Enabling Efficient Caching in High Mobility UAV Communications Network under Limited Backhaul

Yan Wu,Jiandong Li,Junyu Liu,Min Sheng,Chenxi Zhao

State Key Laboratory of Integrated Service Networks,Xidian University,Xi’an,Shaanxi 710071,China

*The corresponding author,email:jdli@mail.xidian.edu.cn

Abstract:Due to fexible deployment,unmanned aerial vehicle(UAV)mounted aerial access points are capable of expanding the coverage capabilities of existing terrestrial base stations(TBSs).Different from TBSs,however,UAV access points(UAPs)are of high mobility in horizontal and vertical dimensions,which may deteriorate the coverage performance.Worsestill,the mobility of UAPs would as well increase the pressure of wireless backhaul.In this light,we investigate the performance of the cache-enabled UAV communications network(CUCN)in terms of network spatial throughput(ST)by analyzing the line of sight(LoS)connections and non-line of sight(NLoS)connections.It is found that the network ST is exponentially decreased with the square of UAP altitude.Furthermore,contrary to intuition,a large cache size may deteriorate the network ST when UAPs are over-deployed.The reason is that a large cache size increases the hit probability,which may increase the activation of UAPs and consequently result in complicated interference.Aiming to maximize the network ST,we optimize the cache strategy under limited backhaul.Remarkably,the results show that network ST could be substantially improved by the optimized cache strategy and the performance degeneration brought by UAP high mobility could be even eliminated especially when the UAP altitude is high.

Keywords:caching optimization;UAV communications network;spatial throughput;stochastic geometry;aerial access point

I.INTRODUCTION

1.1 Motivation

With the development and innovation of unmanned aerial vehicle(UAV)technology,UAV mounted aerial access points are capable of expanding the coverage range of wireless communication system and further provide on-demand service for terrestrial users(TUs)[1,2].Despite the potential benefts,limitation of the wireless backhaul capacity may become the bottleneck of the system performance[3].Accordingly,the cache-enabled UAV communications network(CUCN)becomes a promising method to alleviate backhaul pressure and reduce transmission delay.In CUCN,content is pre-cached in the UAV access point(UAP)to relieve the wireless backhaul pressure and increase the hit probability,which refers to the probability that the associated UAP contains the requested content[4].However,a high hit probability would as well increase the number of active and interfering UAPs,which may degrade the performance of CUCN.Moreover,the mobility of UAPs would infuence the stability of transmission and deteriorate the network performance.Hence,in this paper,we investigate the following two critical problems,1)the impact of high mobility and cache strategy on the performance of CUCN with limited backhaul,and 2)how to optimize cache strategy to maximize the network ST in CUCN.

1.2 Related Work

At present,the application of UAPs in wireless communication has been widely investigated.UAV is used to improve communications and network performance.UAV assisted wireless network is studied in[5,6],where the relay selection problem is transformed into a multi-objective optimization problem.Furthermore,the performance of transmission rate,transmission delay and signaling overhead is improved through Q-learning or storage-carry-forward method.Meanwhile,the network performance is improved by optimizing the mobile characteristics of UAPs in existing literatures,e.g.,the trajectory of UAP on the horizontal plane and vertical plane[7–10].In[7],the UAP acts as a relay to provide services for multiple users.The optimal horizontal position of UAP is obtained through information such as the topology and distributed traffc demand of multiple nodes on the ground,thereby improving the network throughput.In[11],authors propose a general framework for solving the optimal height of UAV by employing the golden section method.Meanwhile,a low complexity iterative algorithm is designed to solve the joint channel and power allocation problem to maximize the total data rate of cell edge users under UAV coverage.User equipment(UE)independent model and UE dependent model for the mobility of UAP are analyzed in[8].Meanwhile,it is proved that the coverage probability of UE independent model is the lower bound of more general model.In[9],authors derive the upper bound of the UAP altitude,beyond which the network performance will decrease.The infuence of hovering radius on the fxed-wing UAV network is analyzed in[10].Furthermore,the beamwidth of UAP is adjusted by the projection area equivalent rule to reduce the interference.

Caching contents in wireless networks can enhance the coverage and reduce transmission delay[12,13].However,due to resource constraints such as cache size,content cannot be all cached on the UAP.Therefore,the cache strategy is optimized to maximize resource utilization[14–17].A cache-enabled UAP framework is proposed in[14],which utilizes edge caching at UAPs to reduce the content acquisition delay of users in cellular networks.In addition,the delay performance is optimized by jointly designing user association and cache strategies.In[15],it is proved that caching most popular contents in limited cache size can reduce transmission delay and energy consumption.A hybrid cache strategy is proposed in[16],where the contents are divided into the popularity set and the less popularity set.The contents in the popularity set are cached in all UAPs and the contents in the less popularity set are only cached in one UAP.Furthermore,the overall spectral effciency is maximized by obtaining an optimal popularity threshold between the two subsets.It is demonstrated that the proactive caching technology has great potential in overcoming the endurance issue in UAV communication[17].Moreover,a trade-off is shown to exist between fle caching cost and fle retrieval cost.

As discussed above,the mobility of UAP and the cache strategy in CUCN is fully analyzed and considered in the existing literature.There are some critical challenges for the application of CUCN.On the one hand,the literature on caching in CUCN generally focuses on how to optimize the cache strategy to meet user requests as much as possible,and improve the network performance by increasing the hit probability.However,a high hit probability may increase the number of active UAPs,resulting in complex interference.Consequently,with the increase of UAP deployment,it is worth discussing whether the improvement of hit probability can continue to improve the network performance.On the other hand,whether the introduction of pre-cache can improve the impact of high mobility on the network is also the focus of this work.

1.3 Outcome and Contributi on

In this work,we consider a downlink CUCN,in which aerial access points are mounted by fxed-wing UAVs and provide on-the-move service to TUs.Compared with rotary-wing UAVs,fxed-wing UAVs are of great potential in carrying aerial access points owing to longer fight endurance and greater payload versatility[18,10].The outcome and main contribution of this paper are summarized as follows.

·We frst derived the semi-closed form expression of network ST.It is found that the network ST is exponentially decreased with the square of UAP altitude.Moreover,aided by the semi-closed form expression,we fnd that there is a critical cache size,beyond which the performance of network ST will degrade.The reason is that a large cache size will increase the number of active UAPs and introduce severe interference,thereby reducing the network ST.

·In order to improve the network ST,the cache strategy is optimized under different UAP density and deployment altitude,subject to the limitation of cache size and backhaul.The optimal cache probability decreases with the increase of UAP density or deployment altitude.Numerical and simulation results show that the optimal cache strategy can greatly improve network ST and the performance degeneration brought by UAP high mobility could be even eliminated especially when the UAP altitude is high.

The rest of this paper is organized as follows.The system model is described in Section II.Afterwards,we analyze the network ST of CUCN in Section III and the optimization of the cache strategy is presented through the upper bound analysis of network ST in Section IV.Finally,we conclude our work in Section V.

II.SYSTEM MODEL

2.1 Network Model

As shown in fgure 1,we consider a downlink CUCN consisting of UAPs and TUs,in which the aerial access points with constant transmit powerPare mounted by fxed-wing UAVs.UAPs and TUs are assumed to be deployed according to two independent homogeneous Poisson Point Processes(HPPPs)ΠUAP={UAPi}with densityλand ΠU={Uj}(i,j∈N)with densityλU,in different planes,respectively.UAPs and TUs are equipped with single antennas.Meanwhile,each UAP is deployed with a variable altitudehUAPand a hovering radiusRhov.TUs are with a fxed altitudehU.Due to the limited movement speed,TUs are less likely to move in a short period of time.For practical concerns,we generally haveΔh=hUAP-hU＞0.

Combined with the hovering characteristics of UAVs,we apply a semi-real-time nearest association strategy(Semi-RTNA)[10],where the each TU is associated with the UAP whose hovering center is geometrically closest to the TU.Note that Semi-RTNA strategy cannot guarantee that a TU is always associated with the closed UAP due to the mobility of UAPs.Furthermore,each UAP would randomly select one of the associated TUs to serve in each time slot for scheduling fairness.To reduce interference,a UAP is active only when the UAP delivers content to the associated TU.

Figure 1.Illustration of a UAV communications network.

2.2 Channel Model

The channel gain is assumed to consist of Rayleigh fadingh～CN(0,1)and pathloss,which is characterized by a dual slope pathloss model(DSPM)[19]

whereddenotes the three-dimensional distance from TU to UAP,RCdenotes the corner distance,andρ=ensures the continuity of(1).In addition,α0andα1denote the pathloss exponents within and out ofRC,respectively.In practice,when the distance between TU and UAP is small,line of sight(LoS)paths will appear between them.As the UAP altitude increases,non-line of sight(NLoS)paths will appear since obstacles may exist between them.Therefore,the pathloss model in(1)can accurately describe the air-to-ground pathloss varying with the transmission distance.

2.3 Content Caching and Delivery Model

We consider that each UAP could probabilistically cacheNCout ofNtotcontents in the content libraryC={Ck:k=1,...,Ntot}.Each content is assumed to be of identical size.If denotingpCk(0≤pCk≤1)as the cache probability for Ck,we haveNC[20].Note that the equal probability cache(EPC)is applied such thatpCkequalqh=,whereqhdenotes the equal cache probability.

Table 1.Summary of notations.

Each TU independently requests a content out ofNtotcontents with the popularityqCkwhich is characterized by the Zipf distribution[21].Until the current requested content is successfully conveyed from the associated UAP,the TU will not make another content request.

Each TU can directly obtain the requested content from the associated UAP when the content is precached in the UAP.Otherwise,the requested content will be obtained from the core network to the UAP through backhaul link.Meanwhile,assuming that the backhaul probability of successful acquisition isqrand the limited backhaul capacity is a constant Cb.

2.4 Performance Metrics

In this paper,the network ST is applied for evaluating the performance of CUCN.Specifcally,the network ST is defned as[22]

In(2),Vdenotes the transmission rate of air-ground link,the defnition of which will be given in Section III.CP(λ)denotes the coverage probability of the typical TU,represented by TU0,which is defned as

In(3),SIRTU0denotes the SIR at TU0,τdenotes the decoding threshold,rdenotes the two-dimensional distance from TU0to the hovering center of associated UAP andθdenotes the hover angle.The probability density function(PDF)ofrandθare given by[23]

Notation:If2F1(·,·,·,·)is defned as the standard Gaussian hypergeometric function,denoteandω2(x,y)=in the rest of the paper.In addition,the symbols involved in this work are shown in Table 1.

III.PERFORMANCE ANALYSIS IN CUCN

3.1 Analysis of Network ST

As shown in(2),the network ST is determined by UAP density,coverage probability and transmission rate of air-ground link.As a result,we will analyze these three parts separately to obtain the fnal expression of network ST.

First,for the density of activated UAPs,only the UAPs that associated with the TUs and have the requested content whether it is pre-cached or successfully obtained through the backhaul link is active.Hence,the activated UAP density can be expressed as

whereqa=1-(1+(bη)-1)-bis the probability that the UAP is connected by at least one TU[24]withη=,b=3.575,andqc=qh+(1-qh)qrindicates the probability that the requested content is contained by UAP.In this work,qcandqaqcrepresent the content hit probability and UAP activation probability,respectively.

Then,the transmission rateVcan be given by

In(7),Vis dependent on the decoding thresholdτand the backhaul capacity Cb.If the requested content is cached in the associated UAP,the transmission rate can be obtained according to Shannon formula.If not,the requested content need to be obtained through the backhaul link and the transmission rate will be affected by the backhaul capacity simultaneously.

Finally,the coverage probability is analyzed.When TU0is associated with the UAP0,SIRTU0is given by

in(8)denotes the interference stemming from the set of interfering UAPs with densityλa.Denoting‖·‖as the Euclidean norm operation,‖UAPi-TU0‖is the distance from UAPito TU0.In addition,Hidenotes the corresponding channel power gain due to small-scale fading.

Proposition 1.Supposing thatCk(k=1,...,Ntot)is requested byTU0inCUCN,the coverage probabilityCP(λ)is given by

where

andCP0(λ)denotes the the coverage probability whenCk is contained byUAP0.Furthermore,d and d0are three-dimensional distances thatand d0=,respectively.The PDF of r andθ are given by(4)and(5).

Proof.Please refer to Appendix.

Combining(6)(7)and(9),the expression of network ST can be given by

As shown in(10),the network ST is impacted by multiple parameters,including cache sizeNC,hovering radiusRhov,UAP altitudehUAPand UAP densityλ,etc.To verify the accuracy of(10),we provide simulation results of network ST in fgure 2 and fgure 3.It can be observed that the numerical results match the simulation results well,verifying the correctness of the existing analysis.

In fgure 2,the network ST is plotted as a function of UAP densityλunder different hovering radiusRhovand TU densityλU.Asλincreases,it can be observed that the network ST frst increases and then decreases whenλU?λ.Moreover,the decrease ofλUcan improve the network ST.In particular,whenλUis small,network ST eventually stabilizes instead of attenuating to 0 with the growingλ.These phenomena can be explained as follows.Asλincreases,the distance between TU0and UAP0is shortened,and the probability of successful transmission is improved.However,asλcontinues to increase,the activated UAP densityλawill increase and the network ST will deteriorate due to complex interference.Meanwhile,whenλUis small,althoughλis increasing,λais limited byλUwhich causes the network ST to stabilize eventually.Furthermore,we noticed that the network ST showed a downward trend as the hovering radius increased.The reason is that a small hovering radius can reduce the number of strong interfering UAPs whose distance to TU0is less than the distance between TU0and UAP0,and further increase the possibility of successful transmission.

Figure 2.Network ST with varying UAP density under different hovering radius and TU density.For system settings,we set α0=3,α1=4,τ=10dB,hUAP=150m,hU=2m,RC=500m,NC=50,Ntot=100 and qr=0.1.Note that lines and markers denote numerical and simulation results,respectively.

Figure 3.Network ST with varying UAP density under different cache size and UAP altitude.For system settings,we set α0=3,α1=4,τ=10dB,λU?λ,hU=2m,Rhov=50m,RC=500m,Ntot=100 and qr=0.1.

Figure 3 explores the impact of the cache sizeNCand UAP altitudehUAPon the network ST.It can be observed that the network ST degenerates ashUAPincreases.The reason is that a high altitude increases the distance between TU0and UAP0,which reduces the received useful signal power.Furthermore,the network ST increases with the increase of cache size whenλis small.Inversely,whenλ＞λ*,the increase of cache size will decrease the network ST.The reason is that a large cache size generally results in a high hit probability,which increases the number of active and interfering UAPs.In other words,the excess cache size will introduce too much interference while increasing the hit probability.Consequently,cache strategy should be optimized by designing cache size to compromise the content hit probability and UAP activation probability,especially for largeλ.

3.2 Upper Bound Analysis of Network ST

In order to reveal the impact of cache size on network ST,we derive the upper bound of network ST in this section.From the analysis in Section 3.1,it can be observed that the network ST decreases with the increase of hovering radius.In order to obtain the upper bound,the following analysis is based onRhov=0.

Proposition 2.Under the Semi-RTNA strategy,the coverage probability in CUCN is given byCPupper

Proof.Please refer to Appendix.

As shown in(11),the closed-form expression of CPupperis obtained and it consists of two parts.The frst part corresponds tod≤RCand the second part corresponds tod＞RC.Furthermore,the result will be simplifed under the given conditions.

Whenλis suffciently large,the probability ofd≤RCincreases and approaches 1.Meanwhile,the case ofd＞RCcan be ignored at this time ande-πλ(R2C-Δh2)(1+Υ)→0 can be obtained due to the largeλ.Therefore,the upper bound of CP in(12)is simplifed to

Remark 1.Assuming that λU?λ,the probability of UAP being associated can be regarded as qa=1.Inthis case,shows a negative exponential rela-tionship with λ and independent of λU.In addition,assuming that λU<λ,the probability of UAP be-ing associated can be regarded as.In thiscase,is independent of λ and inversely pro-portional to λU.

Remark 2.first increases and then decreaseswith the increase of qc.That is to say,coverage probability can not be improved and even worsened under large cache size when λ is large.

Remark 3.WhenΔh=RC,the pathloss model issimplified to a single slope andis the sameas[22].Furthermore,in this case,is propor-tional to e-πqaqcλψΔh2and decreases exponentially asΔh2increases.

In order to obtain the upper bound of network ST,the result in(7)is expanded.Therefore,the the upper bound ofVis simplifed to

Figure 4.Network ST with varying UAP density under magnification in different degrees.For system settings,we set α0=3,α1=4,τ=10dB,λU?λ,hU=2m,Rhov=50m,RC=500m,hUAP=150m,NC=50,Ntot=100 and qr=0.1.

Figure 5.Network ST with varying cache size under different UAP density.For system settings,we set α0=3,α1=4,τ=10dB,λU?λ,hU=2m,Rhov=50m,RC=500m,hUAP=150m and qr=0.1.

According to(7),the result isV=log2(1+τ)when log2(1+τ)≤CbandV=log2(1+τ)+Cb

Hence,the upper bound of network ST can be obtained according to(6)(12)and(14)as follows

For UAP ultra-densely deployed networks,the upper bound of network ST can be represented by(13)as

Combining(10)(15)and(16),the exact result and upper bound of network ST is plotted in fgure 4.It can be observed that the network ST upper bound closely fts the exact result whenλis small.More importantly,the upper bound of network ST under largeλhas the same trend as the exact result.Consequently,for the analysis in the ultra-dense case,we will conduct numerical analysis on the basis ofto obtain the optimal network ST.

Figure 6.q*c with varying UAP density under different decoding thresholds and UAP altitudes.For system settings,we set α0=3,α1=4,λU?λ,hU=2m,RC=500m,NC=50,Ntot=100 and qr=0.1.

IV.OPTIMIZATION OF NETWORK ST

Through the analysis of network ST in Section III,it is shown that the network ST is dependent ofqc.Sinceqc=qh+(1-qh)qr,the essence of inquiry is the infuence ofon the network ST whenqris a constant.The trend of the network ST with varying cache probabilityqhis plotted in fgure 5.It can be obtained intuitively that the network ST increases asqhincreases whenλ=100UAP/km2.However,network ST frst increases and then decreases asλincreases whenλ=101UAP/km2.The reason is that the increase ofqhwill increase the hit probability,however,asqhcontinues to increase,too much interference will be introduced,especially whenλis large.At this time,the deterioration caused by the interference is greater than the gain brought by the content hit,thereby degrading the network ST.Meanwhile,it can be observed from the fgure 5 that asλincreases,the critical cache probability decreases.Therefore,in order to obtain the optimal network ST,the cache probability should be appropriately reduced with the increase ofλ.

Taking the upper bound of ST in(16)as the carrier,the optimal cache probabilityq*his analyzed to optimize network ST under large UAP density.

Corollary 1.When the UAP density λ is large,theoptimal cache probabilityand

where q?c=

Proof.Please refer to Appendix.

It is shown in Corollary 1 that the optimal cache probabilityq*hdepends onAandB.Increasing the cache size can effectively improve the hit probability whenλis small.However,asλincreases,the cache size should be appropriately reduced to control the number of interfering UAPs.From another perspective,with the increase ofλ,the growth trend of interference signal power is greater than that of useful signal power,which will lead to the sharp deterioration of network ST,especially when UAP is densely deployed.

According to(18),the curve ofq*cwith UAP densityλis plotted in fgure 6.Asλincreases,q*cappears unchanged at frst,then decreases and fnally stabilizes atqr.The reason is that the content cannot be completely cached and the performance is mainly limited by the cache size whenλis small.Meanwhile,with the increase ofλ,a large cache size will introduce more interference and deteriorate the network ST.In order to improve network performance,the cache size needs to be appropriately reduced.Finally,due to the ultradense deployment of UAP,the case without caching is optimal and the performance is mainly limited by backhaul.Furthermore,we can observe from fgure 6 thatq*cdecreases as decoding thresholdτor UAP altitudehUAPincreases.The reason is that a largeτis sensitive to interference and the activation probability of UAP needs to be reduced to control interference.Meanwhile,highhUAPwill increase the airground link length and cause the useful signal power to be weakened.In short,reducingq*ccan guarantee SIR under the conditions of largeτand highhUAP,thereby increasing the network ST.

Figure 7.Network ST with varying UAP density under different UAP altitudes.For system settings,we set α0=3,α1=4,τ=20dB,λU?λ,hU=2m,Rhov=50m,RC=500m,NC=50,Ntot=100 and qr=0.1.

Due to each UAP would randomly select one of the associated TUs to serve in each time slot,requests from the associated TUs may not be responded quickly whenλU?λ.In order to improve the quality of service and reduce the queuing delay of TUs,the intuitive solution is to increase the UAP density.Combining the result in Section 3.1 and Corollary 1,variation of the network ST with varying UAP density under different UAP altitudes is portrayed in fgure 7.It can be observed that with the increase of UAP density,the optimal cache strategy can improve the network ST by dynamically adjusting the cache size and maintain it within a certain range,while the cache strategy with fxed cache size will make the network ST decay exponentially[15,16].In addition to the discussion of cache strategies,the control and elimination of interference is also the focus of attention of scholars in the future development of CUCN.Meanwhile,with the increase of UAP altitude,the improvement effect of optimal network ST is more signifcant.Therefore,the optimal cache strategy can improve the network performance,reduce or even reverse the impact of UAP altitude on network ST.

V.CONCLUSION AND PROSPECT

In this work,we investigate the performance of CUCN in terms of the network ST.In particular,the infuence of UAP density,mobility and cache strategy on network ST is revealed.The results show that when UAP is densely deployed and the TU density is large,network ST will decrease as the UAP density increases due to the deterioration of interference.Meanwhile,the high mobility of UAP,such as fight altitude and hover radius,will also cause attenuation of network ST.Worsestill,the network ST is exponentially decreased with the square of UAP altitude.In addition,by optimizing the cache strategy,the content hit probability is adjusted to control the number of activated UAPs and obtain the optimal network ST under different network parameters.The results show that the optimal cache probability can improve the network ST performance in CUCN and even eliminate the impact of mobility,especially when TU density is large and UAP altitude is high.Consequently,the cache strategy proposed in this work could provide helpful insight on the deployment and optimization of CUCN.

In the future,dense deployment of UAPs as one of the key technologies to improve the quality of service has been widely concerned by the academic community.However,the ultra-dense deployment of UAPs may lead to overwhelming interference,which will deteriorate the network performance.Consequently,the control and elimination of interference should be the research focus of dense access wireless network.Furthermore,with the addition of UAP,the air ground integrated network will be formed in the future to provide better services for users.

ACKNOWLEDGEMENT

This work is supported in part by National Key Research and Development Program of China(Grant No.2020YFB1807001),in part by Natural Science Foundation of China(Grant No.62171344,62121001,61725103,61931005),in part by Young Elite Scientists Sponsorship Program by CAST,and in part by Key Industry Innovation Chain of Shaanxi(Grant No.2022ZDLGY05-01,2022ZDLGY05-06).

APPENDIX

Proof for Proposition 1

Givenr,θand ΠaUAV,we have

where(a)follows sinceH0～exp(1)ands=τl2(‖UAP0-U0‖)/P.In(19),LIIΠaUAV(s)=e-sIΠaUAVdenotes the Laplace Transforms ofIIΠaUAVats.

Under the givenrandθ,we can obtain the three-dimensional distanced=. Since this work considers the DSPM,it is necessary to separately discuss whether the useful signal link length is greater than the corner distanceRC.

Whend≤RC,UAP0is within LoS of the TU and the interference in this case is divided into two parts,i.e.,LoS interference and NLoS interference.The result is obtained as follows

Whend＞RC,UAP0is outside the LoS of the TU and only NLoS interference exists in this case.The result is obtained as follows

In(20)and(21),integrations start fromd0=since TU0is associated with the UAP whose hovering center is closest to it.Therefore,the length of the interference link must be greater thand0.d0andddenote the distance from UAP0and interference UAPs to TU0.

whereθfollows the uniform distribution within the interval[0,2π]and the probability density function ofrcan be readily obtained by(4).Hence,we complete the proof.

Proof for Proposition 2

Under the condition that the hovering radius is 0,there isd=d0.Therefore,LIΠaUAV|d≤RC(s)andLIΠaUAV|d＞RC(s)are expressed as

and

Through integrating(4)into(23)and(24),the upper bound of CP can be obtained by

where CP1(λa)

and CP2(λa)

Hence,we complete the proof.

Proof for Corollary 1

In order to investigate the trend of network ST withqc,we differentiatein(15)withqc.

whereMeanwhile,qc∈[qr,qcmax]and

By solvingΘ(qc)=0,we can obtainqc0=0 directly. IfAB/=0,we haveandqc2=.IfAB=0 andA/=B,we have

IfAB＞0,it is easy to get thatqc1<0

IfAB<0 and,we can obtain that 0

IfAB<0 andthere is noqc1andqc2.Meanwhile,andqc*=qcmax.

IfAB=0 andB＞A,we can obtain thatqc3＞0.increases with the increase ofqcsincewhenqr

IfAB=0 andB

Hence,we complete the proof.