Maximum Network Throughput Based on Cross-Technology Communication for Sensor Networks

Demin Gao,Zhihao Guan,Shuo Zhang,Bin Hu

1 College of Information Science and Technology,Nanjing Forestry University,Nanjing 210037,China

2 Department of Computer Science and Engineering,University of Minnesota,55414,USA

3 College of Artificial Intelligence,Nanjing Agriculture University,Nanjing 210095,China

Abstract:The exponentially increasing number of heterogeneous Internet of Things(IoT)devices(e.g.,WiFi and ZigBee)crowed in the same ISM band(2.4G)and recent advances in Cross-Technology Communications(CTC)motivate us to explore more efficient data collection and maximize network throughput.CTC enables WiFi and ZigBee devices to communicate directly without any hardware changes or gateway equipment,which sheds light on a more efficient data collection design.In this work,we propose a distributed algorithm,named MaxBee,to compute the maximum network throughput,which is formulated as a linear programming problem.Considering that the problem turns out to be non-convex and hard to solve exactly,we propose a distributed algorithm to solve nonlinear programming by using the dual decomposition method and gradient/subgradient algorithms.Through extensive simulations on different sets of deployed ZigBee and WiFi devices,we observe that the proposed algorithm significantly increases the network throughput based on CTC for Sensor Networks.

Keywords:network throughput;cross-technology communication;sensor networks

I.INTRODUCTION

With the prosperity of various embedded sensors,lowpower wireless communication,and efficient signal processing techniques,the Internet of Things(IoT)has achieved explosive development and proliferation in recent years[1].According to Gartner’s report,there are approximately 8 billion connected things providing smart services in our daily life[2].The wireless performance in such a large-scale networked system,however,could be severely degraded when weak IoT devices operate with heterogeneous wireless technologies[3](e.g.,WiFi,ZigBee and Bluetooth)that share the same spectrum competing to access the same spectrum[4],e.g.,2.4G ISM(industrial,scientific and medical)and generate cross-technology interference as no direct communications between these technologies are allowed in traditional designs:the wireless traffic generated by different techniques will collide with each other and result in frequent transmission failure[5—9].

Nowadays,Cross-Technology Communications(CTC),e.g.,Freebee[10],HoWiEs[11]opens a new direction of direct communication among different wireless technologies when they operate in the same spectrum band.For example,both WiFi and ZigBee reside on the 2.4 GHz ISM band and thus WEBee[12],TwinBee[13],LongBee[14]uses a high-speed WiFi radio to emulate the desired signals of a lowspeed ZigBee radio.ZigFi[15]uses the channel state information of the overlapped ZigBee packets with WiFi packets to convey data from ZigBee to WiFi.BlueBee[16]emulates legitimate ZigBee frames using a Bluetooth radio for achieves dualstandard compliance.These technologies are purely a software-based solution,requiring no modification on WiFi or ZigBee or Bluetooth hardware[17].It can work on Off-the-Shelf WiFi,ZigBee and Bluetooth devices.

Traditionally,communication between different wireless technologies is achieved indirectly via gateways equipped with multiple radio interfaces[18],which will become a bottleneck when the exponentially increasing number of heterogeneous IoT devices are deployed.CTC[19]lets a WiFi AP(Access Point)undertake the tasks of the gateway and using WiFi devices as gateways are particularly effective from the consumer point of view since 1)Existing WiFi hotspots can be employed and we do not need special action to launch it,2)the WiFi AP can be implemented simply collecting the packets from ZigBee devices and communicating with other WiFi APs together via TDMA for high throughput,and 3)for the user,the incurred cost is almost negligible but obtains significant benefits.

The advances in these CTC technologies provide new opportunities for maximum throughput for sensor networks.Henceforth,in this paper,we address the maximum network throughput problem based on packet-layer CTC for sensor networks,a distributed algorithm named MaxBee,(Maximum throughput using WiFi to assist ZigBee).Specifically,we seek to calculate the upper bound of network throughput by resolving linear programming and output maximum throughput.

Our contributions are summarized as follows:

·We propose MaxBee,maximum network throughput based on CTC technology,where,the packets generated by ZigBee devices will be forwarded to any one WiFi device rather than a sink.

·We formulate the network throughput problem as an optimization model considers the effective packets distribution in a heterogeneous network,where WiFi APs collect the packets from ZigBee devices.

·A mixed integer linear programming optimization model is proposed for maximum network throughput.Due to the NP-Hardness of the optimization models,an efficient algorithm with polynomial run-time complexity are designed to cope with a practically implementable scenarios.

The remainder of this paper is organized as follows.Section II discusses the related work,while in Section III specify the network and system model.Then we elaborate on the design of MaxBee for maximum network throughput in Section IV.Section V presents the analysis and evaluation results.We conclude this work in Section VI.

II.RELATED WORK

A significant quantity of works were done to maximize network throughput for sensor networks;e.g.,[20—23].A degree-constrained routing tree is proposed for enhancing the data collection rate[24].Chawla et al.[25]provide topology control algorithms based on transmission-power and duty-cycle schemes to reduce energy consumption and heighten throughput.Yatika et al.[26]improve the SPIN protocol with a congestion vector to increase network throughput and life.In[27],a novel optimization problem is formulated for maximizing the network throughput.MaxPut[28]employs the appropriate combination of random and scheduled duty cycling schemes to maximize network throughput.Other examples can be found in[29—35],and references therein.These algorithms encounter critical tradeoffs between network throughput and lifetime due to energy constraints.

Optimization techniques generally utilize maximum network throughput to find the best available values of some objective function given a defined domain[36],which is widely used for achieving maximum network lifetime by formulating it as a linear programming problem[37].In[38],three protocols are introduced for maximizing the sum-throughput,namely,the timedivision,hybrid power splitting/data decoding protocol,and backscatter- enabled combination protocol.In[39],a general framework is proposed for network throughput maximization problem by optimizing practically feasible parameters and a mixed-integer linear programming optimization model is introduced for the problem formulation.In[40],a fine-grained backpressure message ferrying algorithm is provided for optimal throughput with motion and transmission control of robots.

CTC technique is proposed in ESence[41]firstly by injecting data packets of specified length and encoding CTC symbols via packet lengths.GSence[42]perpends a customized preamble in front of legacy data packet to implement CTC.FreeBee[10]encodes CTC symbols in the timings of mandatory beacons without introducing dedicated packets.HoWiES[11]uses ZigBee radio to wake up a WiFi interface when it detects that a WiFi network is available.WEBee[12]allows direct communications from WiFi to ZigBee without any modification on hardware,and NetCTC[43]introduces a real-time interaction mechanism which achieves reliable,transmission efficient and concurrent interactive communication among heterogeneous devices,besides many others such as ZigBeeto-WiFi[15],Bluetooth-to-WiFi[44],Bluetooth-to-ZigBee[16]and LTE-to-WiFi[45].With the rapid development,direct communication(e.g.,Bluetooth-to-WiFi,ZigBee-to-Bluetooth,ZigBee-to-LTE)among all these heterogeneous devices also will be achieved without a centralized gateway.

For providing efficient communication in heterogeneous networks,researchers have also proposed various techniques to improve the spectrum utilization and the performance of different wireless systems[46,47].Based on CTC,Chiron[18]proposes a strategy for physical layer concurrent high throughput communication to heterogeneous IoT devices.In[48],the authors present a light-weight and self-adapting Cross Technology Interference(CTI)mitigation strategy for improving the packet reception rate.For achieving efficient spectrum communication,a framework is introduced to address the CTI[49].Amphista[50]presents a novel cross-layer design for IoT communication and data forwarding that can more efficiently utilize the 2.4G spectrum.ECT[51]uses collaborative and concurrent cross-technology communication between WiFi and ZigBee devices for reducing packet delivery delay.B2W2[52]enables N-way concurrent communication among WiFi and Bluetooth Low Energy(BLE)devices.DopplerFi[53]enables a twoway communication channel between BLE and WiFi by injecting artificial Doppler shifts.

These earlier works for maximum network throughput,one part of them focus on utilizing single communication technology and all nodes are homogenous,while another part of them provides the details about the algorithm to establish link paths for data transmission.Compared to these earlier works,the main difference is that we seek to calculate the upper bound of network throughput theoretically based on CTC for heterogeneous IoT devices rather than design communication links and provide the details of communication for packets delivery.In our work,we formulate the maximum throughput of sensor networks as optimization Linear Programming(LP)firstly.Secondly,the LP problem is converted to a dual problem with Lagrange multipliers for the power and flow constraint.Finally,we use the subgradient algorithm to solve the dual problem.

III.SYSTEM MODEL

3.1 Network Model

The Internet of Things model can be represented as an undirected graph Γ =(V,A),where the set of verticesVcorresponds to network nodes.We letZandWdenote the set ofnZigBee nodes andkWiFi APs,respectively,where,V=Z ∪W.Ais the set of links,and,A={A|(Zi,Zj),(Zi,Wk)∈A;Zi,Zj,Wk ∈V}.Γ consists of a finite nonempty vertex setVand edge setAof ordered pairs of distinct vertices ofV.Each ZigBee deviceZi ∈Zhas a transmission rangeℜZand rechargeable battery with capacityCv.One ZigBee deviceZican transmit packets to the otherZjsuccessfully and directly if the Euclidean distance||Zi-Zj||betweenZiandZjsatisfies the necessary condition||Zi-Zj||≤ℜZ.

We note that the transmission distance of a WiFi AP is longer than that of ZigBee devices.The maximum transmission power of the WiFi device is up to 100 mw(20 dBm)and the transmission range is near 300 m[14],while that of ZigBee device(i.e.,MICAz)is generally 1 mw(0 dBm)and the maximum transmission distance is below 70 m[54].Therefore,a WiFi node can easily check a region of ZigBee nodes,where,within the transmission range of ZigBee nodes,the links between ZigBee devices and WiFi devices are bi-directional,while,within the transmission range of WiFi nodes and beyond the transmission range of Zig-Bee nodes,the communications between ZigBee devices and WiFi devices are unidirectional.The maximum bandwidth of nodeZiis set to beRi.

3.2 Energy Consumption Model

Letwidenote the fraction of energy expenditure per unit time for ZigBee nodeZi,which can be written as:

For a ZigBee device,its energy expenditure is always to a lesser degree than the remaining energy.For achieving maximum network lifetime,the energy expenditure is managed strictly to achieve energy-neutral operation in an energy-harvesting network,where energy can be republished with a lower rate due to the sporadic availability of energy.Therefore,the scheme of maximum network throughput is constrained by the limited energy supplement.In the following section,we introduce the method and design for maximizing network throughput.

IV.METHOD AND DESIGN

4.1 Description of the Data Flow Model

In this section,the data flow model is described for a heterogeneous network,where,ZigBee and WiFi devices coexist in the group and interact with each other.As a receiver,the ZigBee device can receive packets from WiFi APs and adjacent ZigBee devices.For data forwarding,the ZigBee device will send its packets to other nodes or WiFi APs directly.The packets from a ZigBee device will be forwarded to anyone WiFi AP by one or multiple-hops finally.Thus,our network has three types of data transmission:i)WiFi to Zig-Bee(W2Z);ii)ZigBee to ZigBee(Z2Z);iii)ZigBee to WiFi(Z2W).For each device,the outflow equals the sum of generation data and inflow data.Figure 1 plots the process of packet transmission for a ZigBee device.We note that Figure 1 is only utilized for illustrating the principle of data flow model rather than the number of devices.In Figure 1,data from ZigBee device or WiFi AP will be forwarded to another ZigBee device until reaches the destination.In the scenario,WiFi AP,as the final destination,is used for collecting the data.Therefore,in general scenario,the packets generated by a ZigBee devices will be forwarded to another ZigBee device.For the sake of simplicity,the phenomenon of packet leak is not considered in our model.

Figure 1.The data flow for a ZigBee device.

where,μj= 0,1,ωj= 0,1,μj ⊕ωj= 1,which indicates whenμj=0,ωj=1,vice versa.Eq.(3)demonstrates that if ZigBee deviceZiconstructs a routing path with a WiFi AP,it will forward all packets to the WiFi AP,while the WiFi AP is out of connection,it has to send data to neighbor nodes.The ZigBee device’s energy consumption includes sensing/generating packets,idle,channel listening,packet transmission,and reception.The energy utilized for generating packets,idle and listening channel is fixed and small,which is neglected in our work.In our work,we maximize the data generation rate of each sensor to achieve maximum network throughput.The problem as a linear programming can be formulated as:

where,the first constraint indicates that the outflow equals the sum of generation data and inflow data.The second constraint ensures that the total energy expenditure for receiving and sending packets over the lifetime must not exceed the total available energy.The third constraint represents the throughput for a ZigBee device per unit time must not exceed its capacity.The fourth constraint ensures that the data traffic is integers.The linear programming problem for Eq.(5)is NP-hard and considerably challenging to be solved directly.Therefore,we propose a dual model to replace the original problem.

4.2 Optimization Technology for Maximizing Throughput

Based on the Eq.(5),the first constraint ensures the data traffics conservation at each device.The optimal objective can be adjusted toQ= 1/Gand an equivalent linear programming formulation is obtained.The linear programming problem in Eq.(5)can be rewritten as:

The optimization problem of maximum network throughput is converted to the minimum problem of minimizing the data collection rate.We can interpret the above problem as minimizing the maximum power consumption ratio to energy supply at a ZigBee device.The linear programming problem for Eq.(5)is NP-hard.Therefore,we introduce a dual model to replace the original problem and the dual model can be solved in a distributed manner.

Consider a convex optimization problem(also referred to as a mathematical programming problem or minimization problem),with variablex ∈D ⊆Rn,whereDis a convex set andD ⊆Rnis the feasible set.w0(x):Rn →Ris the objective,is called convex ifDis a closed convex set andw0(x)is convex onRn.

wherew0,wi’s are convex functions andvi’s are affine functions ofx.Thew0is strictly convex,andDis polyhedral.The Lagrangian,forλ ≥0,is given by

4.3 Dual Problem

The dual problem is constructed by bring Lagrange multipliersλifor the power constraint andνifor the conservation constraint of data traffic at each devicei.Thus,the results in the Lagrangian can be formulated as:

Since the Eq.(11)is equals with the Eq.(5),the original minimum problem can be converted to the lagrangian minimax problem.In the process of resolving the lagrangian minimax problem,whose optimal value is no more than that of the original minimum problem.We define the original issue as follows for a dual problem,whose optimal value is no higher than that of the original problem if they have the optimal value.

Proof.we assume the optimal value for the original minimum problem is

According to Eq.(11)and Eq.(13),for any values ofQ,f,λ,ν,the following inequality holds

For resolve the original problem by formulating the dual problem,we should ensure the objective function is strictly convex in the primal variables of original problem in Eq.(5),which can guaranteep*=d*=L(Q,f,λ,ν).

4.4 Subgradient Algorithm

Therefore,when the value ofσis enough small,for the regularized problem,the data generation rateGiis closer to the optimal value given by problem Eq.(5).The range ofQis set to be[0,Q′],where,Q′is a loose upper bound on the value ofQ.For givenλ,ν,the dual problem is given by,

Algorithm 1.in ZigBee device i(distributed).Algorithm1:in ZigBee device i(distributed)Input:iteration step k;Output:1: k ←1 2:While not converged do 3:Solve the problem in(18)to get QkZi,fZ2ZZiZj for Zj ∈Si 4:Exchange the values of fZ2ZZiZj with neighboring nodes Zj ∈Si 5:set λ(k+1)i=(λ(k)i -θkγ(k)λi )+,ν(k+1)i=(ν(k)i -θkδ(k)νi )+6:Exchange the values of λ(k+1)iand ν(k+1)iwith neighbor nodes j ∈Si.7: k ←k+1 8:End While 9:Return λ*i and ν*i.

V.SIMULATIONS

In this section,we extensively evaluate the performance of MaxBee across various domains under a wide range of settings,such as CTC performance comparison,communication reliability and low-duty cycle,and two demo applications of coexistence between ZigBee and WiFi devices.

5.1 Experimental Setup

To evaluate the theoretical derivations of MaxBee,we implement it on Matlab software firstly,with up to100-200 ZigBee devices and 1-4 WiFi devices are randomly deployed in a 500m*500msquare field.The maximum communication ranges for ZigBee and WiFi are set to be 50mand 300m,respectively.The intensive set of simulation is performed based on the parameter illustrated in Table.1.

Table 1.Simulation Parameters.

5.2 System Implement

We can currently exploit the existing WiFi infrastructure expediently and employ these existing WiFi devices to collect data generated by ZigBee devices with nearly zero cost.Therefore,the gateway,known as a protocol converter,can be replaced by the WiFi APs and the cost can be reduced significantly.In the experiment scenario,WiFi AP has a long transmission range and can check multiple ZigBee devices in a large square of area.Therefore,WiFi AP distributes commands to these ZigBee nodes directly without data forwarding.ZigBee devices generate packets by sensing the surroundings and send them to WiFi AP with one or multiple-hops transmission.The connectivity graph for multiple ZigBee nodes and one WiFi AP is shown in Figure 2.

All packets collected by ZigBee devices will be forwarded to this WiFi AP,which perhaps forms a bottleneck in a large-scale network.In our practical application,e.g.,smart home,it is typical for a ZigBee node to be checked by multiple WiFi APs.Figure 3 plots that multiple WiFi APs were deployed in the scenario.The advantages of multiple WiFi APs deployed contain that,one the one hand,all packets generated by ZigBee devices only needed to be forwarded to anyone closest WiFi AP.On the other hand,it will relieve packets transmission pressure,avoiding a bottleneck for data forwarding,which will finally reduce energy expenditure and improve the data generation rate.

Figure 2.Connectivity graph with 200 ZigBee nodes and 1 WiFi device.

Figure 3.Connectivity graph with 200 ZigBee nodes and 3 WiFi device.

Since the network throughput is on the influence of data generation rate significantly,we first analyze the data generation rate of each source node with distinct duty-cycles.The high duty-cycle indicates that a ZigBee device stays in an active state for more time and wakes up more frequently for sending or receiving packets or sensing information from the environment.Therefore,the data generation rate increases significantly with the duty cycle improved,as is shown in Figure 4.Figure 4 plots the data generation rates for a source node when multiple WiFi APs are deployed and the duty cycles are set from 1%to 10%.The scheme with multiple WiFi APs can improve the efficiency of data collection.Hence,the data generation rates in a scenario with multiple WiFi APs are higher than that of single WiFi AP.

Figure 4.The data generation rates for a sensor when the duty cycle from 1%to 10%.

Figure 5 and Figure 6 show the network throughput with a distinct number of ZigBee devices and different duty-cycles when multiple WiFi APs are deployed.Since high sensor density means more packets will be sensed or generated,these packets will be transmitted to the WiFi devices finally.Therefore,the network throughput will be improved if more ZigBee devices are appended to the scenario.We have to note that,even though,for saving energy,ZigBee devices always operate with low duty cycles and data collations are scarce generally,in a dense network,when a larger number of nodes are deployed in a narrow field,the data collision will be severe and frequent retransmission is unavoidable,which leads to serious energy waste in return.For the sake of simplicity,in our work,an ideal scenario that a reasonable number of sensors are deployed is considered and the data collision is ignored.

In a dense-network or larger-scale network,for reducing data-collision probability,multiple WiFi APs are generally deployed in the monitoring field for sharing the responsibility of data transmission.From Figure 5 and Figure 6,we can observe that the network throughput can be improved significantly with multiple WiFi APs.The network throughput is taller about 39% with two WiFi APs and 70% with three WiFi APs than that of the one WiFi AP when the duty-cycles are about 8%,respectively.Notably,in a higher duty-cycle network or dense-network,the strategy with multiple WiFi APs provides better data collection performance.

Figure 5.Network throughput with number of nodes increasing when multiple WiFi APs are deployed.

Figure 6.Network throughput for the duty cycle from 1%to 10%.

For presenting the significant benefit of MaxBee,the spectrum utilization is evaluated in heterogeneous networks(ZigBee and WiFi devices coexist)in a realistic scenario,where 10 ZigBee compliant MICAz nodes are randomly deployed in a 100m ×100msquare field with 10%duty cycle for each ZigBee node.Figure 7 shows the comparison between the traditional multi-radio gateway and MaxBee.In the traditional multi-radio gateway approach,the gateway has to allocate WiFi and ZigBee packets into different time slots based on TDMA mode,which yields a relatively low spectrum utilization when there multiple ZigBee senders.For MaxBee,a WiFi AP can directly coordinate the spectrum allocation without extra gateway,and the spectrum utilization is slightly higher than the traditional multi-radio gateway.Especially,when more ZigBee devices are appended in the networks,MaxBee can provide better performance of the spectrum utilization.

Figure 7.Spectrum utilization for MaxBee and traditional gateway when the number of ZigBee devices increasing.

Figure 8.The topology of IoT with 7 ZigBee devices and 2 WiFi devices.

5.3 Numerical Result

In this section,the distributed algorithm is evaluated over an IoT network that is composed of seven Zig-Bee devices and two WiFi devices,as is shown in Figure 8.The USRP-N210 platform with 802.11 b/g PHY is utilized for simulating WiFi AP.Two eZ430-RF2500T target boards are used for simulating the ZigBee device and the CC2500 radio transceiver operates in the 2.4GHz band,whose MAC header and footer are the same as defined in IEEE 802.15.4.The intensive set of simulations is performed based on the parameter illustrated in Table.1.We assume that the distance between any two adjacent devices are equal,i.e.,d(Z1,Z2)=d(Z2,Z3).

Figure 9.The evolution of dual variables.

Figure 9 plots the convergence property of our algorithm.Here,we setλ= 0.1.For any ZigBee device,there are multiple routes for data delivery before the data reaching a WiFi AP.For instance,for Zig-Bee deviceZ7,there are at least two routes to reach a WiFi AP,which contains that{(Z7,Z1),(Z1,Ww1)}and{(Z7,Z5),(Z5,Ww2)}.Due to the symmetry in the position of ZigBee deviceZ7andZ6to with respect to the WiFi device,they have almost the same optimal dual variables.Figure 9 presents that dual variables converge fast despite the oscillations at beginning(after 100 iterations).

Figure 10 shows the data generation rate converging to the optimal value with number of iteration.For Zig-Bee deviceZ1,the generated data will be forwarded toW1with obvious reason,where the degree from Zig-Bee deviceZ1to WiFi deviceW1is 1,while it is at least 3 to WiFi deviceW2.For the ZigBee devicesZ1,Z4,Z3,Z5,they are also the symmetry in the positions.Thus,they also have almost the same dual variable and data generation rate and we observe that any four curves associated these four ZigBee devices in Figure 9 and Figure 10 almost coincide with them.

From Figure 8,we can see that the packets from ZigBee deviceZ2can be transmitted to WiFi deviceW1orW2with the same distance.The ZigBee deviceZ2can choose anyone as its destination.However,when it selects the deviceW1as its terminal point,the ZigBee devicesZ1andZ4are the bottleneck nodes that have the higher power dissipation than the other nodes.Therefore,after multiple iteration,part of packets from ZigBee deviceZ2will be forwarded to the deviceW1and remain packets will be transmitted to the deviceW2.

Figure 10.The evolution of data generation rates.

5.4 System Performance Comparison

We evaluate the performance of the optimization approach for maximizing network throughput based on CTC under network settings,where an algorithm is provided for performance comparison,called MaxPut[28]that aims to improve network throughput through the appropriate combination of random and scheduled duty cycling schemes.As a layer-based model,Max-Put attempts to identify risky nodes and enables risky nodes to maximize the utilization of active periods of their neighbors for alleviating the pressure of bursty data to some nodes.This method proposed in Max-Put was widely recognized and used in routing protocols design for improving network throughput in WSNs,which is attributed to the consideration for performance comparison in the work.

The network throughput between MaxBee and Max-Put is compared under the distinct number of ZigBee and WiFi devices,where for the ZigBee devices,the average duty-cycles are 1%,5%,10%,respectively,and the WiFi devices keep in active states perpetually with 100% duty-cycle.The result is shown in Figure 11.The network throughput of both algorithms increases gradually and our algorithm provides a slightly better performance comparing to that of MaxPut when more ZigBee devices appended in the scenario for all distinct duty-cycles.From Figure 11a,the sum network throughput of MaxBee is about 6% and 11%higher than that of MaxPut.We note that MaxBee calculates the upper bound of network throughput with ideal result and rather than building a real data path.The result calculated by MaxBee can be used for measuring the performance of design about network throughput.

Figure 11.Network throughput with number of nodes increasing when duty cycle is 1%,5%,10%.

VI.CONCLUSION

With the exponentially increasing number of IoT devices,more and more attention and efforts foster to more efficiently improve data collection efficiency and maximize network throughput in the crowed ISM band.Therefore,in the work,we propose a distributed algorithm to compute the upper of network throughput based on CTC for a heterogeneous network,where ZigBee and WiFi devices coexist.We first define the network system for ZigBee devices and WiFi APs.Hereafter,the linear programming for maximum network throughput is illustrated.Considering the linear programming is NP-hard,a dual problem by introducing Lagrange multipliers is constructed.Finally,a distributed algorithm is proposed to resolve the LP problem.Our simulation results show that MaxBee is efficient in maximizing network throughput based on CTC for Sensor Networks.

ACKNOWLEDGEMENT

This work was supported by The Project funded by China Postdoctoral Science Foundation(Grant No.2018T110505,2017M611828)and The Priority Academic Program Development(PAPD)of Jiangsu Higher Education Institutions.