). \( {\mathrm{Y}}_k=\left\{{v}_{i_1},{v}_{i_2},{v}_{i_3},\dots {v}_{i_n}\right\} \) is the set of nodes, and the \( \mid {\Lambda}_k^{i_1}\mid \)<\( \left|{\Lambda}_k^{i_2}\right| \)<\( \mid {\Lambda}_k^{i_3}\mid \)<\( \mid {\Lambda}_k^{i_4}\mid \)<… \( {\Lambda}_k^{i_n} \) and ∣Γ k ∣ = N k , where the \( {\Gamma}_k=\left\{{\Lambda}_k^{i_1},{\Lambda}_k^{i_2},{\Lambda}_k^{i_3},\dots {\Lambda}_k^{i_n}\right\}. \) Ω = {Υ1, Υ2, Υ3 … ΥT}, considering the transmission delay is ℷ, it can be calculated as follows.

$$ \mathrm{\gimel}=\sum_{i\subseteq \Omega}{\mathfrak{H}}_i\kern2em \left\{\begin{array}{c}{\mathfrak{H}}_i={\tau}_a^i-{\tau}_{\beta_1}\kern1.25em if\kern1em {\tau}_a^i<{\tau}_{\epsilon}^i\ \\ {}{\mathfrak{H}}_i={\tau}_c-{\tau}_{\epsilon}^i+{\tau}_a^i\kern2em else\end{array}\right. $$
(16)

Proof. In the ADCFD scheme, the broadcast backbone is built based on the minimum covering node sets. The total minimum covering nodes is ∣Ω∣.

If one node wants to send program codes, the node senses the status of neighbor nodes. If some neighbor nodes are in the active time slots, program codes can be transmitted to neighbor nodes. If active time slots of neighbor nodes are early than active time slots of itself nodes, program codes can be transmitted to neighbor nodes in the next working cycle, or program codes are sent to neighbor nodes in this working cycle. For example, for node v i in Ω, if \( {\tau}_a^i \)<\( {\tau}_{\epsilon}^i \), the time for transmitting program codes is \( \Big({\tau}_a^i \)-\( {\tau}_{\beta_1} \)). But if \( {\tau}_a^i \)>\( {\tau}_{\epsilon}^i \), the time for transmitting program codes is \( \Big({\tau_c-\tau}_{\epsilon}^i+{\tau}_a^i \)). Thus the transmission delay of the network is the time for transmitting program codes of nodes in the network. The total number of minimum covering nodes in the set is Ω. Thus, the number of transmission is as follows.

$$ \gimel =\sum_{i\subseteq \Omega}{\mathfrak{H}}_i\kern2em \Big\{\begin{array}{c}{\mathfrak{H}}_i={\tau}_a^i - {\tau}_{\beta_1}\kern1.25em \mathrm{if}\kern1em {\tau}_a^i<{\tau}_{\epsilon}^i\ \\ {}{\mathfrak{H}}_i={\tau}_c - {\tau}_{\epsilon}^i+{\tau}_a^i\kern2em \mathrm{else}\end{array}\operatorname{} $$

5.3 Analysis of energy and network lifetime

Considering node v i at l m away from the sink in a data collection period τ c , the energy consumption E l of node v i is as follows.

$$ {E}^i={e}_{s en}^i+{\varpi}_r^i{\mathrm{Q}}_r^i+{\varpi}_s^i{\mathrm{Q}}_s^i $$
(17)

Theorem 5. Considering node v i at l m away from the sink in a data collection period τ c , the initial energy of node v i is E init , the network lifetime l i of node v i is as follows.

$$ {l}_i={E}_{i nit}/\left({e}_{s en}^i+{\varpi}_r^i{\mathrm{Q}}_r^i+{\varpi}_s^i{\mathrm{Q}}_s^i\right) $$
(18)

Proof. It can be seen that the energy consumption of node v i at l m away from the sink is E l, and the initial energy consumption of node v i at l m away from the sink is E init . The network lifetime is the rounds of transmitting data of node v i . Thus, the network lifetime l i of node v i is as follows.

$$ {l}_i={E}_{i nit}/{E}^i={E}_{i nit}/\left({e}_{s en}^i+{\varpi}_r^i{Q}_r^i+{\varpi}_s^i{Q}_s^i\right)\square $$

Inference 1. Considering effective energy utilization rate in the ADCFD scheme is φ A , and effective energy utilization rate in the ADCFD scheme is φ L . The ratio of effective energy utilization rate in the ADCFD scheme and effective energy utilization rate in the LABS scheme is as follows.

$$ \uppsi =\frac{\upvarphi_A}{\upvarphi_L}=\frac{\sum_{1\le i\le n}\left({e}_{sea}^i+{\varpi}_{ra}^i{Q}_{ra}^i+{\varpi}_{sa}^i{Q}_{sa}^i\right)}{\sum_{1\le i\le n}\left({e}_{sel}^i+{\varpi}_{rl}^i{Q}_{rl}^i+{\varpi}_{sl}^i{Q}_{sl}^i\right)} $$
(19)

Proof: \( {\varpi}_{ra}^i \) is the power used by node v i to receive a packet in the ADCFD scheme, \( {e}_{sea}^i \) is the energy consumption for sensing data in the ADCFD scheme, and \( {\varpi}_{sa}^i \) is the power used by node v i to send a packet in the ADCFD scheme. \( {\varpi}_{rl}^i \) is the power used by node v i to receive a packet of node v i in the LABS scheme, \( {e}_{sel}^i \) is the energy consumption in the sleep status in the LABS scheme, and \( {\varpi}_{sl}^i \) is the power used by node v i to send a packet in the LABS scheme. It is calculated as follows.

$$ \psi =\frac{\varphi_A}{\varphi_L}=\frac{\left({\sum}_{1\le i\le n}\ {e}_A^i\right)/\left({\sum}_{1\le i\le n}\ {E}_{\mathrm{init}}\right)}{\left({\sum}_{1\le i\le n}\ {e}_L^i\right)/\left({\sum}_{1\le i\le n}\ {E}_{\mathrm{init}}\right)}=\frac{\sum_{1\le i\le n}\left({\varpi}_{sea}^i+{\varpi}_{ra}^i{Q}_{ra}^i+{\varpi}_{sa}^i{Q}_{sa}^i\right)}{\sum_{1\le i\le n}\left({\varpi}_{sel}^i+{\varpi}_{rl}^i{Q}_{rl}^i+{\varpi}_{sl}^i{Q}_{sl}^i\right)} $$

The energy consumption in two schemes is given in the Fig. 15. It can be seen that (1) the energy consumption of nodes near to the sink is higher than the energy consumption of nodes far from the sink. (2) The maximum energy consumption in the ADCFD scheme is the same as the maximum energy consumption in the LBAS scheme. (3) The energy consumption of nodes in the non-hotspots area in the ADCFD scheme is higher than the energy consumption of nodes in the non-hotspots area in the LBAS scheme. The reason is that the duty cycle of nodes in the ADCFD scheme is adjusted according to energy left of nodes. If nodes far from the sink have much energy left, the duty cycle of nodes can be increased a lot. In this paper, the duty cycle is the ratio of active period to the working period. The energy consumption may be higher if duty cycle of nodes is bigger. But the energy consumption of nodes near to the sink in the ADCFD scheme isn’t higher than the energy consumption of nodes near to the sink in the LBAS scheme. It can be known that ADCFD scheme can retain network lifetime.

Fig. 15
figure 15

The energy consumption in two schemes

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The energy consumption and network lifetime under different duty cycle in two schemes are given in the Fig. 16 and Fig. 17, respectively. The total energy consumption increases with the increases of duty cycle. Network lifetime in the ADCFD scheme isn’t smaller than network lifetime in the LBAS scheme, but the effective energy utilization in the ADCFD scheme is greater than that of the LBAS scheme (seen from Fig. 18). The reason is that in an ADCFD scheme, duty cycle of nodes far from the sink is increased according to energy left of nodes, and duty cycle of nodes near to the sink in the ADCFD scheme are the same as duty cycle of nodes near to the sink in the LBAS scheme. Thus, the effective energy utilization in the ADCFD scheme is greater than that of LBAS scheme. It shows that ADCFD scheme has better performance.

Fig. 16
figure 16

The total energy consumption under different duty cycle in two schemes

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Fig. 17
figure 17

The network lifetime under different duty cycle in two schemes

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Fig. 18
figure 18

The effective energy utilization rate under different duty cycle in two schemes

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The energy consumption and network lifetime under different r in two schemes are given in Fig. 19 and Fig. 20, respectively. From Fig. 10 and Fig. 20, (1) the energy consumption decreases with the increase of r. (2) The total energy consumption in the ADCFD scheme is higher than the total energy consumption in the LBAS scheme. (3) Network lifetime in the ADCFD scheme isn’t lower than network lifetime in the LBAS scheme. The main invention of this paper is to increase duty cycle of nodes far from the sink. In the above analysis, there are much energy left of nodes far from the sink, the energy left of nodes is used to improve duty cycle of nodes. Thus, the energy consumption of nodes far from the sink in the ADCFD scheme is increased. But the energy consumption of nodes near to the sink in the ADCFD scheme isn’t higher than the energy consumption of nodes near to the sink in the LBAS scheme. Thus, the total energy consumption in the ADCFD scheme is higher than that of LBAS scheme, while the ADCFD scheme can retain the network lifetime.

Fig. 19
figure 19

The total energy consumption under different r in two schemes

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Fig. 20
figure 20

The network lifetime under different r in two schemes

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The effective energy utilization rate is showed in Fig. 21. The effective energy utilization rate in the ADCFD scheme is higher than that of LBAS scheme. The improved effective energy utilization rate in the ADCFD scheme is 1.83713–1.93563 greater than that of LBAS scheme (see from Fig. 22).

Fig. 21
figure 21

the effective energy utilization rate under different r in two schemes

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Fig. 22
figure 22

the ratio of effective energy utilization rate under different r in two schemes

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The energy consumption and network lifetime under different R in two schemes are given in the Fig. 23 and Fig. 24, respectively. From Fig. 23 and Fig. 24, the total energy consumption in the ADCFD scheme is higher than the total energy consumption in the LBAS scheme. Network lifetime in the ADCFD scheme isn’t smaller than network lifetime in the LBAS scheme. The effective energy utilization rate is showed in Fig. 25. The effective energy utilization rate in the ADCFD scheme is higher than that of LBAS scheme.

Fig. 23
figure 23

the total energy consumption under different R in two schemes

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Fig. 24
figure 24

the network lifetime under different R in two schemes

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Fig. 25
figure 25

the effective energy utilization rate under different R in two schemes

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5.4 Experimental results and analyses

The experiments were conducted using Omnet++ platform, which is widely recognized in academia [22, 37]. OmneT++ is popular in academia for its extensibility (due to its open source model) and plentiful online documentation. Omnet++ represents a framework approach. Instead of containing explicit and hardwired support for computer networks or other areas, it provides an infrastructure for writing such simulations. Without loss of generality, the network parameters are as follows: radius R =400 m, packet transfer radius r =50 m, event generation probability of node in one round λ =0.1, 1000 nodes are deployed, i.e., τ a =0.1. Other parameters settings refer to Table 1.

5.5 Analysis of broadcast times

The number of broadcast under |T| = 20 and |T| = 60 are showed in Fig. 26 and Fig. 27, respectively. As can be seen, the number of transmission in ADCFD scheme is smaller than that of LBAS scheme. And the number of broadcast of two schemes generally increases with the network size increases. That is obvious because the |T| is determined in advance, with the number of node increases, a node may be selected as covering node at many duty cycle. Program codes is spread to all nodes in network, thus the number of broadcast increases. The number of transmission in the ADCFD scheme is improved by 44.776%- 118.519% (see from Fig. 28).

Fig. 26
figure 26

The impact of network size with fixed value of |T| = 20 on the total number of broadcast

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Fig. 27
figure 27

The impact of network size with fixed value of |T| = 60 on the total number of broadcast

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Fig. 28
figure 28

The ratio of the number of broadcast for ADCFD scheme VS LBAS scheme with fixed value of |T| = 60

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The total number of broadcast under |T| = 80 and |T| = 120 are given in Fig. 29 and Fig. 30, respectively. It can be seen that the number of broadcast in ADCFD scheme is smaller than that of LBAS scheme. The reason is that in the LBAS scheme, in order to make the number of transmission minimum, a broadcast tree is built. The first phase is to find the minimum covering node set, then constructing broadcast backbone. A broadcast backbone contains all covering nodes. A covering node has many neighbor nodes, those neighbor nodes have different active state. Thus, when program codes are spread to all nodes along the broadcast backbone, each node has different active status, one node will be broadcast several times to neighbor nodes. But duty cycle of nodes in the ADCFD scheme are adjusted, the covering sets are selected using the method as the LBAS scheme. The active time of neighbor nodes of a node is increased, and the active time is earlier. Thus one node can only broadcast once to broadcast program codes to all neighbor nodes to reduce the number of transmission.

Fig. 29
figure 29

The impact of network size with fixed value of |T| = 80 on the total number of broadcast

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Fig. 30
figure 30

The impact of network size with fixed value of |T| = 120 on the total number of broadcast

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The total number of broadcast under fixed 200 and 400 network nodes are showed in Fig. 31 and Fig. 32, respectively. Whether the network nodes is 200 or 400, the number of broadcast in ADCFD scheme is smaller than that of LBAS scheme. In Fig. 31, when |T| = 60, the number of broadcast in ADCFD scheme is 0.69072 times than that in LBAS scheme. But when |T| = 20, the number of broadcast in ADCFD scheme is 0.45763 times than that in LBAS scheme. In Fig. 32, when |T| = 40, the number of broadcast in ADCFD scheme is 0.54335 times than that in LBAS scheme. But when |T| = 80, the number of broadcast in ADCFD scheme is 0.67925 times than that in LBAS scheme.

Fig. 31
figure 31

The impact of |T| with fixed network nodes = 200 with fixed on the total number of broadcast

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Fig. 32
figure 32

The impact of |T| with fixed network nodes = 400 with fixed on the total number of broadcast

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The number of broadcast under fixed 600 and 800 network nodes are given in Fig. 33 and Fig. 34, respectively. Whether the network nodes is 600 or 800, the number of broadcast in ADCFD scheme is smaller than that of LBAS scheme. It shows that the ADCFD scheme can reduce the number of broadcast, it shows that ADCFD has better performance.

Fig. 33
figure 33

The impact of |T| with fixed network nodes = 600 with fixed on the total number of broadcast

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Fig. 34
figure 34

The impact of |T| with fixed network nodes = 800 with fixed on the total number of broadcast

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5.6 Analysis of broadcast delay

The broadcast delay under |T| = 20 and |T| = 60 are showed in Fig. 35 and Fig. 36, respectively. The broadcast delay in ADCFD scheme is smaller than that in LBAS scheme.

Fig. 35
figure 35

The impact of network size with fixed value of |T| = 20 on the broadcast delay

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Fig. 36
figure 36

The impact of network size with fixed value of |T| = 60 on the broadcast delay

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The impact of network size with fixed value of |T| = 80 and |T| = 120 on the broadcast delay are shown in Fig. 37 and Fig. 38, respectively. It can be seen that the delay in ADCFD scheme is smaller than that in the LBAS scheme.

Fig. 37
figure 37

The impact of network size with fixed value of |T| = 80 on the broadcast delay

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Fig. 38
figure 38

The impact of network size with fixed value of |T| = 120 on the broadcast delay

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The reason is that in the LBAS scheme, in order to make the number of broadcast minimum, a broadcast tree is built. The first phase is to find the minimum covering node sets, then build broadcast backbone. A broadcast backbone contains all covering nodes. A covering node has many neighbor nodes, those neighbor nodes have different active states. Thus, when program codes are spread to all nodes along broadcast backbone, one node will be broadcast several times to neighbor nodes due to different active states of neighbor nodes. But duty cycle of nodes in the ADCFD scheme are adjusted, the covering sets are selected using the method in the LBAS scheme. The active time of neighbor nodes of a node are adjusted, the active time of neighbor nodes of a node are the same, and the active time are earlier. Thus, one node can only broadcast program codes once to reduce broadcast time.

The transmission delay under fixed value of 200 and 400 network nodes are given in Fig. 39 and Fig. 40, respectively. As can be seen, the delay in ADCFD scheme is smaller than that of LBAS scheme. And the delay of two schemes generally increases as the network size increases. That is obvious because the |T| is determined in advance, with the number of node increases, a node may be selected as covering node at many duty cycle, and program codes is spread to all nodes in network, thus, the delay increases.

Fig. 39
figure 39

The impact of |T| with fixed network nodes = 200 with fixed on the delay

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Fig. 40
figure 40

The impact of |T| with fixed network nodes = 400 with fixed on the delay

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The transmission delay under fixed value of 200 and 400 network nodes and network nodes are shown in Fig. 41 and Fig. 42, respectively. It can be seen that the delay in ADCFD scheme is smaller than that of LBAS scheme. The reason is the same with the above. One node can only broadcast once to broadcast program codes to all neighbor nodes to reduce broadcast time. Transmission delay from dominating program codes is reduced by 17.895%- 107. 527% (see from Fig. 43).

Fig. 41
figure 41

The impact of |T| with fixed network nodes = 600 with fixed on the total number of transmissions

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Fig. 42
figure 42

The impact of |T| with fixed network nodes = 800 with fixed on the total number of transmissions

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Fig. 43
figure 43

The impact of |T| with fixed network nodes = 800 with fixed on the total number of transmissions

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6 Conclusion

There are many researches about Software defined wireless networks (SDWNs), it has great significance for configuring automatically and upgrading software in industrial and environmental monitoring fileds. The fast transmission for program codes becomes a important content. In the previous schemes, the methods are adopted to optimize scheduling algorithm in the newtork with the determined duty cycle, which reduce the transmission delay and the energy consumption. An adjustable duty cycle based fast disseminate (ADCFD) scheme is proposed for minimum-transmission broadcast (MTB) in smart wireless software-define sensor network. The difference of ADCFD scheme with the previous scheme is to increase the duty cycyle of nodes using energy left of nodes, thus, the transmission times and transmission delay are reduced while retaining network lifetime. The more important is that this method in this paper can be applied to many methods, which can improve the performance of previous schemes. Thus, it has a very good meaning.