Game-Theoretic Spectrum Trading in RF Relay-Assisted Free-Space Optical Communications

Transcription

1 Game-Theoretc Spectrum Tradng n RF Relay-Asssted Free-Space Optcal Communcatons 1 arxv: v1 [cs.it] 27 Jun 2018 Shenje Huang, Student Member, IEEE, Vahd Shah-Mansour, Member, IEEE, and Majd Safar, Member, IEEE Abstract Ths work proposes a novel hybrd RF/FSO system based on a game theoretc spectrum tradng process. It s assumed that no RF spectrum s preallocated to the FSO lnk and only when the lnk avalablty s severely mpared by the nfrequent adverse weather condtons,.e. fog, etc., the source can borrow a porton of lcensed RF spectrum from one of the surroundng RF nodes. Usng the leased spectrum, the source establshes a dual-hop RF/FSO hybrd lnk to mantan ts throughout to the destnaton. The proposed system s consdered to be both spectrum- and power-effcent. A marketequlbrum-based prcng process s proposed for the spectrum tradng between the source and RF nodes. Through extensve performance analyss, t s demonstrated that the proposed scheme can sgnfcantly mprove the average capacty of the system, especally when the surroundng RF nodes are wth low traffc loads. In addton, the system benefts from nvolvng more RF nodes nto the spectrum tradng process by means of dversty, partcularly when the surroundng RF nodes have hgh probablty of beng n heavy traffc loads. Furthermore, the applcaton of the proposed system n a realstc scenaro s presented based on the weather statstcs n the cty of Ednburgh, UK. It s demonstrated that the proposed system can substantally enhance the lnk avalablty towards the carrer-class requrement. Index Terms Free-space optcal communcaton, hybrd RF/FSO lnk, spectrum tradng, market equlbrum. S. Huang and M. Safar are wth the School of Engneerng, the Unversty of Ednburgh, Ednburgh (e-mal: {shenje.huang, majd.safar}@ed.ac.uk). V. Shah-Mansour s wth the School of Electrcal and Computer Engneerng, Unversty of Tehran, Tehran (e-mal: vmansour@ut.ac.r).

2 2 I. INTRODUCTION In recent decades, the scarcty n the rado frequency (RF) spectrum becomes the bottleneck n the expanson of wreless communcaton networks. As a potental canddate for the longrange wreless connectvty, free-space optcal (FSO) communcaton has attracted wdespread and sgnfcant nterest n both scentfc communty and ndustry because of ts hgh achevable data rates, lcense-free spectrum, outstandng securty level and low nstallaton cost. FSO has numerous applcatons and n partcular t s consdered as a cost-effectve wreless backhaul soluton of the future 5G systems [1]. However, there exst some lmtatons and challenges n practcal FSO systems ncludng the pontng and msalgnment loss due to buldng sways [2] and unpredctable connectvty n the presence of atmosphere due to the turbulence-nduced ntensty fluctuaton (also known as scntllaton) and adverse weather condtons such as ran, snow and fog [3]. Beam msalgnment fadng n terrestral FSO systems has been accurately modelled [4] and several effectve methods have been proposed to mtgate ts effects on system performance such as the utlzaton of beamwdth optmzaton [5] and adaptve trackng systems [6]. On the other hand, multple technques have also been proposed to mtgate the performance degradaton caused by scntllaton ncludng the spatal dversty at the transmtter [7], at the recever [8] or at both transcevers [9], mult-hop relayng [10] and adaptve optcs [11]. However, all these mentoned technques are only useful n the presence of spatally dynamc channel fluctuatons. Adverse weather condtons on the other hand have farly statc characterstcs both n tme and space, whch makes the above technques neffectve [12]. Studes have shown that the adverse weather condtons can sgnfcantly deterorate FSO lnk by ntroducng an optcal power attenuaton of up to several hundreds of decbels per klometer [13]. Recently, the socalled hybrd RF/FSO lnk has been proposed to effectvely mprove the lnk avalablty of the FSO lnk by employng an addtonal RF lnk [14]. The motvaton behnd ths dea s that because of the dstnct carrer frequences, FSO lnks are more susceptble to scatterng due to fog and turbulence-nduced scntllaton whereas RF lnks are more senstve to ran condtons (especally for frequences above 10 GHz). Therefore, hybrd RF/FSO lnks can combne the benefts of the two lnks to combat the effects of adverse weather. In the lterature, there are bascally two man types of hybrd RF/FSO systems based on

3 3 ether swtch-over or smultaneous transmsson. In swtch-over transmsson (also called hardswtchng transmsson) scheme, the RF lnk s smply a backup lnk and data s transmtted through ether of the channels. In [15], a low-complexty hard-swtchng hybrd RF/FSO system wth both sngle-threshold and dual-threshold for FSO lnk operaton s proposed. Besdes the theoretcal studes, several expermental works focusng on ths type of hybrd lnk have also been reported [16]. Although swtch-over hybrd RF/FSO lnk s smple and has also been employed n some commercal FSO products [3], the preallocaton of RF spectrum to a backup lnk wth occasonal use s nherently spectrum-neffcent [17]. In another type of hybrd RF/FSO lnks, smultaneous data transmsson s consdered where both the FSO and RF lnks are smultaneously actve. One smple mplementaton of such hybrd lnks s sendng the same data on both channels concurrently and decodng the sgnal at the recever based on the more relable channel [18] or the maxmal rato combnng of two channels [19]. Some other works focus on the desgns of jont channel codng and decodng over the two channels n the hybrd lnk. In partcular, the hybrd rateless codng s employed so that the codng rate for each channel n the hybrd lnk can be adapted to the data rate that the channel can provde and no channel knowledge at the transmtter s requred [20], [21]. Furthermore, the hybrd RF/FSO systems are also modelled as two ndependent parallel channels to further mprove the total throughout [22], [23]. Although hybrd RF/FSO lnks wth smultaneous transmsson outperform those wth swtch-over transmsson, they requre both FSO and RF lnk to be actve contnuously even when the FSO lnk s n good condtons and tself s able to support the requred data throughput. Therefore, n the absence of power allocaton strategy, hybrd RF/FSO lnks wth smultaneous transmsson are power-neffcent and may also generate unneeded RF nterference to the envronment [15], [19]. In ths work, we propose a novel hybrd RF/FSO system based on the game theoretc spectrum tradng. We assume that there exsts a prenstalled FSO lnk between the source and destnaton, however, no RF spectrum s preallocated to ths lnk. When the lnk avalablty s sgnfcantly mpared by the nfrequent long-term adverse weather condtons, the source attempts to borrow a porton of RF spectrum from one of the surroundng RF nodes, whch have lcensed spectrum to communcate wth the destnaton, to establsh a dual-hop RF/FSO hybrd lnk and mantan ts throughput to the destnaton. A market-equlbrum-based prcng process s proposed for the spectrum tradng between the source and RF nodes. Compared to above-mentoned hybrd

4 4 RF/FSO systems n the lterature, the proposed system s consdered to be spectrum-effcent snce no preallocaton of RF spectrum to the lnk s necessary and the source borrows the RF spectrum only when t s needed. In addton, the nvestgated system s consdered to be power-effcent snce the hybrd lnk s only establshed durng the nfrequent adverse weather condtons. Furthermore, the proposed system s also cost-effectve by borrowng RF spectrum from surroundng RF nodes rather than establshng and always mantanng a hgh-cost RF lnk 1. Game theory has been wdely employed n the context of wreless networks for resource management especally n cogntve rado networks. For nstance, n [24] the prce-based power allocaton strateges for a two-ter femtocell network wth a central macrocell underlad wth multple femtocells s nvestgated usng the Stackelberg game. The frequency spectrum tradng between lcensed and unlcensed users n the cogntve rado networks s nvestgated n [25] where three prcng models ncludng market-equlbrum, compettve and cooperatve prcng models are consdered. In addton, the game theoretcal dynamc spectrum sharng between prmary and secondary strategc users s nvestgated n [26]. However, to the best of authors knowledge, ths work s the frst tme the game-theoretc spectrum tradng based hybrd RF/FSO systems are proposed and analysed. The rest of ths paper s organzed as follows. The channel model and system descrpton are shown n Secton II. Secton III presents the dervatons of demand and supply functons and descrbes the spectrum tradng scheme and relay selecton strategy n detal. The numercal results and dscusson are presented n Secton IV. Fnally, we conclude ths paper n Secton V. II. SYSTEM MODEL Fg. 1 shows the schematc of the proposed system for the applcaton of wreless backhaulng n a heterogeneous network conssts of a large macro-cell and numerous small cells. The source S denotes the small cell base staton (SBS) whch requres hgh data throughput to the macrocell base staton (MBS),.e., the destnaton D and the RF nodes R wth {1,...,N} are the surroundng SBSs that have wreless backhaul connectvty to D usng lcensed sub-6 GHz spectrum W. The source S would lke to send ts nformaton to the destnaton D and there exsts an already nstalled FSO lnk between them where a mnmal data rate s requred. When 1 Ths could be any type of RF lnk: f sub-6 GHz RF lnk s used the cost of lcensng s hgh; whereas f hgh-frequency lne-of-sght RF lnk s used the lnk tself s costly [1]).

5 5 Fg. 1. The proposed system model n practcal applcaton for wreless backhaulng. S: the source; R: the surroundng RF nodes; D: the destnaton; b : the leased bandwdth; W : the total lcensed bandwdth the th RF node s allocated for backhaulng. the data rate of FSO lnk goes below the requred data rate, the source broadcasts a request sgnal to the surroundng N dstrbuted RF nodes for the sake of buyng a porton of ther spectrum and establsh a hybrd dual-hop RF/FSO lnk to mprove ts data rate to the MBS. Compared to the lne-of-sght (LoS) RF backhaulng wth unlcensed hgh-frequency spectrum (e.g., mllmeter-wave), the RF backhaulng wth low-frequency lcensed spectrum s also wdely nvestgated [20], [22], [23] due to ts advantage of non-los property and lower nstallaton cost, whch makes t more attractve on small-cell networks especally n urban areas [1], [27]. The proposed novel communcaton scheme can also be employed n other applcatons where the surroundng RF nodes are any types of relay nodes n LTE-based wreless backhaul archtecture [28]. A. Channel Model Snce both FSO and RF lnks are nvolved n the proposed system, the channel models for both channels need to be nvestgated. 1) FSO lnk: Consderng that adaptve trackng systems are employed to properly address the msalgnment fadng, the FSO lnk S D suffers from two man channel mparments ncludng the turbulence-nduced scntllaton and adverse weather condton whereas at very dfferent tme scales. The scntllaton s a short-term effect wth coherence tme T s on the order of several mllseconds [6], however, the weather condton s a long-term effect wth tme-scale T c on the order of hours [20]. Assumng that the FSO lnk employs the ntensty modulaton drect

6 6 detecton (IM/DD), the channel expresson can be wrtten as s o = ρg o h o x o +z o, (1) where ρ s the the responsvty of the photodetector, g o refers to the average power gan, h o denotes the random turbulence-nduced ntensty fadng, x o s the transmtted optcal ntensty, s o s the receved electrcal sgnal and z o s zero-mean real Gaussan nose wth varance σ 2 o. Note that we use the subscrpt o to denote the optcal lnk. The sgnal-ndependent Gaussan nose z o n (1) arses from thermal nose as well as the shot nose nduced by the ambent lght. The average gan g o can be expressed as [22], [29] g o = [ ( )] 2 πd erf 2 e κl SD, (2) 2φL SD where the frst and second term denote the geometrc loss due to the dvergence of the transmtted beam and weather-related atmospherc loss due to scatterng and absorpton, respectvely, d s the recever aperture dameter, φ s the beam dvergence angle, L SD s the dstance between the source and the destnaton, and κ s a weather-dependent attenuaton coeffcent determned based on the Beer-Lambert law. The relatonshp between κ and vsblty V n km can be expressed as [13] κ = 3.91 V ( ) ζ λ o, (3) where λ o s optcal wavelength and ζ s the sze dstrbuton of the scatterng partcles equal to 1.6, 1.3 and 0.585V 1/3 when V > 50, 6 V 50 and V < 6, respectvely. There are several ways to model the turbulence-nduced ntensty fluctuaton h o such as log-normal dstrbuton and Gamma-Gamma dstrbuton. In ths work, we employ the Gamma-Gamma dstrbuton whch can descrbe the ntensty functon wthn a wde range of turbulence condtons as [29] f ho (x) = 2(αβ)(α+β)/2 x (α+β)/2 1 K α β (2 ) αβx, (4) Γ(α)Γ(β) where Γ( ) s the Gamma functon, K p ( ) s the modfed Bessel functon of the second knd. The parameter α and β are gven by [ ( α= exp 0.49χ 2 (1+0.18ϑ χ 12/5 ) 7/6 ) 1 [ ( 1] ( 0.51χ χ 12/5) 5/6 ) 1,β= exp 1], (5) (1+0.9ϑ ϑ 2 χ 12/5 ) 5/6

7 7 respectvely, where χ 2 = 0.5C 2 nk 7/6 L 11/6 SD, ϑ2 = kd 2 /4L SD and k = 2π/λ o. Note that C 2 n s the turbulence refracton structure parameter. For IM/DD FSO channel gven n (1), the achevable rate (channel capacty lower bound) n the presence of average transmtted optcal power constrant,.e., E[x o ] P o, condtoned on the random channel gan h o can be expressed as [30] C o = W ( ) o 2 log 2 1+ eρ2 h 2 o P2 o, (6) 2πσo 2 where W o s the bandwdth of the FSO lnk. Note that dfferent from the tradtonal AWGN channel capacty expresson n RF, the SNR term n (6) s proportonal to the squared optcal power due to the employed ntensty modulaton. In the proposed system, settng a reasonable performance metrc to decde whether the FSO lnk s satsfactory or not s crucal for defnng the trgger of swtchng between FSO-only lnk and hybrd RF/FSO lnk. In ths work, we employ the sldng wndow averagng strategy wth a relatvely long wndow nterval compared to the scntllaton coherence tme T s to smooth out the quck FSO lnk capacty fluctuatons ntroduced by scntllaton, therefore the measured average lnk capacty can accurately reflect the current long-term weather condton [31]. The measured average FSO lnk capacty over the wndow nterval s selected as the performance metrc whch s approxmated as C o = E[C o ] where E[ ] denotes the ensemble expectaton. The source keeps montorng the FSO lnk condton and calculate C o every wndow nterval. When C o s lower than the mnmal data rate requrement C th, the spectrum tradng process s trggered to establsh the dual-hop RF relay lnk, whereas when t exceeds C th, the source wll stop buyng the RF spectrum. Note that to ensure the source could make quck response to the change of weather condtons, the wndow nterval should be set much less than the tme-scale of the weather changes T c. Also note that C th can be any postve values and larger C th ndcates hgher data rate requrement, whch results n more frequent spectrum tradng events between the relays and source and also hgher system complexty. 2) RF lnk: In ths work, t s assumed that the source does not establsh and mantan a drect RF lnk along the FSO lnk to the destnaton, but nstead t uses the borrowed spectrum from surroundng RF nodes to relay ts data to the destnaton only when needed. When a RF node R s selected by the source as the relay to realze the dual-hop S R D lnk, ths RF

8 8 relay channel can be expressed as s (t) R, = g (t) R, h(t) R, x(t) R +z(t) R,, (7) where the subscrpt R s used to denote the RF lnk, t = {1,2} refer to the RF lnk from the source to the relay (S R lnk) and that from the relay to the destnaton (R D lnk), respectvely, s (t) R, and x(t) R average power gan, h (t) R, are the receved and transmtted sgnal, respectvely, g(t) R, s the RF fadng coeffcent and z(t) R, denotes the refers to the zero-mean complex Gaussan nose wth power spectrum densty N 0. For RF sgnal transmsson t s consdered that Gaussan codebooks are employed at transmtters, whch means at every symbol duraton the transmtted symbol x (t) R s generated ndependently based on a zero-mean rotatonally nvarant complex Gaussan dstrbuton [22]. The RF transmtted power,.e., E[ x (t) R 2 ], at the source and the relay are denoted as P S R and PR R as [22], respectvely. The average power gan g(t) R, can be expressed [ ] 2 ) δ g (t) R, = GTX G RX λ R L ref (, (8) 4πL ref L (t) R, where G TX and G RX refer to the RF transmtter and recever gan, respectvely, λ R denotes the RF wavelength, L ref s the reference dstance for the antenna far-feld, L (t) R, denotes the lnk dstance wth t = 1,2 for S R and R D lnk, respectvely, and δ refers to the RF path-loss exponent. Note that the average power gan g (t) R, s assocated wth the specfc dstance between the source (relay) and relay (destnaton) and the fadng coeffcent h (t) R, can be descrbed by dstnct models such as Raylegh, Rcan or Nakagam-m fadng n dfferent applcaton scenaros [1], [19], [32]. Accordng to the channel expresson of RF lnks gven n (7), the channel capacty for the S R lnk and R D lnk can be expressed as ( ) C (1) R, (b ) = b log 2 1+ g(1) R, h(1) R, 2 PR S N 0 b, C (2) R, (b ) = b log 2 ( ) 1+ g(2) R, h(2) R, 2 PR R N 0 W, (9) respectvely, where b s the sze of spectrum borrowed from the relay and W s the total lcensed spectrum that the th relay possesses for R D lnk. From (9) one can also see that the spectral effcency of S R lnk s related to the sze of the leased spectrum b. Ths s because the transmtted RF power s assumed to be fxed, hence ncreasng b wll ntroduce more nose and

9 9 reduce the receved SNR whch results n the decrease of the spectral effcency [33]. On the other hand, the spectral effcency of the R D lnk s not assocated wth the sze of leased bandwdth b, snce the sgnal-to-nose rato (SNR) of the R D lnk s determned by the fxed total transmtted power and total bandwdth. Wth the expressons of the RF lnk capacty gven n (9), the capacty of the decode-and-forward RF lnk S R D can be wrtten as [27] { } C R, (b ) = mn q C (1) R, (b ),(1 q )C (2) R, (b ), (10) where q (0,1) refers to a tme sharng varable. We use q to ndcate the fracton of tme when S R lnk s actve and hence 1 q denotes the tme fracton when R D lnk s actve. 2 The capacty of the RF relay lnk can be expressed as n (10) on the condton that the RF relay s operated on a half-duplex mode [22], [23]. Snce the frst term n the mn functon of (10) s a monotoncally ncreasng functon of q and the second term s a monotoncally decreasng functon of q, for a gven shared spectrum b the optmal q to maxmze C R, s gven by the cross pont of the two functons as q = r y(b )+r. (11) where y(b ) and r s the spectral effcency of the S R lnk and R D lnk gven by ( y(b ) = log 2 1+ v ) (, r = log b 2 1+ g(2) R, h(2) R, 2 PR R N 0 W ), (12) respectvely, wth v = g (1) R, h(1) R, 2 P S R /N 0. Ths optmal tme sharng varable (11) ndcates that the capactes of S R and R D lnks are dentcal, whch means all the data transmtted to the relay can be successfully transferred to the destnaton. By substtutng (11) nto (10) one can hence get the maxmal capacty of the dual-hop S R D lnk as C R, (b ) = b r y(b ) y(b )+r. (13) To check the monotoncty of C R, (b ), one can take the frst dervatve of t as T (b ) = dc R,(b ) db = r [y(b )] 2 [ +ry(b 2 ) ] r2 ln2 1 2 y(b ) [y(b )+r ] 2. (14) 2 It s worth notng that the RF node contnuously transmts ts own backhaul data usng the remanng W b spectrum.

10 10 It can be proved that T (b ) > 0 holds for all b 0 and approaches to zero for b +, thus C R, (b ) s a monotoncally ncreasng functon of b wth a saturated value v /ln2 at hgh b. B. Spectrum Tradng Game We consder that each RF node has data traffc and hence has ts own backhaul data to transmt to the destnaton usng a maxmum allocated bandwdth W va the R D lnk. Partcularly, n wreless backhaul applcaton presented n Fg. 1 ths data traffc comes from the connected user equpments (UEs) n the small cell. In addton, t s assumed that dfferent RF nodes are allocated non-overlappng frequency spectrum for data transmsson so that there s no nterference between them at the destnaton [34]. When RF nodes are not n hgh traffc load, they mght be wllng to lend part of ther spectrum to the source and obtan some revenues at the expense of self-data transmsson lmtaton. In the wreless backhaul applcaton, ths lmtaton may effect the QoS performance of the connected UEs n the small cells. When the RF nodes notce that the source requests to buy ther spectrum due to ts nonfunctonal FSO lnk condton, a two-player game wll be operated between the source and each RF nodes. In ths paper, we consder a market-equlbrum-based prcng approach for the spectrum tradng game [25], [35], [36] where the source s treated as the buyer and the RF relay nodes are treated as the sellers. It s assumed that dfferent RF nodes are not aware of each other 3 and each seller has to negotate wth the source and sets ther prce ndependently to meet the buyer s demand accordng to ther own utltes. In the spectrum tradng games (dscussed later n Secton III), both source and RF nodes n the system are consdered to be ratonal and selfsh so that they only focus on ther own payoff and always follow the best strateges whch maxmze ther own utlty. When all games (or negotatons) are fnshed, the source wll receve the dstnct unt spectrum prces proposed by dfferent RF nodes and the szes of spectrum that they want to lend. Based on ths receved nformaton, the source s able to decde on the best RF node. By notfyng the selected RF node, a dual-hop RF relay lnk S R D can be establshed and the prevous FSO lnk S D turns to a hybrd dual-hop RF/FSO lnk whch s desgned to provde a data rate hgher than the requred data rate. The proposed system wth sngle RF 3 In practce, ths assumpton s justfed due to the lack of any centralzed controller or nformaton exchange among RF nodes. When the RF nodes have more nformaton about each other, more complcated spectrum tradng games can be nvestgated such as compettve and cooperatve games [25], [37].

11 11 Fg. 2. Tme scales: T s, the coherence tme of the scntllaton; T c, the coherence tme of the weather condton; T u, tme duraton of usng the leased RF spectrum before repeatng the game. relay selecton benefts from ts smplcty. However, t s also possble for the source to borrow RF spectrum from multple RF nodes smultaneously rather than routng ts traffc from a sngle node (smlar to the system nvestgated n [38] n the context of RF relayng system), whch can be an nterestng work to address n the future. The result of the spectrum tradng game manly depends on the condton of the RF relay lnk and the traffc load supported by surroundng RF nodes. As a result, the source should repeat the relay selecton when these condtons sgnfcantly change. Denote the coherence tme of the fadng n RF lnk and traffc loads n surroundng nodes as T f and T t, respectvely. For wreless RF lnks wth fxed transcevers the temporal behavour of the fadng s more stable compared to the moble channels and s tghtly related to the exstence of LoS and envronmental condtons (e.g., the vehcular traffc). It s concluded thatt f n the fxed wreless RF lnks s typcally on the order of seconds [32], [39]. On the other hand, T t s relatvely longer whch depends on dfferent applcaton scenaros and could vary from several seconds to hours. Therefore, t s reasonable to assume that the source uses the leased RF spectrum for a tme perod of T u = mn{t f,t t } before nvolvng nto a new spectrum tradng game and possbly updatng the selected RF relay. The tme scales consdered n ths work are plotted n Fg. 2. Invokng the dscusson of the coherence tme of scntllaton T s and weather change T c n Secton II-A, one can get that T u s much shorter than T c but much longer than T s. Therefore, n every T c nterval the source should repeat the RF relay selecton based on the updated nformaton every T u nterval.

12 12 III. SOLUTION OF SPECTRUM TRADING In ths secton, we present the soluton of the game-theoretc spectrum tradng process for the proposed communcaton setup, whch s trggered under the condton C o < C th. we consder a market-equlbrum-based prcng model n whch, for a gven unt spectrum prce, the buyer (source) chooses ts spectrum demand based on ts demand functon and the sellers (the surroundng RF nodes) set the amount of spectrum they would lke to offer accordng to ther supply functon. The market-equlbrum refers to the prce n whch the spectrum demand of the source equals to the spectrum supply of the relays and no excess supply exsts n the market [35], [36]. In the proposed system, the same spectrum tradng process (game) should be appled between the source and each surroundng RF node. Wthout loss of generalty, we wll frstly focus on the spectrum tradng between the source S and the th RF nodes R. Later n Secton III-C, the relay selecton ssue wll be dscussed. A. Utlty of Source and the Demand Functon To quantfy the spectrum demand of the source when choosng R as the relay node, the utlty ganed by the source should be determned whch can be expressed as [33] U = λc R, (b ) b p, (15) where λ s the constant weght ndcatng the obtaned revenue per unt transmsson rate and p (0,+ ) s the unt spectrum prce offered by the RF node R. Equaton (15) ndcates that the source gans revenue from the data rate mprovement by borrowng spectrum from the relay at the cost of payng the leased spectrum. By substtutng (13) nto (15), the utlty functon can be rewrtten as U = λb r y(b ) y(b )+r b p. (16) The so-called demand functon refers to the spectrum demand that maxmzes the utlty functon (16) when the spectrum prce p s gven [35]. To establsh a RF relay lnk S R D, the optmzaton problem at S can be expressed as follows: max b U = λb r y(b ) y(b )+r b p, (17) s.t. C R, (b ) C th C o, b > 0, U > 0,

13 13 The frst condton n (17) ensures that the capacty of the RF dual-hop relay S R D lnk should be larger than the C th C o so that by establshng the hybrd lnk, the total data rate between the source and destnaton,.e., C R, (b ) + C o, s hgher than the mnmum data rate requrement C th. The second condton n (17) ndcates that the leased bandwdth s postve and the last condton guarantees that the source can acheve postve utlty. The soluton of the optmzaton problem (17) gves the optmal spectrum demand denoted as D (p ) as a functon of the prce p. Proposton 1. The soluton to the optmzaton problem (17) can be expressed as b root 0 < p < λt (b mn ), D (p ) = b mn λt(b mn ) p < λr y(b mn 0 p λr y(b mn ) y(b mn )+r, ) y(b mn )+r, (18) where b mn denotes the postve root of the equaton (C th C o )[y(b )+r ] = b r y(b ), (19) and b root refers to the postve root of the equaton T(b ) = p /λ, (20) when C th C o v /ln2 s satsfed. However when C th C o > v /ln2 holds, the source wll qut the spectrum tradng game resultng n zero spectrum demand,.e., D (p ) = 0, Proof. To solve the optmzaton problem (17), we should frstly check the convexty of the objectve functon. The second dervatve of (16) wth respect to b can be expressed as [ d 2 U = λ d2 C R, (b ) λr 2 = v2 db 2 db 2 ln2b (y(b )+r ) 3 (b +v ) 2 y(b )+r + 2 ], (21) ln2 One can see that for b > 0, d2 U < 0 always holds, therefore the objectve functon n (17) s db 2 concave. The optmal b can therefore be calculated through the KKT condtons. The Lagrangan assocate wth ths optmzaton problem can be wrtten as L = λc R, (b )+γ 1 [ Cth C o C R, (b ) ] γ 2 b +b p γ 3 [λc R, (b ) b p ], (22)

14 14 and the correspondng KKT condtons can be expressed as b > 0, C R, (b ) C th C o, λc R, (b ) > b p, γ 1,γ 2,γ 3 0, (23) γ 2 b = 0, γ 3 [λc R, (b ) b p ] = 0, dl/db = 0, γ 1 [ Cth C o C R, (b ) ] = 0. To solve the KKT condtons (23), let s frstly focus on the gven condtons except the last two equaltes. Snce the capacty C R, (b ) gven n (13) s a monotoncally ncreasng functon of b, the condton C R, (b ) C th C o ndcates that b b mn where b mn s gven by the root of the nonlnear equaton (19). It can be proved that wth the constrant C th C o v /ln2, a sngle postve root for the nonlnear equaton (19) denotng b mn calculated numercally. The bandwdth b mn always exsts, whch can be hence refers to the mnmum bandwdth that the source requests from the relay R. On the other hand, f C th C o > v /ln2 holds, there s no postve root for (19). It means that the source requests a data rate too hgh so that the RF relay lnk cannot provde even when nfnte bandwdth s leased. In ths scenaro, the source wll not borrow spectrum from the relay and qut the game wth D (p ) = 0. When the condton b b mn s met, the condton γ 2 b = 0 necesstates γ 2 = 0. In addton, the equalty γ 3 [λc R, (b ) b p ] = 0 and nequalty λc R, (b ) > b p together ndcates γ 3 = 0. One can further calculate that the nequalty λc R, (b ) > b p necesstates y(b ) > p r /(λr p ) when p < λr. Note that consderng the defnton of the source utlty gven n (16), f p λr, one has U < 0 whch aganst our condton of postve utlty. Therefore, the constrant on the unt prce p < λr has to be satsfed. Based on the condtons dscussed above, we have the constrant for b gven by b mn ( ) b < y 1 p r, (24) λr p wth the condton on p that p < λr where y 1 ( ) refers to the nverse functon of y( ) gven n (12). To further justfy (24) we also need to put a constrant on the unt prce p so that b mn < y 1 (p r /(λr p )) always holds, whch ndcates p < λr y(b mn ) y(b mn )+r. (25) Note that ths constrant on p s even strcter that the prevous constrant p < λr notng that λr y(b mn )/ [ y(b mn )+r ] < λr holds.

15 15 Untl now we have derved the constrants on the leased spectrum as well as the unt prce based on the condtons n (23) except the last two equaltes. Now let s nvolve the last two equaltes n (23),.e., dl/db = 0 and γ 1 [ Cth C o C R, (b ) ] = 0. The equalty dl/db = 0 can be expressed as (λ+γ 1 )T(b ) = p, where T (b ) s the frst dervatve of C R, (b ) wth b gven n (14). When C R, (b ) = C th C o s satsfed,.e., b = b mn, dl/db = 0 can be rewrtten as (λ+γ 1 )T (b mn ) = p. Ths equaton should hold under the condton that γ 1 0 and therefore p λt(b mn ). Thus, nvokng the prevous constrant on p gven n (25) we can get the spectrum demand D (p ) = b mn when λt(b mn ) p < λr y(b mn )/[y(b mn )+r ]. On the other hand, when C R, (b ) > C th C o holds,.e., b > b mn, the equalty γ 1 [ Cth C o C R, (b ) ] = 0 necesstates γ 1 = 0. By substtutng γ 1 = γ 2 = 0 nto (λ+γ 1 )T(b ) = p, the equalty dl/db = 0 can be rewrtten as (20). Hence the spectrum demand D (p ) should be the root of the non-lnear equaton (20) (f any) wthn the range of b gven n (24). Invokng the defnton of T (b ) gven n (14) and ts frst dervatve gven n (21), one can get that T (b ) s a monotoncally decreasng functon wth respect to b wth T (0) = r and T (+ ) = 0. Therefore a sngle root of (20), denotng as b root, always exsts wthn the range gven n (24) as long as the followng nequalty s satsfed ( ( )) T y 1 p r < p λr p λ < T(bmn ). (26) ( ) By substtutng b = y 1 p r λr p nto (14), one can get T ( ( )) ( ) y 1 p r = p 2 (λr p ) 1 2 p r λr p λr p λ. (27) ln2λ 2 r 2 From the constrant of p gven n (25) we know p < λr holds, thus the second term n the rght hand sde of (27) s postve, whch ndcates that the frst nequalty n (26) always holds. Therefore, as long as the second nequalty n (26),.e., p < λt(b mn ) s satsfed, a sngle postve root of (20) b root numercally. can be found wthn the range (24), whch can be calculated So far the optmzaton problem (17) s solved completely as summarzed n (18). It s worth notng that when the unt prce p s above λr y(b mn )/ [ y(b mn )+r ] or when Cth C o > v /ln2 holds, the KKT condtons n (23) cannot be all satsfed and the source wll qut the spectrum

16 16 tradng resultng n zero spectrum demand D (p ) = 0. It s nterestng to nvestgate the behavour of the spectrum demand D (p ) wth respect to the ncrease of the unt prce p. Snce T (b ) s a monotoncally decreasng functon, larger p wll result n smaller root of equaton (20),.e., b root. Therefore, the demand functon gven n (18) s a monotoncally decreasng functon of the unt prce n low p regme. Ths s a reasonable result consderng that wth the ncrease of unt spectrum prce, the source wll requre less spectrum demand due to the ncrease of the cost. However, wth the further ncrease of p, the demand saturates at a fxed value b mn whch s the mnmal bandwdth that the source wants to borrow n order to acheve the data rate requrement. In ths stage, the source wll only request to borrow ths mnmum requred bandwdth, snce the unt prce s too hgh so that borrowng a bt more bandwdth wll result n the reducton of ts utlty. When the prce keeps ncreasng so that even wth the mnmal leased bandwdth the achevable utlty s negatve, the source wll qut the game and demand spectrum wll drop to zero. B. Utlty of the Relays and the Supply Functon As mentoned n Secton II, the RF nodes consdered n ths work have ther own data to transmt to the destnaton. Although they can gan extra revenue by lendng spectrum to the source, they may experence the QoS degradaton of ther connected UEs. Assumng that a RF node serves M UEs and each wth a constant data rate requrement of R ur, the utlty of each RF node can be expressed as [25] ( P = b p +c 1 M c 2 M R ur (W ) 2 b )r, (28) M where c 1 denotes the constant weght for the revenue of servng each local connecton whereas c 2 denotes the constant weght for the cost of QoS degradaton, the frst term refers to the spectrum lendng revenue based on lnear prcng, the second term denotes the ncome of provdng the servce of local data transmsson, and the thrd term s the cost due to the QoS degradaton. In (28), t s assumed that the RF node charges a fxed fee for servng every connected UE to communcate wth MBS so that ts ncome of offerng local servce can be expressed as c 1 M. In addton, wth the assumpton that the remanng data rate s unformly allocated to M UEs, the avalable data rate for each UE s (W b )r /M. Therefore, the cost nduced by the QoS

17 17 degradaton of each UE can then be expressed as c 2 (R ur (W b )r /M ) 2. Ths cost can be treated as the dscount offered to the UE because of the spectrum lendng to the source. The quadratc form of the QoS degradaton cost has been wdely used n lterature [25], [26], [37], whch ndcates that the dssatsfacton of the UEs ncreases quadratcally wth the gap between the requred data rate and the actual data rate. To establsh a RF relay lnk S R D, n our proposed spectrum tradng game each RF has to solve an optmzaton problem to get the optmal sze of the leased spectrum for any gven prce p, whch s the so-called spectrum supply functon. Ths optmzaton problem s gven by where P no max b to the source gven by P = b p +c 1 M c 2 M s.t. b > 0, b W, P > P no, ( R ur (W ) 2 b )r, (29) M denotes the RF node s ganed utlty when the RF node does not lend ts spectrum [ ( P no = c 1 M c 2 M R ur W ) ] + 2 r. (30) M Note that x + = max(x,0). The frst condton n (29) ndcates that the leased spectrum s postve, the second nequalty condton means that the leased spectrum should not exceed the total lcensed spectrum of the RF node,.e., W, and the thrd nequalty represents that by lendng the spectrum the RF node s able to enhance ts utlty, otherwse the RF node wll qut the game due to the loss of utlty. The soluton of the optmzaton problem (29) gves the optmal spectrum supply denoted as S (p ) as a functon of the prce p. Proposton 2. The soluton to the optmzaton problem (29) s gven by W M R ur r + M p 2c 2, p r 2 L < p < 2c 2 r R ur, S (p ) = W, 0, otherwse, p 2c 2 r R ur, (31) where p L = [ (2c 2 r R ur r ]) + W. (32) M

18 18 Proof. To solve the optmzaton problem gven n (29), we should frstly check the convexty of the objectve functon. By takng the second dervatve of the objectve functon wth respect to b, one can get d 2 P /db 2 = 2c 2 r 2 /M whch means that the objectve functon P s concave. Therefore, we can agan use KKT condtons to get the optmal spectrum supply b. The Lagrangan assocate wth problem s L = P +γ 1 (P no P ) γ 2 b γ 3 (W b ). (33) The correspondng KKT condtons for ths optmzaton problem can be expressed as b > 0, b W, P > P no, γ 1,γ 2,γ 3 0, (34) γ 1 (P no P ) = 0, dl db = 0, γ 2 b = 0, γ 3 (W b )= 0. Snce the nequalty b > 0 should hold, the equalty γ 2 b = 0 necesstates γ 2 = 0. Smlarly, P > P no and γ 1 (P no P ) = 0 necesstate γ 1 = 0. After some algebrac manpulatons, the equalty dl/db = 0 can be expressed as ( dl = [p 2c 2 r R ur (W )] b )r db M +γ 3 = 0. (35) Assumng b < W, n order to make sure the equalty γ 3 (W b ) = 0 n KKT condtons (34) s satsfed, the parameter γ 3 should be zero. Substtutng γ 3 = 0 nto (35), one can get the optmal spectrum supply functon denoted as S (p ) gven by S (p ) = W M R ur r + M p. (36) 2c 2 r 2 In order to ensure that the rest KKT condtons are all satsfed, ths spectrum supply should also satsfy 0 < S (p ) < W and P > P no. By substtutng (36) nto these two nequaltes and after some manpulatons, we can get that a constrant on the gven unt prce p should be satsfed as p L < p < 2c 2 r R ur where p L s gven n (32). However, outsde ths range of gven prce the KKT condtons cannot be all guaranteed and hence no optmal spectrum supply exsts when b < W. Secondly, let s consder the case when the optmal spectrum supply equals to the total bandwdth,.e., b = W. Substtutng ths nto (35), one can get (p 2c 2 r R ur )+γ 3 = 0. Consderng that γ 3 s non-negatve, ths equalty results n a constrant for the prce as p 2c 2 r R ur.

19 19 It can be easly shown that when ths constrant on p s satsfed, the nequalty P > P no also holds. Therefore, as long as p 2c 2 r R ur s met, we have the optmal spectrum supply S (p ) = W. Note that when the unt prce p s less than p L, the KKT condtons n (34) cannot be all satsfed and the relay wll qut the spectrum tradng resultng n zero spectrum supply,.e., S (p ) = 0. The optmal spectrum supply gven n (31) llustrates that wth the ncrease of the unt prce, the spectrum supply wll frstly be zero and then ncrease wth the ncrease of the unt prce. Fnally when the prce s hgh enough, the relay s pleased to lend all of ts lcensed spectrum W to gan hgher revenue. C. Spectrum tradng Process and Relay Selecton We have derved the demand functon of the source gven n (18) and the supply functon of the relays gven n (31). The remanng problem s how to reach the market-equlbrum prce usng these functons. The market-equlbrum s defned as the stuaton where the supply of an tem s exactly equal to ts demand so that there s nether surplus or shortage n the market and the prce s hence stable n ths stuaton [25], [35]. As dscussed n Secton III-A, the spectrum demand functon s a decreasng functon wth respect to the prce n low unt prce regme, whch means hgher prce wll result n less spectrum demand due to hgh cost. One the other hand, as dscussed n Secton III-B, supply functon s an ncreasng functon wth respect to the prce, whch n turn mples that hgher prce wll lead to more spectrum supply due to the hgher revenue. Therefore, the cross pont of these two functons (f exsts),.e., D (p ) = S (p ), (37) gves the market-equlbrum prce p where the spectrum demand and supply are balanced. In ths equlbrum, both source and RF node are happy wth the prce and sze of the leased spectrum. Snce an analytcal expresson of the root for (37) cannot be derved, we solve ths equaton numercally usng bsecton method. It s possble that the root of (37) does not exst whch means no market-equlbrum prce can be reached and thus the spectrum tradng between the source and the RF node cannot be establshed. So far we manly focused on the spectrum tradng process between the source S and the

20 20 th RF relay R. However, as llustrated n Fg. 1, there may be multple surroundng RF node canddates that can act as the relay for the source. Therefore, a relay selecton method needs to be developed. When C o < C th s satsfed due to the presence of adverse weather condtons, the source wll notfy the dstrbuted N avalable RF nodes and broadcast some nformaton ncludng ts mnmal requred data rate and the current average FSO lnk capacty. Assumng that all RF nodes know the form of the source demand functon gven n (18), each RF node s able to generate the exact demand functon of the source. In addton, based on ts own traffc load nformaton such as the number of connected UEs and ther data rate requrements, every RF node can also generate ts own supply functon usng the equaton gven n (31). Wth both calculated demand and supply functons, every RF node could calculate ts proposed marketequlbrum unt spectrum prce p usng (37) as well as the correspondng optmal porton of leased spectrum to offer. The RF nodes wll then send ths nformaton back to the source. Havng the proposed prce and the optmal leased spectrum from each RF node, the source wll be able to calculate ts own utlty usng (15) to fnally select the RF node whch provdes t wth the maxmal utlty. After notfyng the selected RF node, the source s able to use the leased bandwdth to establsh RF dual-hop relay lnk. Based on the desgn of the spectrum tradng game, ths RF relay lnk together wth the FSO lnk enables the source to transmt data to the destnaton wth a rate above the data rate requrement C th. Note that the proposed spectrum tradng game and relay selecton wll be restarted after each tme nterval T u. IV. NUMERICAL RESULT ANALYSIS In ths secton, we present some smulaton results for our proposed system n the applcaton of wreless backhaulng as plotted n Fg. 1. Unless otherwse stated, the values of the system parameters used for the numercal smulatons are lsted n the Table I [20], [22], [29]. Note that for smplcty t s assumed that each RF relay has the same amount of total lcensed bandwdth W = W, {1,...,N} and the RF fadng coeffcent h (t) R, s modelled as Raylegh dstrbuton [1], [27]. In addton, the transmtted RF power at the source and the relays are consdered to be equal,.e., P S R = PR R = P R. The property of the market-equlbrum prcng n the absence of RF channel fadng wll be frstly presented and the communcaton performance mprovement of employng the proposed spectrum tradng strategy wll then be dscussed.

21 21 TABLE I THE PARAMETER SETTING [20], [22], [29] FSO Lnk Symbol Defnton Value d Recever aperture dameter 5 cm ρ Responsvty of FSO photodetector 0.5 V 1 L SD Dstance between the source and destnaton 1000 m φ Beam dvergence angle 3.5 mrad Cn 2 Refracton structure ndex m 2/3 λ o Laser wavelength 1550 nm σo 2 Nose varance at FSO recever A 2 P o Optcal transmsson power at the source 20 mw λ Utlty gan per unt data rate 1 Mbps 1 W o Bandwdth of FSO lnk 1 GHz C th Mnmal data rate requrement 80 Mbps RF Lnk Symbol Defnton Value λ R RF wavelength 85.7 mm G TX, G RX Antenna Gan (10,10) db L ref Reference dstance of the RF lnk 80 m L (1) R, Dstance between source and RF nodes 600 m L (2) R, Dstance between RF nodes and destnaton 600 m P S R, PR R RF transmtter power at the source and the RF nodes 0.2 W N 0 Nose power spectral effcency at RF recever 114 dbm/mhz W Lcensed spectrum for the R D lnk 20 MHz R ur Data rate requrement per user equpment 3 Mbps δ RF path-loss exponent 3.5 c 1,c 2 Constant weghts for the utlty of the RF nodes (1,0.5) A. Market-Equlbrum Prcng Fgure 3(a) plots the source demand functon (18) wth respect to the unt spectrum prce p wth varous average FSO lnk capactes (C o ). Ths fgure shows that wth the ncrease of the unt prce p, the bandwdth demand frstly decreases and saturates at a fxed value whch denotes the mnmum requred bandwdth. In addton, wth the ncrease of C o less bandwdth s requred to acheve C th whch results n reducton of the saturaton level. For nstance, when C o = 30 Mbps, the mnmum requred bandwdth s 8.94 MHz, however, when C o = 70 Mbps, the correspondng mnmum requred bandwdth s only 1.64 MHz. Wth further ncrease of the unt spectrum prce, snce the source cannot acheve a postve utlty by even buyng the mnmal bandwdth, t wll qut the game and the bandwdth demand wll reduce to zero. One can observe that when the FSO lnk s n a better condton, the source has a hgher tolerance to the ncrease of the unt spectrum prce. For example, the spectrum demand drops to zero on p = 5.57 when C o = 30 Mbps, but the correspondng prce ncreases to 6.03 when C o = 70 Mbps. Ths s because better FSO lnks requre less leased bandwdth and the cost of borrowng ths amount of bandwdth reduces accordngly and hence the source s able to accept hgher unt spectrum prce. Besdes the condton of FSO channel, the behavour of the demand functon s also assocated wth the locatons of the RF nodes, whch determnes the data rate of the RF relay lnk. Fgure 3(b) shows the effect of RF node locatons to the demand functon. The dstance between the source and RF node s fxed at L (1) R, = 600 m, whereas the the dstance between the RF node

22 spectrum demand [MHz] spectrum demand [MHz] unt spectrum prce unt spectrum prce Fg. 3. The spectrum demand versus the unt spectrum prce: (a) for dfferent FSO lnk condtons; (b) for dfferent dstance between the RF node and destnaton wth average FSO lnk capacty C o = 40 Mbps. and destnaton L (2) R, vares. The same results can also be observed when L (1) R, changes. Ths fgure shows that, for a gven low unt spectrum prce, lower L (2) R, results n hgher spectrum demand. For example, the spectrum demand when L (2) R, = 600 m s 18 MHz wth a unt prce 5, however, the correspondng bandwdth demand when L (2) R, = 700 m s only 11 MHz. Ths s because smaller dstance between the RF node and destnaton results n better channel condton n R D lnk. The source s then happy to buy more bandwdth, whch can result n hgher data rate to the destnaton and hence hgher utlty. From the supply functon gven n (31), we know that the behavour of the supply functon s assocated wth both the locaton of the RF node and the number of connected UEs n RF node. Fgure 4(a) plots the supply functon versus the unt prce wth varous number of connected UEs and dstance between the RF node and destnaton. It s presented that wth the ncrease of the unt prce, the bandwdth supply ncreases. For nstance, when L (2) R, = 600 m and M = 25, the bandwdth supply s 13.1 MHz wth a gven spectrum prce 1, whereas the correspondng bandwdth ncreases to 13.7 MHz when the spectrum prce ncreases to 3.5. In addton, t s also obvous that wth the ncrease of the dstance L (2) R, and the number of connected local UEs M, the spectrum supply for a gven prce decreases due to the less total data rate to the destnaton and hgher traffc load, respectvely. Fgure 4(b) shows both supply and demand functons wth respect to the unt spectrum prce under dfferent number of connected UEs n the small cell. The market-equlbrum prce s

23 23 spectrum supply [MHz] spectrum demand/supply [MHz] market-equlbrum no market-equlbrum unt spectrum prce unt spectrum prce Fg. 4. (a) The spectrum supply versus the unt spectrum prce for dfferent numbers of connected UEs n the small cell and lnk dstance between the relay and destnaton; (b) The demand and supply functons wth respect to the unt spectrum prce for dfferent number of connected UEs n the small cell where C o = 25 Mbps and L (1) R, = L(2) R, = 700 m. gven by the cross-pont when the supply functon meets the demand functon. One can observe that the market-equlbrum only exsts for a range of connected UE numbers. For example, market-equlbrum prces for M = 20 and M = 30 are p = 4.68 and p = 4.75, respectvely. However, no market-equlbrum prce exsts for M = 40 due to the hgh traffc load for RF node. We would lke to emphasze that besde the number of connected UEs, the exstence of market-equlbrum s also assocated wth many other condtons,.e., the FSO lnk condton, the dstance between the source (RF node) and RF node (destnaton), and also the fadng condtons of RF lnks. B. Performance Improvement Now let s consder the performance mprovement acheved by the proposed system n the wreless backhaulng applcaton consderng Raylegh fadng n RF lnks. We assume that the data rate requrement of each UE n the small cell R ur s a constant, however, the traffc load of the SBS vares due to the random number of connected UEs M, whch s modelled as a Posson dstrbuted random varable wth mean υ M. Fgure 5(a) presents the average capacty versus weather-dependent attenuaton coeffcent κ wth varous υ M wth and wthout the spectrum tradng. Note that n ths fgure a sngle RF node locatng at L (1) RF, = L(2) RF, = 600 m s consdered,.e., N = 1, and n order to get accurate average capacty performance, samples of channel realzatons are generated

24 average capacty [Mbps] average capacty [Mbps] weather-dependent attenuaton coeffcent [db/km] weather-dependent attenuaton coeffcent [db/km] Fg. 5. The average capacty versus the weather-dependent attenuaton coeffcent of the FSO lnk κ: (a) wth a sngle RF node nvolved n the spectrum tradng,.e., N = 1, and varous average number of connected UEs υ M n the small cell; (b) wth varous υ M under dfferent number of surroundng small cells N. The vertcal dashed-dotted lne represents the κ whch results n an average capacty of FSO lnk equal to the threshold C th. ST: spectrum tradng. for each κ. From Fg. 5(a) one can observe that wth the ncrease of κ the average capacty for FSO-only lnk decreases sgnfcantly and when κ s above 20 db/km the capacty of FSO lnk becomes neglgble. However, n the presence of proposed spectrum tradng, as long as the average FSO capacty s less than the data rate requrement C th = 80 Mbps (or equvalently when κ > db/km), the average capacty of the system can be sgnfcantly mproved by means of establshng the dual-hop RF/FSO hybrd lnk. For nstance, an average capacty of 130 Mbps can be acheved when κ = 15 db/km and υ M = 1 when spectrum tradng s employed, however, for FSO system wthout spectrum tradng the correspondng average capacty s only 40 Mbps. In addton, t s also presented that the performance of systems wth smaller υ M outperforms those wth larger υ M as expected, because of the lower probablty of hgh traffc loads. Furthermore, wth the ncrease of attenuaton coeffcent κ, the average capactes of systems wth spectrum tradng decrease and fnally saturate at fxed values. For example, the asymptotc average capactes at hgh κ are and for systems wth υ M = 5 and υ M = 10, respectvely. Ths s because when κ, the FSO lnk s totally non-functonal and the throughput from the source to the destnaton thoroughly reles on the dual-hop RF relay lnk, whch provde the source wth a fxed average capacty. Fgure 5(b) plots the performance of the average capacty wth varous number of surroundng RF small cells N. It s shown that wth the ncrease of N, better average capacty performance can be acheved. For nstance, by ncreasng N = 1 to N = 3, the average capacty ncreases