I Introduction
In recent decades, the scarcity in the radio frequency (RF) spectrum becomes the bottleneck in the expansion of wireless communication networks. As a potential candidate for the longrange wireless connectivity, freespace optical (FSO) communication has attracted widespread and significant interest in both scientific community and industry because of its high achievable data rates, licensefree spectrum, outstanding security level and low installation cost. FSO has numerous applications and in particular it is considered as a costeffective wireless backhaul solution of the future 5G systems [1]. However, there exist some limitations and challenges in practical FSO systems including the pointing and misalignment loss due to building sways [2] and unpredictable connectivity in the presence of atmosphere due to the turbulenceinduced intensity fluctuation (also known as scintillation) and adverse weather conditions such as rain, snow and fog [3].
Beam misalignment fading in terrestrial FSO systems has been accurately modelled [4] and several effective methods have been proposed to mitigate its effects on system performance such as the utilization of beamwidth optimization [5] and adaptive tracking systems [6]. On the other hand, multiple techniques have also been proposed to mitigate the performance degradation caused by scintillation including the spatial diversity at the transmitter [7], at the receiver [8] or at both transceivers [9], multihop relaying [10] and adaptive optics [11]. However, all these mentioned techniques are only useful in the presence of spatially dynamic channel fluctuations. Adverse weather conditions on the other hand have fairly static characteristics both in time and space, which makes the above techniques ineffective [12]. Studies have shown that the adverse weather conditions can significantly deteriorate FSO link by introducing an optical power attenuation of up to several hundreds of decibels per kilometer [13]. Recently, the socalled hybrid RF/FSO link has been proposed to effectively improve the link availability of the FSO link by employing an additional RF link [14]. The motivation behind this idea is that because of the distinct carrier frequencies, FSO links are more susceptible to scattering due to fog and turbulenceinduced scintillation whereas RF links are more sensitive to rain conditions (especially for frequencies above GHz). Therefore, hybrid RF/FSO links can combine the benefits of the two links to combat the effects of adverse weather.
In the literature, there are basically two main types of hybrid RF/FSO systems based on either switchover or simultaneous transmission. In switchover transmission (also called hardswitching transmission) scheme, the RF link is simply a backup link and data is transmitted through either of the channels. In [15], a lowcomplexity hardswitching hybrid RF/FSO system with both singlethreshold and dualthreshold for FSO link operation is proposed. Besides the theoretical studies, several experimental works focusing on this type of hybrid link have also been reported [16]. Although switchover hybrid RF/FSO link is simple and has also been employed in some commercial FSO products [3], the preallocation of RF spectrum to a backup link with occasional use is inherently spectruminefficient [17].
In another type of hybrid RF/FSO links, simultaneous data transmission is considered where both the FSO and RF links are simultaneously active. One simple implementation of such hybrid links is sending the same data on both channels concurrently and decoding the signal at the receiver based on the more reliable channel [18] or the maximal ratio combining of two channels [19]. Some other works focus on the designs of joint channel coding and decoding over the two channels in the hybrid link. In particular, the hybrid rateless coding is employed so that the coding rate for each channel in the hybrid link can be adapted to the data rate that the channel can provide and no channel knowledge at the transmitter is required [20, 21]. Furthermore, the hybrid RF/FSO systems are also modelled as two independent parallel channels to further improve the total throughout [22, 23]. Although hybrid RF/FSO links with simultaneous transmission outperform those with switchover transmission, they require both FSO and RF link to be active continuously even when the FSO link is in good conditions and itself is able to support the required data throughput. Therefore, in the absence of power allocation strategy, hybrid RF/FSO links with simultaneous transmission are powerinefficient and may also generate unneeded RF interference to the environment [15, 19].
In this work, we propose a novel hybrid RF/FSO system based on the game theoretic spectrum trading. We assume that there exists a preinstalled FSO link between the source and destination, however, no RF spectrum is preallocated to this link. When the link availability is significantly impaired by the infrequent longterm adverse weather conditions, the source attempts to borrow a portion of RF spectrum from one of the surrounding RF nodes, which have licensed spectrum to communicate with the destination, to establish a dualhop RF/FSO hybrid link and maintain its throughput to the destination. A marketequilibriumbased pricing process is proposed for the spectrum trading between the source and RF nodes. Compared to abovementioned hybrid RF/FSO systems in the literature, the proposed system is considered to be spectrumefficient since no preallocation of RF spectrum to the link is necessary and the source borrows the RF spectrum only when it is needed. In addition, the investigated system is considered to be powerefficient since the hybrid link is only established during the infrequent adverse weather conditions. Furthermore, the proposed system is also costeffective by borrowing RF spectrum from surrounding RF nodes rather than establishing and always maintaining a highcost RF link^{1}^{1}1This could be any type of RF link: if sub GHz RF link is used the cost of licensing is high; whereas if highfrequency lineofsight RF link is used the link itself is costly [1])..
Game theory has been widely employed in the context of wireless networks for resource management especially in cognitive radio networks. For instance, in [24] the pricebased power allocation strategies for a twotier femtocell network with a central macrocell underlaid with multiple femtocells is investigated using the Stackelberg game. The frequency spectrum trading between licensed and unlicensed users in the cognitive radio networks is investigated in [25] where three pricing models including marketequilibrium, competitive and cooperative pricing models are considered. In addition, the game theoretical dynamic spectrum sharing between primary and secondary strategic users is investigated in [26]. However, to the best of authors’ knowledge, this work is the first time the gametheoretic spectrum trading based hybrid RF/FSO systems are proposed and analysed.
The rest of this paper is organized as follows. The channel model and system description are shown in Section II. Section III presents the derivations of demand and supply functions and describes the spectrum trading scheme and relay selection strategy in detail. The numerical results and discussion are presented in Section IV. Finally, we conclude this paper in Section V.
Ii System Model
Fig. 1 shows the schematic of the proposed system for the application of wireless backhauling in a heterogeneous network consists of a large macrocell and numerous small cells. The source denotes the small cell base station (SBS) which requires high data throughput to the macrocell base station (MBS), i.e., the destination and the RF nodes with are the surrounding SBSs that have wireless backhaul connectivity to using licensed sub GHz spectrum . The source would like to send its information to the destination and there exists an already installed FSO link between them where a minimal data rate is required. When the data rate of FSO link goes below the required data rate, the source broadcasts a request signal to the surrounding distributed RF nodes for the sake of buying a portion of their spectrum and establish a hybrid dualhop RF/FSO link to improve its data rate to the MBS. Compared to the lineofsight (LoS) RF backhauling with unlicensed highfrequency spectrum (e.g., millimeterwave), the RF backhauling with lowfrequency licensed spectrum is also widely investigated [22, 23, 20] due to its advantage of nonLoS property and lower installation cost, which makes it more attractive on smallcell networks especially in urban areas [1, 27]. The proposed novel communication scheme can also be employed in other applications where the surrounding RF nodes are any types of relay nodes in LTEbased wireless backhaul architecture [28].
Iia Channel Model
Since both FSO and RF links are involved in the proposed system, the channel models for both channels need to be investigated.
IiA1 FSO link
Considering that adaptive tracking systems are employed to properly address the misalignment fading, the FSO link suffers from two main channel impairments including the turbulenceinduced scintillation and adverse weather condition whereas at very different time scales. The scintillation is a shortterm effect with coherence time on the order of several milliseconds [6], however, the weather condition is a longterm effect with timescale on the order of hours [20]. Assuming that the FSO link employs the intensity modulation direct detection (IM/DD), the channel expression can be written as
(1) 
where is the the responsivity of the photodetector, refers to the average power gain, denotes the random turbulenceinduced intensity fading, is the transmitted optical intensity, is the received electrical signal and
is zeromean real Gaussian noise with variance
. Note that we use the subscript ‘’ to denote the optical link. The signalindependent Gaussian noise in (1) arises from thermal noise as well as the shot noise induced by the ambient light. The average gain can be expressed as [29, 22](2) 
where the first and second term denote the geometric loss due to the divergence of the transmitted beam and weatherrelated atmospheric loss due to scattering and absorption, respectively, is the receiver aperture diameter, is the beam divergence angle, is the distance between the source and the destination, and is a weatherdependent attenuation coefficient determined based on the BeerLambert law. The relationship between and visibility in km can be expressed as [13]
(3) 
where is optical wavelength and is the size distribution of the scattering particles equal to , and when , and , respectively. There are several ways to model the turbulenceinduced intensity fluctuation
such as lognormal distribution and GammaGamma distribution. In this work, we employ the GammaGamma distribution which can describe the intensity function within a wide range of turbulence conditions as
[29](4) 
where is the Gamma function, is the modified Bessel function of the second kind. The parameter and are given by
(5) 
respectively, where , and . Note that is the turbulence refraction structure parameter. For IM/DD FSO channel given in (1), the achievable rate (channel capacity lower bound) in the presence of average transmitted optical power constraint, i.e., , conditioned on the random channel gain can be expressed as [30]
(6) 
where is the bandwidth of the FSO link. Note that different from the traditional AWGN channel capacity expression in RF, the SNR term in (6) is proportional to the squared optical power due to the employed intensity modulation.
In the proposed system, setting a reasonable performance metric to decide whether the FSO link is satisfactory or not is crucial for defining the trigger of switching between FSOonly link and hybrid RF/FSO link. In this work, we employ the sliding window averaging strategy with a relatively long window interval compared to the scintillation coherence time to smooth out the quick FSO link capacity fluctuations introduced by scintillation, therefore the measured average link capacity can accurately reflect the current longterm weather condition [31]. The measured average FSO link capacity over the window interval is selected as the performance metric which is approximated as where denotes the ensemble expectation. The source keeps monitoring the FSO link condition and calculate every window interval. When is lower than the minimal data rate requirement , the spectrum trading process is triggered to establish the dualhop RF relay link, whereas when it exceeds , the source will stop buying the RF spectrum. Note that to ensure the source could make quick response to the change of weather conditions, the window interval should be set much less than the timescale of the weather changes . Also note that can be any positive values and larger indicates higher data rate requirement, which results in more frequent spectrum trading events between the relays and source and also higher system complexity.
IiA2 RF link
In this work, it is assumed that the source does not establish and maintain a direct RF link along the FSO link to the destination, but instead it uses the borrowed spectrum from surrounding RF nodes to relay its data to the destination only when needed. When a RF node is selected by the source as the relay to realize the dualhop link, this RF relay channel can be expressed as
(7) 
where the subscript ‘’ is used to denote the RF link, refer to the RF link from the source to the relay ( link) and that from the relay to the destination ( link), respectively, and are the received and transmitted signal, respectively, denotes the average power gain, is the RF fading coefficient and refers to the zeromean complex Gaussian noise with power spectrum density . For RF signal transmission it is considered that Gaussian codebooks are employed at transmitters, which means at every symbol duration the transmitted symbol
is generated independently based on a zeromean rotationally invariant complex Gaussian distribution
[22]. The RF transmitted power, i.e., , at the source and the relay are denoted as and , respectively. The average power gain can be expressed as [22](8) 
where and refer to the RF transmitter and receiver gain, respectively, denotes the RF wavelength, is the reference distance for the antenna farfield, denotes the link distance with for and link, respectively, and refers to the RF pathloss exponent. Note that the average power gain is associated with the specific distance between the source (relay) and relay (destination) and the fading coefficient can be described by distinct models such as Rayleigh, Rician or Nakagamim fading in different application scenarios [1, 19, 32].
According to the channel expression of RF links given in (7), the channel capacity for the link and link can be expressed as
(9) 
respectively, where is the size of spectrum borrowed from the relay and is the total licensed spectrum that the th relay possesses for link. From (9) one can also see that the spectral efficiency of link is related to the size of the leased spectrum . This is because the transmitted RF power is assumed to be fixed, hence increasing will introduce more noise and reduce the received SNR which results in the decrease of the spectral efficiency [33]. On the other hand, the spectral efficiency of the link is not associated with the size of leased bandwidth , since the signaltonoise ratio (SNR) of the link is determined by the fixed total transmitted power and total bandwidth. With the expressions of the RF link capacity given in (9), the capacity of the decodeandforward RF link can be written as [27]
(10) 
where refers to a time sharing variable. We use to indicate the fraction of time when link is active and hence denotes the time fraction when link is active.^{2}^{2}2It is worth noting that the RF node continuously transmits its own backhaul data using the remaining spectrum. The capacity of the RF relay link can be expressed as in (10) on the condition that the RF relay is operated on a halfduplex mode [22, 23]. Since the first term in the min function of (10) is a monotonically increasing function of and the second term is a monotonically decreasing function of , for a given shared spectrum the optimal to maximize is given by the cross point of the two functions as
(11) 
where and is the spectral efficiency of the link and link given by
(12) 
respectively, with . This optimal time sharing variable (11) indicates that the capacities of and links are identical, which means all the data transmitted to the relay can be successfully transferred to the destination. By substituting (11) into (10) one can hence get the maximal capacity of the dualhop link as
(13) 
To check the monotonicity of , one can take the first derivative of it as
(14) 
It can be proved that holds for all and approaches to zero for , thus is a monotonically increasing function of with a saturated value at high .
IiB Spectrum Trading Game
We consider that each RF node has data traffic and hence has its own backhaul data to transmit to the destination using a maximum allocated bandwidth via the link. Particularly, in wireless backhaul application presented in Fig. 1 this data traffic comes from the connected user equipments (UEs) in the small cell. In addition, it is assumed that different RF nodes are allocated nonoverlapping frequency spectrum for data transmission so that there is no interference between them at the destination [34]. When RF nodes are not in high traffic load, they might be willing to lend part of their spectrum to the source and obtain some revenues at the expense of selfdata transmission limitation. In the wireless backhaul application, this limitation may effect the QoS performance of the connected UEs in the small cells.
When the RF nodes notice that the source requests to buy their spectrum due to its nonfunctional FSO link condition, a twoplayer game will be operated between the source and each RF nodes. In this paper, we consider a marketequilibriumbased pricing approach for the spectrum trading game [25, 35, 36] where the source is treated as the buyer and the RF relay nodes are treated as the sellers. It is assumed that different RF nodes are not aware of each other^{3}^{3}3In practice, this assumption is justified due to the lack of any centralized controller or information exchange among RF nodes. When the RF nodes have more information about each other, more complicated spectrum trading games can be investigated such as competitive and cooperative games [25, 37]. and each seller has to negotiate with the source and sets their price independently to meet the buyer’s demand according to their own utilities. In the spectrum trading games (discussed later in Section III), both source and RF nodes in the system are considered to be rational and selfish so that they only focus on their own payoff and always follow the best strategies which maximize their own utility. When all games (or negotiations) are finished, the source will receive the distinct unit spectrum prices proposed by different RF nodes and the sizes of spectrum that they want to lend. Based on this received information, the source is able to decide on the best RF node.
By notifying the selected RF node, a dualhop RF relay link can be established and the previous FSO link turns to a hybrid dualhop RF/FSO link which is designed to provide a data rate higher than the required data rate. The proposed system with single RF relay selection benefits from its simplicity. However, it is also possible for the source to borrow RF spectrum from multiple RF nodes simultaneously rather than routing its traffic from a single node (similar to the system investigated in [38] in the context of RF relaying system), which can be an interesting work to address in the future.
The result of the spectrum trading game mainly depends on the condition of the RF relay link and the traffic load supported by surrounding RF nodes. As a result, the source should repeat the relay selection when these conditions significantly change. Denote the coherence time of the fading in RF link and traffic loads in surrounding nodes as and , respectively. For wireless RF links with fixed transceivers the temporal behaviour of the fading is more stable compared to the mobile channels and is tightly related to the existence of LoS and environmental conditions (e.g., the vehicular traffic). It is concluded that in the fixed wireless RF links is typically on the order of seconds [32, 39]. On the other hand, is relatively longer which depends on different application scenarios and could vary from several seconds to hours. Therefore, it is reasonable to assume that the source uses the leased RF spectrum for a time period of before involving into a new spectrum trading game and possibly updating the selected RF relay. The time scales considered in this work are plotted in Fig. 2. Invoking the discussion of the coherence time of scintillation and weather change in Section IIA, one can get that is much shorter than but much longer than . Therefore, in every interval the source should repeat the RF relay selection based on the updated information every interval.
Iii Solution of Spectrum Trading
In this section, we present the solution of the gametheoretic spectrum trading process for the proposed communication setup, which is triggered under the condition . we consider a marketequilibriumbased pricing model in which, for a given unit spectrum price, the buyer (source) chooses its spectrum demand based on its demand function and the sellers (the surrounding RF nodes) set the amount of spectrum they would like to offer according to their supply function. The marketequilibrium refers to the price in which the spectrum demand of the source equals to the spectrum supply of the relays and no excess supply exists in the market [36, 35]. In the proposed system, the same spectrum trading process (game) should be applied between the source and each surrounding RF node. Without loss of generality, we will firstly focus on the spectrum trading between the source and the th RF nodes . Later in Section IIIC, the relay selection issue will be discussed.
Iiia Utility of Source and the Demand Function
To quantify the spectrum demand of the source when choosing as the relay node, the utility gained by the source should be determined which can be expressed as [33]
(15) 
where is the constant weight indicating the obtained revenue per unit transmission rate and is the unit spectrum price offered by the RF node . Equation (15) indicates that the source gains revenue from the data rate improvement by borrowing spectrum from the relay at the cost of paying the leased spectrum. By substituting (13) into (15), the utility function can be rewritten as
(16) 
The socalled demand function refers to the spectrum demand that maximizes the utility function (16) when the spectrum price is given [35]. To establish a RF relay link , the optimization problem at can be expressed as follows:
(17)  
The first condition in (17) ensures that the capacity of the RF dualhop relay link should be larger than the so that by establishing the hybrid link, the total data rate between the source and destination, i.e., , is higher than the minimum data rate requirement . The second condition in (17) indicates that the leased bandwidth is positive and the last condition guarantees that the source can achieve positive utility. The solution of the optimization problem (17) gives the optimal spectrum demand denoted as as a function of the price .
Proposition 1.
The solution to the optimization problem (17) can be expressed as
(18) 
where denotes the positive root of the equation
(19) 
and refers to the positive root of the equation
(20) 
when is satisfied. However when holds, the source will quit the spectrum trading game resulting in zero spectrum demand, i.e., ,
Proof.
To solve the optimization problem (17), we should firstly check the convexity of the objective function. The second derivative of (16) with respect to can be expressed as
(21) 
One can see that for , always holds, therefore the objective function in (17) is concave. The optimal can therefore be calculated through the KKT conditions. The Lagrangian associate with this optimization problem can be written as
(22) 
and the corresponding KKT conditions can be expressed as
(23)  
To solve the KKT conditions (23), let’s firstly focus on the given conditions except the last two equalities. Since the capacity given in (13) is a monotonically increasing function of , the condition indicates that where is given by the root of the nonlinear equation (19). It can be proved that with the constraint , a single positive root for the nonlinear equation (19) denoting always exists, which can be calculated numerically. The bandwidth hence refers to the minimum bandwidth that the source requests from the relay . On the other hand, if holds, there is no positive root for (19). It means that the source requests a data rate too high so that the RF relay link cannot provide even when infinite bandwidth is leased. In this scenario, the source will not borrow spectrum from the relay and quit the game with .
When the condition is met, the condition necessitates . In addition, the equality and inequality together indicates . One can further calculate that the inequality necessitates when . Note that considering the definition of the source utility given in (16), if , one has which against our condition of positive utility. Therefore, the constraint on the unit price has to be satisfied. Based on the conditions discussed above, we have the constraint for given by
(24) 
with the condition on that where refers to the inverse function of given in (12). To further justify (24) we also need to put a constraint on the unit price so that always holds, which indicates
(25) 
Note that this constraint on is even stricter that the previous constraint noting that holds.
Until now we have derived the constraints on the leased spectrum as well as the unit price based on the conditions in (23) except the last two equalities. Now let’s involve the last two equalities in (23), i.e., and . The equality can be expressed as , where is the first derivative of with given in (14). When is satisfied, i.e., , can be rewritten as . This equation should hold under the condition that and therefore . Thus, invoking the previous constraint on given in (25) we can get the spectrum demand when .
On the other hand, when holds, i.e., , the equality necessitates . By substituting into , the equality can be rewritten as (20). Hence the spectrum demand should be the root of the nonlinear equation (20) (if any) within the range of given in (24). Invoking the definition of given in (14) and its first derivative given in (21), one can get that is a monotonically decreasing function with respect to with and . Therefore a single root of (20), denoting as , always exists within the range given in (24) as long as the following inequality is satisfied
(26) 
By substituting into (14), one can get
(27) 
From the constraint of given in (25) we know holds, thus the second term in the right hand side of (27) is positive, which indicates that the first inequality in (26) always holds. Therefore, as long as the second inequality in (26), i.e., is satisfied, a single positive root of (20) can be found within the range (24), which can be calculated numerically.
It is interesting to investigate the behaviour of the spectrum demand with respect to the increase of the unit price . Since is a monotonically decreasing function, larger will result in smaller root of equation (20), i.e., . Therefore, the demand function given in (18) is a monotonically decreasing function of the unit price in low regime. This is a reasonable result considering that with the increase of unit spectrum price, the source will require less spectrum demand due to the increase of the cost. However, with the further increase of , the demand saturates at a fixed value which is the minimal bandwidth that the source wants to borrow in order to achieve the data rate requirement. In this stage, the source will only request to borrow this minimum required bandwidth, since the unit price is too high so that borrowing a bit more bandwidth will result in the reduction of its utility. When the price keeps increasing so that even with the minimal leased bandwidth the achievable utility is negative, the source will quit the game and demand spectrum will drop to zero.
IiiB Utility of the Relays and the Supply Function
As mentioned in Section II, the RF nodes considered in this work have their own data to transmit to the destination. Although they can gain extra revenue by lending spectrum to the source, they may experience the QoS degradation of their connected UEs. Assuming that a RF node serves UEs and each with a constant data rate requirement of , the utility of each RF node can be expressed as [25]
(28) 
where denotes the constant weight for the revenue of serving each local connection whereas denotes the constant weight for the cost of QoS degradation, the first term refers to the spectrum lending revenue based on linear pricing, the second term denotes the income of providing the service of local data transmission, and the third term is the cost due to the QoS degradation. In (28), it is assumed that the RF node charges a fixed fee for serving every connected UE to communicate with MBS so that its income of offering local service can be expressed as . In addition, with the assumption that the remaining data rate is uniformly allocated to UEs, the available data rate for each UE is . Therefore, the cost induced by the QoS degradation of each UE can then be expressed as . This cost can be treated as the discount offered to the UE because of the spectrum lending to the source. The quadratic form of the QoS degradation cost has been widely used in literature [26, 25, 37], which indicates that the dissatisfaction of the UEs increases quadratically with the gap between the required data rate and the actual data rate.
To establish a RF relay link , in our proposed spectrum trading game each RF has to solve an optimization problem to get the optimal size of the leased spectrum for any given price , which is the socalled spectrum supply function. This optimization problem is given by
(29)  
where denotes the RF node’s gained utility when the RF node does not lend its spectrum to the source given by
(30) 
Note that . The first condition in (29) indicates that the leased spectrum is positive, the second inequality condition means that the leased spectrum should not exceed the total licensed spectrum of the RF node, i.e., , and the third inequality represents that by lending the spectrum the RF node is able to enhance its utility, otherwise the RF node will quit the game due to the loss of utility. The solution of the optimization problem (29) gives the optimal spectrum supply denoted as as a function of the price .
Proposition 2.
Proof.
To solve the optimization problem given in (29), we should firstly check the convexity of the objective function. By taking the second derivative of the objective function with respect to , one can get which means that the objective function is concave. Therefore, we can again use KKT conditions to get the optimal spectrum supply . The Lagrangian associate with problem is
(33) 
The corresponding KKT conditions for this optimization problem can be expressed as
(34)  
Since the inequality should hold, the equality necessitates . Similarly, and necessitate . After some algebraic manipulations, the equality can be expressed as
(35) 
Assuming , in order to make sure the equality in KKT conditions (34) is satisfied, the parameter should be zero. Substituting into (35), one can get the optimal spectrum supply function denoted as given by
(36) 
In order to ensure that the rest KKT conditions are all satisfied, this spectrum supply should also satisfy and . By substituting (36) into these two inequalities and after some manipulations, we can get that a constraint on the given unit price should be satisfied as where is given in (32). However, outside this range of given price the KKT conditions cannot be all guaranteed and hence no optimal spectrum supply exists when .
Secondly, let’s consider the case when the optimal spectrum supply equals to the total bandwidth, i.e., . Substituting this into (35), one can get . Considering that is nonnegative, this equality results in a constraint for the price as . It can be easily shown that when this constraint on is satisfied, the inequality also holds. Therefore, as long as is met, we have the optimal spectrum supply . Note that when the unit price is less than , the KKT conditions in (34) cannot be all satisfied and the relay will quit the spectrum trading resulting in zero spectrum supply, i.e., . ∎
The optimal spectrum supply given in (31) illustrates that with the increase of the unit price, the spectrum supply will firstly be zero and then increase with the increase of the unit price. Finally when the price is high enough, the relay is pleased to lend all of its licensed spectrum to gain higher revenue.
IiiC Spectrum trading Process and Relay Selection
We have derived the demand function of the source given in (18) and the supply function of the relays given in (31). The remaining problem is how to reach the marketequilibrium price using these functions. The marketequilibrium is defined as the situation where the supply of an item is exactly equal to its demand so that there is neither surplus or shortage in the market and the price is hence stable in this situation [25, 35]. As discussed in Section IIIA, the spectrum demand function is a decreasing function with respect to the price in low unit price regime, which means higher price will result in less spectrum demand due to high cost. One the other hand, as discussed in Section IIIB, supply function is an increasing function with respect to the price, which in turn implies that higher price will lead to more spectrum supply due to the higher revenue. Therefore, the cross point of these two functions (if exists), i.e.,
(37) 
gives the marketequilibrium price where the spectrum demand and supply are balanced. In this equilibrium, both source and RF node are happy with the price and size of the leased spectrum. Since an analytical expression of the root for (37) cannot be derived, we solve this equation numerically using bisection method. It is possible that the root of (37) does not exist which means no marketequilibrium price can be reached and thus the spectrum trading between the source and the RF node cannot be established.
So far we mainly focused on the spectrum trading process between the source and the th RF relay . However, as illustrated in Fig. 1, there may be multiple surrounding RF node candidates that can act as the relay for the source. Therefore, a relay selection method needs to be developed. When is satisfied due to the presence of adverse weather conditions, the source will notify the distributed available RF nodes and broadcast some information including its minimal required data rate and the current average FSO link capacity. Assuming that all RF nodes know the form of the source demand function given in (18), each RF node is able to generate the exact demand function of the source. In addition, based on its own traffic load information such as the number of connected UEs and their data rate requirements, every RF node can also generate its own supply function using the equation given in (31). With both calculated demand and supply functions, every RF node could calculate its proposed marketequilibrium unit spectrum price using (37) as well as the corresponding optimal portion of leased spectrum to offer. The RF nodes will then send this information back to the source. Having the proposed price and the optimal leased spectrum from each RF node, the source will be able to calculate its own utility using (15) to finally select the RF node which provides it with the maximal utility. After notifying the selected RF node, the source is able to use the leased bandwidth to establish RF dualhop relay link. Based on the design of the spectrum trading game, this RF relay link together with the FSO link enables the source to transmit data to the destination with a rate above the data rate requirement . Note that the proposed spectrum trading game and relay selection will be restarted after each time interval .
Iv Numerical Result Analysis
In this section, we present some simulation results for our proposed system in the application of wireless backhauling as plotted in Fig. 1. Unless otherwise stated, the values of the system parameters used for the numerical simulations are listed in the Table I [22, 20, 29]. Note that for simplicity it is assumed that each RF relay has the same amount of total licensed bandwidth and the RF fading coefficient is modelled as Rayleigh distribution [1, 27]. In addition, the transmitted RF power at the source and the relays are considered to be equal, i.e., . The property of the marketequilibrium pricing in the absence of RF channel fading will be firstly presented and the communication performance improvement of employing the proposed spectrum trading strategy will then be discussed.


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