An MAC Layer Algorithm Based on Power Line-Wireless Dual Media Channels and Multiplexing

Zhixiong Chen,Leixin Zhi,Peiru Chen,YinCheng Qi,2

1 School of Electrical and Electronic Engineering,North China Electric Power University,Baoding 071003,China

2 Hebei Key Laboratory of Power Internet of Things Technology,North China Electric Power University,Baoding 071003,China

*The corresponding author,email:zxchen@ncepu.edu.cn

Abstract:Hybrid networking of power line and wireless communication can complement each other,save construction costs,improve network robustness,and has important value on the Internet of Things and smart grid.For power line and wireless dual-interface devices,a media access control(MAC)layer algorithm for hybrid communication is proposed,in which the stations can compete for two channels simultaneously.When a station obtains two channels at the same time,the station will randomly select a channel for transmission to ensure fairness,and reset the counters when the transmission is successful.In this paper,the performance of the unsaturated traffic model and counter spatial division is considered,and the throughput under coupling conditions is obtained.Finally,the simulation results show that the MAC layer algorithm based on the hybrid communication network can make full use of the advantages of the two channels,improve system throughput,and reduce transmission delay.

Keywords:hybrid communication;MAC layer;Traffic model

I.INTRODUCTION

With the development of Internet of Things(IoT),the number of connected devices and business volume are substantially increasing,which places higher requirements and challenges on the existing network structure.Wireless communication(WLC)[1–5]is the key communication technology in the application of the IoT,but it has indoor coverage problems because the radio frequency signal in the indoor scene is easily blocked by obstacles and has a shadow effect.Power line communication(PLC)does not need to be rewired,which is a beneficial supplement to WLC[6].Therefore,considering the construction cost and the utilization of existing equipment,the hybrid networking of PLC and WLC can complement each other and improve system performance.

In order to study the hybrid power line and wireless communication(HPWC)performance,the PLC performance should be analyzed.Because most of the existing PLC research focuses on the modeling of the energy exchange[7]and time-varying channel characteristics,such as the influence of load time-varying behavior[8],electromagnetic interference[9],and impulse noise[10,11].In addition,the increase of stations will deteriorate the channel quality.Therefore,it is important to study the MAC layer of PLC,because studying the MAC layer can realize resource sharing and reduce unfair phenomena such as conflict and starvation[12–16].In[10],the author analyzed the bit error rate and outage probability for the wireless and power line two-hop system models,but did not involve the throughput analysis of the MAC layer.In[12],the purpose was to improve the efficiency of PLC MAC layer by reducing time slots in competition and conflict.However,the transmission probability of its MAC layer was not deduced.In IEEE 1901,the introduction of delay counters(DC)will lead to coupling between stations.Christinaet al.analyzed the performance of 1901 through a model that did not rely on decoupling assumption[13].In[14],Christina compared the fairness of the MAC layer between IEEE 802.11 and IEEE 1901 protocols.Because IEEE 1901 introduces delay counter,which reduces short-term fairness and generates instantaneous delays.Pittoloet al.indicated that the PLC was still popular in Internet of Vehicles in[15],because PLC could provide reliable data communication links,improving the reliability.However,these did not consider hybrid communication of WLC and PLC,and there was no analysis of the performance of the vehicle-PLC.The network layout of PLC and the TDMA protocol supported by the Home-Plug AV standard in[16].However,the complexity of performance analysis was greatly increased because relays were introduced.

Hybrid communication is a hot spot in IoT research.In[17],the authors analyzed the characteristics of HPWC architecture for Smart Grid and IoT application,and discussed the advantages of hybrid communication.Qianet al.focused on solving the problem that mobile wireless and PLC sensors cannot communicate with another in IoT in[18].The results of the above paper show that hybrid communication can improve the reliability and robustness of the system.However,the existing research mainly focuses on the physical layer and rarely on the MAC layer protocol in hybrid communication.In the study of the MAC layer of hybrid communication,a selective carrier sense multiple access with collision avoidance(CSMA/CA)algorithm was analyzed[19],and used simultaneous nonlinear equations to calculate system performance.However,its calculation was relatively complex,requiring multiple iterations to solve the equation system,and the scalability of the theoretical model was poor.

Because most papers on hybrid communication are related to the physical layer,and the existing algorithms proposed at the MAC layer are relatively complex and cannot be extended.Therefore,this paper proposes a multiplexing algorithm of HPWC for the MAC layer,which can make full use of the two channels and improve the system performance.The main contributions of this paper are as follows:1)Considering the fairness of the channels,a MAC algorithm based on power line and wireless is proposed.2)The coupling phenomenon between the two channels and its reasons are analyzed.3)Based on the Poisson traffic model,the Markov process and spatial counter partition are used to analyze the performance of the stations and the competition effect of the two channels,including transmission probability,collision probability,and system throughput.

Figure 1.System model.

The remainder of this paper as follows:The MAC layer scheme of WLC,PLC and hybrid communication,the Markov model,and the traffic model are introduced in Section II.The coupling effect and the performance of the system model are analyzed in Section III.The performance of the system is verified by simulation in Section IV.Finally,the work is summarized in Section V.

II.HPWC MAC PROTOCOL ANALYSIS

In this section,the differences between IEEE 802.11 and IEEE 1901 protocols and the MAC layer algorithm of HPWC are discussed.Figure 1 shows a network topology with N stations and a controller G.In this system,the stations have a power line–wireless interface,so they can compete for both power line and wireless channels.The stations use the CSMA/CA protocol on both power line and wireless channels,and all stations can communicate through the coordinator.

2.1 Differences between IEEE 1901 and 802.11 CSMA/CA Protocols

Figure 2.Markov model of backoff process in HPWC.

Although IEEE 1901 and IEEE 802.11 use CSMA/CA protocols in the MAC layer,they have several differences.The CSMA/CA protocol in IEEE 802.11 has only two counters[20],backoff phase counter(BPC)and backoff counter(BC),and the BC is randomly selected from the contention window(CW).The CSMA/CA protocol in IEEE 1901 has three counters[21]:BPC,BC,and deferral counter(DC),which are the specific differences between the two protocols.And the CSMA/CA protocol of IEEE 1901 becomes more complicated because of the DC counter.In this paper,let BPC1,BC1,and DC represent the BPC,BC,and DC in IEEE 1901,respectively,and BPC2 and BC2 represent the BPC and BC in IEEE 802.11,respectively.

The backoff process of stations with a Markov model is shown in figure 2,wherep1represents the collision probability in the PLC,p2represents the collision probability in the WLC,andm1andm2represent the maximum backoff phase of the station in the PLC and WLC,respectively.In the CSMA/CA protocol of IEEE 1901,when a packet arrives,the station will perform the backoff process,then initialize the value of DC and BC1.BC1 will be randomly selected in{0,...,CW10},whereCW10denotes the CW value of backoff phase 0 in the power line.In the power line channel,if the channel is idle,then DC will be unchanged,and BC1 will be decremented by 1;if the channel is busy,then BC1 will be frozen,and DC willbe decremented by 1.If the power line channel becomes idle again,then BC1 will be decremented by 1 again.This is the difference between the IEEE 1901 and IEEE 802.11 CSMA/CA protocols.Similar to the 802.11 protocol,when BC1 backoff to 0,the station will try to transmission,if a collision occurs,then the station will enter the next backoff phase and draw BC1 uniformly at random in{0,...,CW1i},whereCW1idenotes the CW value of backoff phaseiin the PLC channel.For the IEEE 1901 protocol,the value ofCW1iis related to the priority of the stations.Table 1 shows that the stations have four backoff phases and two priority classes(CA0/CA1 and CA2/CA3).

Table 1.CW and DC values of the IEEE 1901 protocol.

Thedc(i)represents the value of DC in each backoff phasei,which decreases only when the power line channel is busy.If DC is reduced to 0 and the channel is busy again or a transmission collides,then the station will enter the next backoff phase(if the station is in the maximum backoff phase,then it will re-enter the last backoff phase)and initial the value of BC1.Therefore,the DC can react in advance to make the stations enter a higher backoff phase in the congested channel,reducing the probability of collisions.

Figure 3.The backoff process in HPWC.

For the IEEE 802.11 protocol,if the channel is idle,BC2 will be decremented by 1;if the channel is busy,then BC2 will be frozen.AndCW2jfollows an exponential distribution,CW2j=2jCW20,whereCW2jdenotes the CW value of backoff phasejin the wireless channel.

2.2 Multiplexing Algorithm for HPWC MAC Layer

In this system,the stations have a power line–wireless dual interface,which can compete for two channels.In order to make full use of the advantages of two channels,we propose a multiplexing algorithm,which has five counters,BPC1,BPC2,BC1,BC2,and DC.When stations need to be transmitted,the five counters will be initialized,assuming that the two channels do not affect each other during the backoff process.The backoff process is shown in Table 2.

In this algorithm,when a station competes for a channel(such as the power line channel),the station will try to transmit in that channel,while the other channel(wireless channel)can still be used by other stations.When the station successfully sends packets,it will immediately initialize the five counters and continue to execute the algorithm.The initialization of all counters aims to ensure the fairness of the other stations and the system.For example,a station successfully sends packets in the WLC channel but does notinitialize BPC1,BC1,and DC of the PLC.If BC1 is reduced to 0 when the next time slot arrives,the station will continue to try to send packets.Thus,the probability that other stations will compete for the channel is changed.

Table 2.Backoff process in hybrid communication.

2.3 Special Case in Hybrid Communication Algorithm

In the previous section,the multiplexing algorithm of HPWC MAC layer was introduced.In step 4 of Table 2,there exist a case where BC1 and BC2 are reduced to 0 simultaneously.The reasons and the effects on the system are analyzed here.

In the HPWC model,the status of the stations are related to the status of the wireless and power line channels,as shown in figure 2.And in the backoff process,the two channels of the station do not affect each other,therefore,BC1 and BC2 may be reduced to 0 at the same time.As shown in figure 3,the backoff process of station A in HPWC is depicted,and BC1 and BC2 are reduced to 0 at timet2.If the station transmission in two channels at the same time,the throughput will be calculated repeatedly.Therefore,the probability that the stations will compete for the two channels simultaneously must be calculated to eliminate the effect of this situation on system performance.

Figure 3 also shows that Station A competes for power line channel att0time.Before the successful transmission,the station continues listening in the wireless channel.If the station still has packets to transmit after a successful transmission,five counters need to be initialized.If the station is not initialized int1,the following conditions may exist:when the station has successfully transmitted the packages through the PLC channel,the next time slot BC2 will be reduced to 0,such that the station will send the packets in the WLC channel.When the station successfully transmits in the WLC channel,if BC1 retreats to 0 again,the station will send packets all the time and may cause other stations to“starve to death”.

2.4 Traffic Model

In this section,the traffic model for stations is introduced.The arrival rate of the packets of the station is assumed subject to the Poisson process with a parameter ofλ.In this system,each station has a packet buffer,and the maximum number of packets that are stored in the buffer of stationnfor transmission is denoted byBn,max;the packets are arranged in the order of arrival and transmitted first in first out at the beginning of the next superframe[22];therefore,the station obeys the M/M/1 queuing model.The stations can sendNBuff packetsat most,and excess packets will be discarded.

Xn,Γdenotes the packets generated by stationnat interarrival timeΓ,and the arrival of the packets obey the Poisson distribution of parameterλ.Thus,Xn,Γfollows the compound Poisson distribution.By analyzing the model of the packet arrival rate and the Poisson process of the stations,the characteristic function ofXn,Γis:

In a system with a total ofNstations,if a station does not generate packets during a time interval ofΓ,the station will not compete for the channel in the next superframe.The average number of stationsN(λ)competing in the channel can be obtained using the characteristic function:

III.SYSTEM PERFORMANCE ANALYSIS

In this section,the transmission probability of the station in the two channels is analyzed,the probability that BC1 and BC2 are reduced to 0 simultaneously is calculated,and derives the transmission probability,throughput and collision probability of the system.

3.1 Transmission Probability in the PLC and WLC

Although the stations use the CSMA/CA algorithm in the PLC and WLC channels,the transmission probability in PLC cannot use the same algorithm as WLC because of the introduction of DC counter.So,the transmission probabilityτ1,τ2and collision probabilityp1,p2of the stations in the independent PLC and WLC channels are discussed by analyzing the Markov process and updating iterative function.Theτ1andp1denote the probability parameters in the PLC,andτ2andp2denote the probability parameters in the WLC,which can be expressed as follows.

Since DC was introduced in IEEE 1901,the transmission probabilityτ1of PLC cannot be calculated directly likeτ2of WLC,which can be calculated using the updating iterative function[21,23].The numerator of the equation ofτ1represents the number of attempts to transmit when the station successfully transmits,and the denominator denotes the total number of time slots spent when the packets are successfully transmission.xkrepresents the probability that the station at backoff phaseijumps to the next backoff phasei+1 inkor fewer time slots due to sensing the medium busydc(i)+1 times.tidenotes the probability that the station will attempt to transmit during backoff phasei,sirepresents the probability that the stations will send packets successfully,andbcirepresents the number of time slots spent.As for the expression ofτ2,CW2minrepresents the minimum CW value in the WLC channel,andm1andm2represent the maximum number of backoff phase in the PLC and WLC,respectively.

3.2 The Probability That Transmission in Two Channels

The analysis in section 2.3 showed that the station may obtain the two channels at the same time.Therefore,the probability of a station using both channels at the same time is analyzed.

We assume that at timet,stationnis in backoff phaseiin the PLC and in backoff phasejin the WLC.k1andk2denote the backoff counter value of BC1 and BC2 in the backoff phaseiof the PLC and in the backoff phasejof the WLC,respectively.Anddc(i)denote the DC value in backoff phasei.

In the HPWC model,since there are five backoff counters,the probability of the station transmission in two channels at the same time cannot be expressed through a closed analytical expression,and the values ofk1andk2will affect the backoff process of the stations.By classifying the values ofk1andk2in the state space,the result is more accurate and easier to calculate.And through division,k1,k2,anddc(i)can be divided into six categories:

Through the analysis of these cases,it can be seen that not all categories will have the situation thatk1andk2are reduced to 0 simultaneously.Becausek1andk2decrease 1 when the channels are idle,and will freeze when the channels are busy.When the power line channel becomes idle again,k1will decrease 1 again.Therefore,ifk1

Because the values of BC1 and BC2 are randomly selected from{0,...,CW1i}and{0,...,CW2j}.Thus,the probability of the values ofk1andk2are 1/CW1iand 1/CW2j,respectively.The three cases in which the stations compete for two channels at the same time are discussed and calculated.Becausedc(0)＞k1≥k2does not exist whendc(0)=0,soi=0 needs to be calculated independently.

1)Wheni=0,the counters ofk1andk2can be reduce to 0 at the same time only whenk1≥k2≥dc(0).At this time,the number of times the station detects that the power line channel is busy needs to be less than or equal todc(0)to prevent the station from entering the next phase.Allk1time slots in the power line channel are required to be free,and the probability is(1-p1)k1,becausedc(0)=0.Whenk1andk2are reduced to 0 at the same time,the number of times the station detects that the wireless channel is busy isk1-k2,the number of idle times isk2times,and the probability that BC2 is reduced to 0 is.Therefore,the probability that the station transmits on two channels is

2)Wheni＞0,the three cases above may happen,so their probabilities need to be calculated separately:

Case 1.When k1≥k2≥dc(i),if BC1 and BC2 are reduced to 0 at the same time,the number of times that the power line channel is busy should be less than or equal to dc(i),and the number of times that the wireless channel is busy is k1-k2.The probability that k1and k2are reduced to0at the same time slot is

Case 2.When k1＞dc(i)≥k2,if k1and k2are reduced to 0 at the same time,the busy time slots in the power line should be less or equal to dc(i),and the number of busy times of the wireless is k1-k2.The probability that k1and k2are reduced to0in the same time slot is

Case 3.When dc(i)≥k1＞k2,k1will be first reduced to0because dc(i)≥k1.If k1and k2are reduced to0simultaneously,the number of busy times of the wireless channel is k1-k2.The probability in this case is

The classification discussion shows that the probability of the stations transmitting simultaneously in both wireless and power line channels is

Thus,the probabilityτdualthat the stations will compete for two channels at the same time is

wherepijis the probability that the stations are in backoff phaseiandjin the PLC and WLC channel,respectively.(The calculation ofpijis shown in the Appendix.)

The throughput of the stations that successfully transmit simultaneously in two channels and the average transmission probabilityτof the hybrid communication system can be calculated using probabilityτdual.

3.3 System Performance Analysis

The average number of stationsN(λ)and the average number of packets in each time slot can be calculated because the arrival of the packets follows the Poisson process.If a station competes for two channels simultaneously and selects one channel for transmission,then the probability of the remainingN(λ)-1 stations competing for the other channel will increase.The station that transmits successfully will initialize all counters,because resetting the BC1 and BC2 values will make other stations access the channel more fairly,increasing the probability of transmission.The analysis of these two cases shows that combined withτdual,the transmission probability of the station in the two channels can be updated.andrepresent the average transmission probability of the HPWC stations in the power line channel and wireless channel,respectively,τdualrepresents the probability that BC1 and BC2 are reduced to 0 at the same time,and the probability can be calculated.

where theτdualis divided by 2 because the stations randomly select one channel after competing for two channels.If the weight ofτdualis changed,then the system performance will not be greatly affected.Taking different weights will only slightly change the throughput of the two channels because the average transmission probability of the stations remains unchanged,and the simulation results show that the effect of theτdualweight on the throughput of the two channels will not exceed 0.03.After calculating the transmission probability and collision probability of the station in the two channels,the average transmission probabilityτand collision probabilitypof the HPWC model can be calculated.

where theandare averaged because the stations have the same priority and probability of competing for the two channels.PtrandPscan be calculated by using the transmission probability in two channels.Ptris the probability that at least one station transmits in the two channels,andPsis the probability of a successful transmission in the system.

Assuming that the stations use Basic to access the channel,the frame structures in the two channels are different.Ts1,Ts2represent the time for successful transmission of the station in the power line and wireless channels,andTc1,Tc2represent the time for collision of the station in power line and wireless channel.

whereE[P*]denotes the longest packet in the collision,E[P]represents the average length of the packets,andEIFSrepresents the slot length when collisions occur in the power line channel.PHYhdrandMAChdrare the physical and MAC header data in the wireless frame,respectively.SIFSandDIFSare the short frame interval and the long frame interval,respectively.PRSis the header data of the stations in the power line,andRIFSandCIFSare the corresponding short frame interval and long frame interval time,respectively.

Although the frame structure is different,we assume that the time slots for successful transmission and collision of stations in the PLC and WLC are the same,both of which areTsandTc.Therefore,the throughputSand delayTof the system can be calculated.

where the numerator ofSis multiplied because two channels exist.The time of the transmission is divided into three parts:the timeTsof successfully transmitting packets,the idle timeσin the successful transmission,and the slotTcwhen a collision occurs.PsPtrindicates the possibility of a successful transmission,(1-Ptr)/PsPtrindicates the time slot when the channel does not transmit packets,and(1-Ps)/Psindicates the probability that the station collisions.

IV.PERFORMANCE ANALYSIS

In this section,a simulation and theoretical comparison of the proposed HPWC algorithm is analyzed.The throughput,collision probability and delay of the system model are verified.Assuming the stations are not drop packets in the physical layer and the collision occurs because more than one station transmission simultaneously.Referring to the IEEE 1901[6,21]and IEEE 802.11 standards[20],we set the parameters as follows.In the system,all stations have the same priority CA0/CA1,the maximum backoff phasem1in the power line channel is 3,CW1minis 8.The wireless channel competition window valueCW2minis 32,and the maximum backoff phasem2is 3.The PLC and WLC set different CW values to make the utilization of the two channels more fairness,because there exist DC in the PLC.Let the total simulation time is 109μs,because the simulation results will not be smooth if the time is small,and the packet is 4092 bits.The system parameter settings are shown in Table 3.

Table 3.Simulation parameters.

In figure 4,we analyze the probability of a station transmitting in two channels simultaneously.Although the value ofτdualis small,it has a greater effect on throughput.The effect of this parameter on throughput is shown in figure 5.The probability ofτdualin the analysis is less than the simulation in figure 4 because the stations and channels have coupling effect.All counters will be reset when the station transmits successfully,making the two channels have a coupling effect.Since the DC was introduced in 1901,coupling existed between stations[22].Therefore,when the number of stations is small,there will be an error inτdual.

As can be seen in figure 5,if the counters are not reset when the transmission successful,the throughput error will increase as the number of stations increases.Because throughput will be calculated repeatedly,and the fairness of the other stations will be changed when two channels send simultaneously.Whenτdualis considered,the transmission probabilities of different channels are updated,making the results more accurate.It can be observed that the throughput decreases as the number of stations increases.This is due to the use of an out-of-order CSMA/CA contention scheme,in which a station can only attempt to transmission when BC1 or BC2 is decremented to zero.As the number of stations increases,the channel will become congested,so that the probability of station collision will increase,and the throughput will decrease.

Figure 4.Dual-interface transmission probability τdual.

Figure 5.Throughput comparison between τdual and nonτdual.

Figure 6.Analysis(markers)and simulation(lines)throughput comparison for different protocols.

Figure 7.Analysis(markers)and simulation(lines)collisions comparison for different protocols.

In figure 6-8,the throughput,collision probability,and delay of the HPWC model are analyzed,and the performances of the WLC,PLC and other hybrid communication schemes are compared.We find the performance of the multiplexing algorithm is substantially improved compared with other communication methods.Figure 6 shows that the throughput of HPWC is much higher than that of WLC and PLC,because the dual channels greatly increases the probability of transmission.When the Poisson traffic model is introduced,only the stations that generate packets compete for the channels.Thus,the number of stations competing for channels and the probability of collisions are reduced,and the throughput of the system is improved.Therefore,the collision probability of the Poisson model is lower than the saturated collision.In figure 7,the collision probability of HPWC is higher than that of PLC and WLC.Because the probability of successful transmission of dual-interface stations in two channels is high,the number of stations in the smaller backoff phase increases,so that the collision probability increases.Figure 8 shows that when the stations make full use of the resources of the two channels,the probability of attempted transmission increases,so that the probability of successful transmission increases and the delay decreases.

Figure 8.Delay of different protocols.

Figure 9.Throughput of different CW and DC values.

It also can be found that the system performance of the selective[19]and parallel[24]scheme is slightly higher than that of WLC and PLC,but lower than that of multiplexing algorithm.Because the parallel scheme uses diversity technology to transmit in two channels at the same time,the reliability increases and the system performance decreases.In the selective scheme,only one channel is randomly selected for transmission,so that the utilization rate of the other channel is low.

Figure 10.Delay of different CW and DC values.

Figure 9 and figure 10 show the effect of different values of CW and DC on system performance.Figure 9 shows that when CW1 or CW2 increase,the throughput increases because the larger CW values increase the backoff time slots of the stations in the channel,thereby reducing channel congestion and improving system performance.However,when stations are fewer,the throughput will not change clearly because the channels are not congested,and the stations compete for the channels more easily.Figure 10 shows that delay decreases as CW increases.When CW is small,collision probability is high,and re-transmission probability increases,such that delay is large.As CW increases,the probability of collision and delay decrease.And when the DC increases,the throughput will decrease,because the DC value is small,the station has a high probability to backoff to a higher phase if BC is small.Therefore,when the DC value is large,the backoff process and the delay will increase,and the throughput will decrease.

The throughput and collision probability of the Poisson distribution with different arrival ratesλare compared in figure 11-12.In this system model,the maximum number of transmitted packets is 6,and the redundant packets are discarded.It can be found that with theλincrease,the throughput decreases,and the collision probability increases.And whenλincreases to a certain extent,throughput and collision probability begin to become saturated.Moreover,the increase in the number of stations will make the channel become congested,and the collision probability of the system tends to the collision probability when the system is saturated.

Figure 11.Throughput of different λ.

Figure 12.Collisions probability of different λ.

V.CONCLUSION

In this paper,the MAC layer of HPWC is studied.Firstly,the throughput and collision probability are calculated by analyzing each counter and state in the backoff process using Markov model and state space classification method.Secondly,the traffic model is introduced to make the channel and stations become more reasonable.The comparison shows that when the Poisson model is introduced,if no packets are generated,the stations will no longer contend for the channels,thus reducing the channel congestion and improving the performance.And the partial coupling effect is eliminated by calculating the probabilityτdualthat stations compete for two channels at the same time.Finally,the MAC layer performance in the dual channels is verified.Compared with the other communication system model,hybrid communication greatly improves throughput and reduces delay.In further,we will study the physical layer,explore the cross-layer performance of physical layer and MAC layer,and make the resource allocation of HPWC more reasonable.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China(Nos.61601182),in part Natural Science Foundation of Hebei Province(F2017502059),and in part by the Fundamental Research Funds for the Central Universities under Grant 2021MS070.

APPENDIX

When calculating the probability that a station will compete for two channels at the same time,the probability that the station in different backoff phasesiandjmust be known.

A.Backoff Process in the PLC

The assumptions are as follows:p1is the collisions probability of stations in the PLC channel,andτ1is the transmission probability.In backoff phasei,the value of BC1 isk1,and the value of DC isdc(i).In the CSMA/CA protocol of IEEE 1901,whenk1＞dc(i),one of the following situations occurs:

(1)In backoff phasei,because the number of times that the channel is busy is greater thandc(i),then the probability of the station enter the next phase(i+1)ispn(i).

(2)When the number of times that the channel is busy is less thandc(i),then BC1 will be reduced to 0,and the probability of the station trying to transmission is[1-pn(i)].

Whenk1≤dc(i),the counter BC1 will decrease to 0 firstly,and the station will transmit.Therefore,the probability ofpn(i)is:

Letpm(i+1)represent the probability that the stations enter the next phase(i+1)from the backoff phaseias

The probability of backoff phase 0 isb1,0,0because the stations will enter backoff phase 0 after successfully transmitting packets:

In the backoff process,the probability of successful transmission will increase as CW increases.Therefore,the probability of different backoff phasesbp(i)is related to the value ofCW1iandpm(i):

And the probability valueb1(i)of the stations at different backoff phases in the PLC isb1(i)=

B.Backoff Process in the WLC

In WLC,when a station successfully transmits the packets,it will enter backoff phase 0.Letb2,0,0represent the probability that the BC2 is reduced to 0 in backoff phase 0 in the WLC.According to the Markov model,b2,0,0can be expressed as:

whereWis the minimum value of the contention window(CW2)in the WLC.

Then the transmission probabilityτ2(j)and the average number of slotsbw(j)in the different backoff phases in WLC can be obtained throughb2,0,0.

Therefore,the probability that the station in different backoff phases in WLC isb2(j)=When the probability of different backoff phases in different channels is obtained,the probabilitypijin the system can be obtained as follows: