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Abstract

Femtocell is a promising technology for wireless communication in the near future, due to its benefits in improving indoor coverage and enhancing system throughput. Access control is a critical issue for both operator and mobile users, and there lie three choices including open, closed and hybrid access. Hybrid access seems to be a promising choice since the femtocell opens part of the resources for macrocell users while reserving the residual part for its own femtocell users. In this way, the femtocell helps improve the network performance without much sacrifice of its own utility. Meanwhile, the macrocell should offer adequate incentive to motivate the femtocells, otherwise the femtocells are unwilling to open their resources. As another challenging obstacle, the macrocell could hardly ask femtocells to truthfully reveal their private information which is important for hybrid access, thus the situation with asymmetric information happens. Contract theory is hereby introduced to deal with such a problem. In this article, we propose a mechanism for hybrid access control, wherein the macrocell remunerates certain amount of spectrum to femtocells according to their contribution of data rate to macrocell users. This design ultimately improves the service quality for macrocell users. After figuring out the sufficient and necessary conditions for feasible contract using contract theory, we further derive the optimal contract. We also discuss several issues due to the distinctive characteristics of our problem.

Keywords

Femtocell hybrid access contract theory asymmetric information

Introduction

Motivation

In the area of wireless communication, demand for data transmission is soaring at an incredible speed. It is estimated that by 2017 there would be over 10 billion mobile-connected devices requesting data traffic at the level of 11.2 exabytes per month.¹ As a significant technology to cope with this increasing demand, femtocell has revealed powerful potential.²

As a practical application of short-range transmission, the femtocell could provide wireless terminals with better service quality due to the proximity between transmitters and receivers. However, the deployment of femtocells reduces operating expenditure for macrocell provider, such as construction of new macrocell base stations (MBSs) and electricity expenses of transmission power. In addition, femtocells enhance spectrum efficiency by great spectrum reuse.

Femtocell access schemes can be classified into three categories:³ open access that all resource is open to public use, closed access that only authorized femtocell users (FUs) could get access, and hybrid access that some of the resource remains reserved while the residual part is open.

All three configurations have their own pros and cons. Open access could improve the overall capacity of network because considerable macrocell users (MUs) locating at poor-coverage area could get access to femtocells.⁴ However, open access substantially increases the handovers between cells and thus the call drop probability rises. On the contrary, closed access guarantees service quality and confidentiality of FUs, whereas it causes strong cross-tier interference between macrocells and femtocells. Hybrid access reaches a compromise between open access and closed access.

From this perspective, it is a reasonable option that macrocell providers turn to femtocell owners and ask for some of femtocells’ resources to be assigned to MUs. Nevertheless, since femtocell owners are obviously uninterested in sharing their own resources without any reward, macrocell providers have to offer adequate incentive to ensure femtocell owners’ contribution is not fruitless.

In order to set up the hybrid access, lots of private information of each femtocell should be shared with the macrocell provider. The private information may include channel conditions, resource constraints, and transmission costs. Although such information could be obtained,^5,6 those femtocell owners might be reluctant to share, or even intentionally conceal such information for the sake of their privacy. Thereby, as a common scene, dealing with problems under asymmetric information is of great significance.

Contract theory⁷ is a useful tool in dealing with circumstances under asymmetric information. Usually a principal–agent model^8,9 is set up, in which the principal gives various contracts, and every agent picks out a suitable contract truthfully according to its private information. A vivid analogy is a labor market where an employer provides the contract and employees work for the reward.

In this article, we concentrate on a problem about interactions between a macrocell provider and several femtocell owners. As mentioned before, the macrocell provider looks forward to some help from femtocell owners such that MUs could enjoy better service quality, while the macrocell provider also offers some refund to make up the performance loss of femtocell owners. Specifically, the amount of such help represents the extent of openness of femtocells. Femtocell owners could choose hybrid access, or even open access, if the reward is inviting. We assume the macrocell provider would pay some spectrum as the refund for femtocell owners. Each femtocell owner would use self-paid spectrum and the refunded spectrum to serve its own users and relevant MUs. The macrocell provider would design different contracts. Then each femtocell owner would pick one out according to its own type that represents its private status and information. Our major goal is to design an optimal contract that makes the macrocell provider maximize its utility.

Related works

Femtocell has attracted growing research interests recently. In Claussen et al.¹⁰ and Knisely et al.,¹¹ the concept of femtocell is introduced. Extended problems consisting of spectrum sharing,¹² interference management,¹³ as well as energy efficiency^14,15 have also been investigated. Moreover, cognitive radio^16,17 is applied to enhance the performance of femtocells.^18,19

For femtocells, how to manage access control is a challenging task.² There are three access mechanisms as listed above: open access, closed access, and hybrid access. De La Roche et al.²⁰ and Feng et al.²¹ discuss the access mechanisms, yet without further specifications on hybrid access. Chandrasekhar and Andrews²² focus on uplink capacity analysis and interference avoidance when closed access is selected. Xia et al.²³ claim that the open and closed access should be matched with non-orthogonal and orthogonal multiple access schemes, respectively. A few recent studies switch view to hybrid access, including Valcarce et al.²⁴ in which hybrid access aims to reduce interference and Chen et al.²⁵ in which utility-aware femtocells are refunded for their open part of resources.

A couple of works have discussed several kinds of refunding patterns. For instance, Chen et al.²⁵ propose that the refund is proportional to open time of each femtocell, which provides a fresh angle to view hybrid access. Besides open time, however, it is more valid to take other pertinent factors into consideration since channel state information also largely influences the achievable data rate. In other words, when resulting in an equal date rate, a femtocell which dedicates more open time because of poorer channel state should obtain no more refund than another femtocell which dedicates less open time with better channel state. In this sense, to be more objective and justifiable, we figure out the refund on the basis of achievable data rate.

Moreover, to cope with circumstances under asymmetric information, auction mechanism is an applicable approach. In auction, bidders could truthfully compete while the seller does not need every aspect of information.^26,27 As another practical method, contract theory has also been utilized widely in the domain of management, marketing, and economics.^28,29 Reaping benefits from its effectiveness to tackle problems under asymmetric information,³⁰ an increasing number of works have brought contract theory into wireless communication problems. Spectrum trading³¹ and spectrum sharing^32,33 are cogent examples.

Contribution

In this article, we uniquely make use of contract theory to design a mechanism about hybrid access for femtocells under asymmetric information, aiming to improve the utility of MUs.

The main contributions of the article are as follows:

We study the access problem of femtocell, using contract theory to cope with the problem of femtocell’s hybrid access with asymmetric information. Without too much information exchange, which also protects the privacy of femtocells, the macrocell provider could pinpoint the type of femtocells. According to different types of femtocells, the macrocell provider is able to give reasonable contracts to them and easily obtain their help to satisfy the demands of MUs. With some spectrum paid by the macrocell in return, femtocells save the cost on self-paid spectrum.

We use the achievable data rate to calculate the refund, which convincingly reflects the contribution of each femtocell. Using data rate as the criterion, we depict how much help a given femtocell exactly offers, thereafter avoiding the ambiguity when other indicators such as service time are taken into consideration. On this line, it leads to a tangible interaction between the macrocell provider and femtocells.

We provide the sufficient and necessary conditions, that is, individual rationality (IR) and incentive compatibility (IC), for a feasible contract. We then prove the optimality of our designed contract. Our final results show that only one type of femtocells would make contribution to the MUs and correspondingly acquire the refunded spectrum. In this way, the interaction protocol could work efficiently by letting the macrocell provider pick out the femtocells which have the suitable type.

We discover that hybrid access is a convincing mechanism for femtocells. The simulation results demonstrate that the femtocells do not have to open too much resource. Instead, a moderate open proportion—no more than 0.3—is viable enough. Hence, the femtocells are able to reserve adequate resource for their own FUs rather than sacrificing a lot. Our work goes forward with a practical application of hybrid access from the previous dual-option circumstance (open access and closed access).

The rest of this article is organized as follows. In section “System model,” we elaborate our system model. We then formulate the problem with contract theory in section “Problem formulation with contract theory.” Feasible contract along with relative sufficient and necessary conditions is analyzed in section “Feasible contract,” followed by the optimal contract design in section “Optimal contract design.” We make several further discussions in section “Further discussions.” Simulation results are presented in section “Simulation results,” and the conclusion is drawn out at last.

System model

In this section, we elaborate the basic system model. Figure 1 illustrates a two-tier network consisting of a single MBS and multiple femtocell base stations (FBSs). In total, there are N FBSs denoted as ${FB S_{n}}_{n = 1}^{N}$ . We assume that the MBS and every FBS use split spectrum and operate on different frequencies, thus no inter-cell interference exists. In this article, we focus on the downlink situation, and the uplink situation could be considered similarly.

Figure 1.

System model.

The MBS is the major wireless service provider of MUs. However, the MBS is not able to constantly meet MUs’ requirement on data rate. To be more explicit, as scores of MUs are asking for service, the MBS would have difficulty dealing with overwhelming requests beyond its capability. Considering the advantages mentioned before, a feasible solution is to turn to the FBSs. Notice that in the rest of the article we no longer intentionally distinguish the MBS and MUs and consider them interchangeable unless specified otherwise since they want to achieve the same goal.

The FBSs, however, play a positive role. In each frame, as long as the incentive from the MBS is sufficiently attractive, each FBS dedicates a portion of spectrum to its own FUs and offers service to MUs using the residual spectrum. Hence, FUs and MUs are distinguished in frequency. Each FBS transmits data to its own FUs via pre-assigned orthogonal channels by time division multiple address (TDMA), such that no intra-cell interference among FUs exists. Similarly, when a MU is admitted by a femtocell, it is also separated from FUs by orthogonal transmissions. As a consequence, the FBSs are able to provide better service quality to MUs and relieve the pressure on the MBS. Note that FUs are not explicitly depicted in Figure 1 as they are integrated with corresponding FBSs.

The incentive, as mentioned above, is the spectrum that the MBS pays to the FBSs. For the routine operation of the network, we assume that the MBS owns a fixed amount of spectrum to serve MUs. Yet, the FBSs are more flexible, and they are privileged to purchase certain amount of spectrum according to their instantaneous demands at the beginning of each frame. Rather than sharing part of its own spectrum with the FBSs, the MBS would pay a part of spectrum to the FBSs as a kind of spontaneous incentive and remuneration because the MBS always exhausts all of its own spectrum to cope with the overwhelming data requests from MUs. However, the idea that the MBS refunds the involved FBSs with spectrum instead of gratuity lies in the lower price of MBS’ preemption on spectrum. In practice, the spectrum deal could be implemented with the notion of cognitive mobile virtual network operator (C-MVNO),^34,35 but we would not discuss it any more as it is out of our main scope.

Problem formulation with contract theory

In this section, we propose the utility functions of the MBS and FBSs and some principles in contract theory.

As specified in section “System model,” the MBS asks FBSs to help serve MUs and pays FBSs some spectrum in return. Meanwhile, the FBSs benefit from the free spectrum obtained by serving MUs.

Since the MBS and FBSs are rational and selfish, all of them care for their own utilities. In other words, the MBS wants to acquire more help from FBSs paying less spectrum, while the FBSs hope to get a larger amount of free spectrum without sacrificing their FUs a lot. In this way, the problem could be proposed. We would next discuss their utility functions.

Utility of the MBS

In this part, we concentrate on MBS’ utility. Having a fixed amount of spectrum, the MBS would fully utilize the spectrum to serve MUs. We denote the achievable data rate solely provided by the MBS as R₀. At the same time, the MBS asks for FBSs’ help, and the FBSs would cooperate to increase available data rate to MUs if possible. Suppose that FBS_n contributes data rate R_n, thereby the total achievable data rate of MUs is calculated as $R_{0} + \sum_{n = 1}^{N} R_{n}$ .

However, in order to remunerate the FBSs, the MBS would pay some spectrum to them. For FBS_n, the amount of spectrum that the MBS would pay is B_n with the unit price d₀. As a result, the MBS has to expend $d_{0} \sum_{n = 1}^{N} B_{n}$ totally.

So the utility of MBS could be expressed as

u_{0} = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{n = 1}^{N} R_{n}) - b_{0}]}} - d_{0} \sum_{n = 1}^{N} B_{n}

(1)

where v₀ is the unit revenue gain, a₀ describes the sensitivity toward the increase in data rate, and b₀ represents the reserved data rate requirement of MUs. Both a₀ and b₀ are positive. Note that Sigmoid function³⁶ is introduced here, which has been widely used to evaluate the satisfactory of users regarding resource allocation or service quality.^37–39 It is obvious that MUs would be hardly satisfied below the threshold b₀, whereas the satisfaction would increase swiftly once b₀ is reached until reaching an asymptotic value. a₀ determines the speed of increment, that is, a larger a₀ means faster growth and vice versa. Notice that equation (1) means that a larger data rate, that is, $R_{0} + \sum_{n = 1}^{N} R_{n}$ , would be of significance to increase the satisfaction of MUs as well as the utility of MBS.

Apparently the MBS would like to maximize its utility u₀ with the largest $\sum_{n = 1}^{N} R_{n}$ and the smallest $\sum_{n = 1}^{N} B_{n}$ .

Utility of the FBS

Driven by the spectrum paid by the MBS, the FBSs offer service to MUs. In this way, the FBSs could save some expenditure. We would next depict the problem from the perspective of FBSs.

Every FBS utilizes pre-assigned orthogonal channels resulted from TDMA to transmit to MUs on corresponding spectrum. We assume log-normal distributed shadowing channel⁴⁰ to model the wireless channels. Suppose Y_n MUs are served by FBS_n, thereafter the received signal-to-noise ratio (SNR) between FBS_n and MUs is given as

α_{n, j} = \frac{P_{n, j} S | h_{n} |^{2}}{N_{0} l_{n, j}^{r}}, j = 1, 2, \dots, Y_{n}

(2)

where P_n,j is the transmission power of FBS_n, S is the log-normal shadowing component which has 0 dB as the mean of 10logS and σ_sdB as the standard deviation, h_n is the Rayleigh-distributed fading magnitude wherein $E (| h_{n} |^{2}) = 1$ , N₀ is the Gaussian noise, l_n,j is the distance between the FBS_n and the jth MU, and r is the path fading exponent. FBS_n would adjust its transmission power to provide all the accessed MUs with an identical SNR α_n. Without loss of generality, each frame is normalized to be 1 in the rest of this article. Thus, the achievable data rate of MUs provided by FBS_n is denoted as

R_{n} = B_{m, n} \log (1 + α_{n})

(3)

where B_m,n is the bandwidth that FBS_n allocates to MUs.

In the meantime, the achievable data rate that FBS_n serves its own FUs via pre-assigned orthogonal channels is

R_{f, n} = B_{f, n} \log (1 + β_{n})

(4)

where B_f,n is the bandwidth FBS_n allocated to FUs, β_n is the received SNR between FBS_n and its own FUs. Similar to equation (2), β_n is derived as an identical value since FBS_n could adjust its transmission power to control β_n,k, which is the SNR between FBS_n and all of the corresponding FUs

β_{n, k} = \frac{P_{n, k} S | h_{n} |^{2}}{N_{0} l_{n, k}^{r}}, k = 1, 2, \dots, Z_{n}

(5)

where P_n,k is the transmission power, l_n,k is the distance between the FBS_n and the kth FU, and Z_n is the total amount of FUs served by the FBS_n.

Then define the open proportion as

x_{n} = \frac{B_{m, n}}{B_{f, n} + B_{m, n}}

(6)

Obviously x_n denotes the proportion of bandwidth contributed to MUs, while 1 − x_n is the residual part of bandwidth reserved for FUs. Closed access, open access, and hybrid access are, respectively, established when x_n = 0, x_n = 1, and $x_{n} \in (0, 1)$ . We point out here that the actual range is $x_{n} \in (0, 1]$ , and we would clarify how x_n = 0 is eliminated later.

Combining equations (3), (4), and (6) would lead to

B_{m, n} = \frac{R_{n}}{\log (1 + α_{n})}, B_{f, n} = \frac{(1 - x_{n}) R_{n}}{x_{n} \log (1 + α_{n})}

(7)

and further

R_{f, n} = \frac{(1 - x_{n}) R_{n}}{x_{n} \log (1 + α_{n})} \log (1 + β_{n})

(8)

To give service to MUs and FUs, the FBSs have to purchase adequate spectrum at the beginning of each frame because the FBSs own no spectrum. FBS_n should have paid for the total amount of spectrum it would occupy, namely, $B_{m, n} + B_{f, n}$ . Nevertheless, the MBS pays part of the bill which is equivalent to B_n and then grants it to FBS_n as refund. Therefore, the actual bandwidth FBS_n that has to pay by itself is

B_{n}^{actual} = B_{m, n} + B_{f, n} - B_{n} = \frac{R_{n}}{x_{n} \log (1 + α_{n})} - B_{n}

(9)

At this moment, the utility of $FB S_{n}, n \in {1, 2, \dots, N}$ could be defined as follows

\begin{matrix} u_{n} = v_{n} R_{f, n} - d_{n} B_{n}^{actual} = v_{n} \frac{(1 - x_{n}) R_{n}}{x_{n} \log (1 + α_{n})} \log (1 + β_{n}) \\ - d_{n} (\frac{R_{n}}{x_{n} \log (1 + α_{n})} - B_{n}) \end{matrix}

(10)

where v_n is the unit revenue gain of FBS_n, and d_n is the unit price per bandwidth for FBS_n. Notice here that $d_{n} > d_{0}$ due to the MBS’ preemption. It is also worthwhile to point out that different kinds of service including phone calls and data services³ lead to the difference of utility functions with respect to the data rate in equations (1) and (10). Since the MBS always provides outdoor users with phone call services, the Sigmoid function in equation (1) is introduced to depict the utility of phone call services given by the MBS as phone calls could take place as long as a certain amount of data rate is reached. Meanwhile, the linear function in equation (10) represents the data services provided by the FBSs because FBSs often offer data services and data users would like to have more data transmissions constantly.

It is straightforward that the utility of FBS_n in equation (10) stands for the difference between the service gain to its own FUs using B_f,n bandwidth and the cost of purchased spectrum $B_{n}^{actual}$ . A reasonable assumption is that the bandwidth MBS pays for is always no larger than the overall bandwidth FBS_n needs, that is, $R_{n} / (x_{n} \log (1 + α_{n})) - B_{n} \geq 0$ . We would validate this assumption in section “Bandwidth constraint.” Note here that u_n is increasing in B_n and is decreasing in x_n.

Dividing both sides of equation (10) by d_n would lead to the normalized utility

{\bar{u}}_{n} = B_{n} - θ_{n} R_{n}

(11)

where

θ_{n} = \frac{d_{n} - v_{n} (1 - x_{n}) \log (1 + β_{n})}{d_{n} x_{n} \log (1 + α_{n})}

(12)

We assume that the unit price d_n is always high enough to ensure that FBS_n prefers the amount of bandwidth paid by the MBS as much as possible. It also guarantees $d_{n} > v_{n} \log (1 + β_{n})$ and further $θ_{n} > 0$ . In the rest of this article, we would call normalized utility as utility for simplicity unless specified otherwise.

Notice here that $θ_{n}$ contains all private information of itself, including SNR $(α_{n}, β_{n})$ , unit price (d_n), and open proportion (x_n). In this sense, $θ_{n}$ is defined as FBS_s’ type representing its status and information. It is usually not revealed to the MBS, which leads to asymmetric information. It is apparent in equations (11) and (12) that a smaller $θ_{n}$ would derive a larger utility.

Obviously each FBS wants to maximize its own utility ${\bar{u}}_{n}$ by adjusting its type $θ_{n}$ to obtain the largest B_n and expend the smallest R_n.

Contract formulation

Both the MBS and the FBSs are eager to maximize their own utilities due to their rationality and selfishness, thus they have conflicting objectives. The conflict could be observed from u₀ and u_n as well. The MBS wants to get larger R_n while spending less B_n, whereas the FBS would like to obtain more B_n with less contribution R_n.

In this part, we introduce contract theory to solve the whole problem. Here, we first introduce the principal–agent model. As the principal, the MBS offers contracts to the FBSs who act as the agents in the model.

The principal’s task is to design a set of rate R_n and bandwidth B_n. Without loss of generality, we assume that the set of types of FBSs is $Θ = {θ_{1}, θ_{2}, \dots, θ_{S}}$ , and we sort them in the order $θ_{1} > θ_{2} > \dots > θ_{S}$ . At the same time, the number of FBS with type $θ_{s}$ is N_s and N_s is a positive integer. In this way, the MBS would design S contract items, that is, different rate–bandwidth combinations of R_s and B_s, for all S types of FBSs. We denote them as $Ψ = {(R_{s}, B_{s}), \forall s \in S}$ in which $S = {1, 2, \dots, S}$ .

After knowing the possible contracts offered by the MBS, each FBS chooses among these contracts according to its type. Obviously, a FBS would never take a contract which makes its utility negative.

Definition 1: IR

A contract that type $θ_{s}$ FBS accepts should provide a non-negative utility

B_{s} - θ_{s} R_{s} \geq 0

(13)

Also, as the MBS offers scores of contracts, each FBS would prefer a given contract to any other contracts.

Definition 2: IC

Type $θ_{s}$ FBS would choose its favorite contract satisfying

B_{s} - θ_{s} R_{s} \geq B_{t} - θ_{s} R_{t}

(14)

for $\forall s, t \in S, s \neq t$ .

The optimal contract design for the MBS is the one maximizing its own utility as follows

{R_{s}^{*}, B_{s}^{*}} = \arg max_{{(R_{s}, B_{s}) \geq 0, \forall s}} u_{0}

(15)

subject to the IR and IC constraints in equations (13) and (14), where u₀ is deducted from equation (1) and rewritten as

u_{0} = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{s = 1}^{S} N_{s} R_{s}) - b_{0}]}} - d_{0} \sum_{s = 1}^{S} N_{s} B_{s}

(16)

In this article, the vector operations are considered component-wise. Specifically, $(R_{s}, B_{s}) \geq 0$ means that $R_{s} \geq 0$ and $B_{s} \geq 0$ unless specified otherwise.

We here explain why x_n = 0 is eliminated. It is obvious that, if a FBS determines x_n = 0 which means that it contributes nothing to MUs, the MBS would definitely not pay any spectrum to this FBS in return. From this perspective, it is actually equivalent to the circumstance that this FBS quits as it is unwilling to help. Hence, the contract would be effective only when $x_{n} \in (0, 1]$

However, the goal of FBS_n is

max {\bar{u}}_{n}

subject to the IR and IC constraints in equations (13) and (14).

Till now, we have proposed the whole problem. We will next derive the optimal contract defined in equation (15).

Feasible contract

In this section, we provide the necessary and sufficient conditions to guarantee a contract to be feasible. It would be useful when we subsequently derive the optimal contract.

Necessary condition 1

For any feasible contract $Ψ = {(R_{s}, B_{s}), \forall s}$ , $R_{s} > R_{t}$ is established if and only if $B_{s} > B_{t}$ .

Necessary Condition 1 demonstrates that a FBS which contributes more to MUs would have more spectrum paid by the MBS, and vice versa. In this sense, FBSs with identical amount of contribution would be refunded equally, and vice versa.

Corollary 1

For any feasible contract $Ψ = {(R_{s}, B_{s}), \forall s}$ , $R_{s} = R_{t}$ is established if and only if $B_{s} = B_{t}$ .

Besides the above necessary condition and corollary, there is another necessary condition required for a contract to be feasible.

Necessary condition 2

For any feasible contract $Ψ = {(R_{s}, B_{s}), \forall s}$ , if $θ_{s} < θ_{t}$ , then $B_{s} \geq B_{t}$ .

Necessary Condition 2 demonstrates that the FBSs with a smaller type should have more spectrum paid by the MBS. Combining it with Necessary Condition 1 shows that the FBS having a smaller type would make more contribution as well as get more remuneration. It rightly corresponds to the discussion after equation (12) that a smaller type is advantageous.

The aforementioned necessary conditions and corollary are typical in contract theory. Referring to Gao et al.,³¹ we omit their proofs here for brevity.

From Necessary Conditions 1 and 2, we could reach the conclusion that for a feasible contract, the following structure should be satisfied

0 \leq R_{1} \leq R_{2} \leq \dots \leq R_{S}, 0 \leq B_{1} \leq B_{2} \leq \dots \leq B_{S}

(17)

with $R_{s} = R_{s + 1}$ if and only if $B_{s} = B_{s + 1}$ .

Also, we could obtain the following Theorem.

Theorem 1

For a contract $Ψ = {(R_{s}, B_{s}), \forall s}$ , it is feasible if satisfying the three conditions listed below:

Condition A. $0 \leq R_{1} \leq R_{2} \leq \dots \leq R_{S}$ and $0 \leq B_{1} \leq B_{2} \leq \dots \leq B_{S}$ ;

Condition B. $B_{1} - θ_{1} R_{1} \geq 0$ ;

Condition C. For any $s = 2, 3, \dots, S$ , $B_{s - 1} + θ_{s} (R_{s} - R_{s - 1}) \leq B_{s} \leq B_{s - 1} + θ_{s - 1} (R_{s} - R_{s - 1})$ .

Proof

Similar to Appendix A in Duan et al.,³³ mathematical induction method is utilized here to prove the theorem.

We denote $Ψ (s)$ as a subset containing the first $s (s < S)$ rate–bandwidth combinations in the whole contract $Ψ$ , that is, $Ψ (s) = {(R_{t}, B_{t}) | t = 1, 2, \dots, s}$ . $Ψ (s)$ could also be deemed as a contract for the network including the FBSs having first s types.

To start with, we show that $Ψ (1)$ is feasible. Since there is only one type, the prerequisite for its feasibility is to satisfy the IR constraint defined in equation (13). Condition B obviously verifies it.

Next we show that, when $Ψ (s)$ is feasible, the new contract $Ψ (s + 1)$ is also feasible after adding new item $(R_{s + 1}, B_{s + 1})$ into $Ψ (s)$ . Here, we need to prove the following two points:

Point 1. The IR and IC constraints for the new item

{\begin{matrix} B_{s + 1} - θ_{s + 1} R_{s + 1} \geq 0 \\ B_{s + 1} - θ_{s + 1} R_{s + 1} \geq B_{t} - θ_{s + 1} R_{t}, \forall t = 1, 2, \dots, s \end{matrix}

(18)

Point 2. For the existing types $θ_{1}, θ_{2}, \dots, θ_{s}$ which are already contained in $Ψ (s)$ , the IC constraints are still satisfied after $θ_{s + 1}$ is added

B_{t} - θ_{t} R_{t} \geq B_{s + 1} - θ_{t} R_{s + 1}, \forall t = 1, 2, \dots, s

(19)

Proof of point 1

Now we prove the IR and IC constraints, that is, equation (18). As $Ψ (s)$ is feasible, the IC constraint is established

B_{s} - θ_{s} R_{s} \geq B_{t} - θ_{s} R_{t}, \forall t = 1, 2, \dots, s

Substituting s by $s + 1$ in the left side in Condition C induces

B_{s + 1} \geq B_{s} + θ_{s + 1} (R_{s + 1} - R_{s})

Combining the above two inequalities, we could obtain

B_{s + 1} - θ_{s} R_{s} \geq B_{t} - θ_{s} R_{t} + θ_{s + 1} (R_{s + 1} - R_{s})

Since $θ_{s} > θ_{s + 1}$ and $R_{s} \geq R_{t}$ , we have

θ_{s} R_{s} - θ_{s} R_{t} \geq θ_{s + 1} R_{s} - θ_{s + 1} R_{t}

Again combining the above two inequalities, we would derive

B_{s + 1} - θ_{s + 1} R_{s + 1} \geq B_{t} - θ_{s + 1} R_{t}

and thus complete the proof of the IC constraint.

As for the IR constraint, due to $θ_{s + 1} < θ_{t}, \forall t \leq s$ , there should be

B_{t} - θ_{s + 1} R_{t} > B_{t} - θ_{t} R_{t}

Combining the above two inequalities with $B_{t} - θ_{t} R_{t} \geq 0$ , we would get

B_{s + 1} - θ_{s + 1} R_{s + 1} \geq 0

and thus complete the proof of the IR constraint.

Proof of point 2

Now we prove the IC constraint in equation (19). As $Φ (s)$ is feasible, there would have

B_{t} - θ_{t} R_{t} \geq B_{s} - θ_{t} R_{s}, \forall t = 1, 2, \dots, s

Substituting s by s + 1 in the right side in Condition C induces

B_{s} + θ_{s} (R_{s + 1} - R_{s}) \geq B_{s + 1}

Combining the above two inequalities, we could obtain

B_{t} - θ_{t} R_{t} + θ_{s} R_{s + 1} - θ_{s} R_{s} \geq B_{s + 1} - θ_{t} R_{s}

Since $θ_{t} \geq θ_{s}$ and $R_{s + 1} \geq R_{s}$ , we have

θ_{t} R_{s + 1} - θ_{t} R_{s} \geq θ_{s} R_{s + 1} - θ_{s} R_{s}

Again combining the above two inequalities, we would derive

B_{t} - θ_{t} R_{t} \geq B_{s + 1} - θ_{t} R_{s + 1}

and thus complete the proof of the IC constraint.

With all of the above-mentioned details, the theorem is proved to be true. ■

It is also necessary to point out the following theorem.

Theorem 2

The sufficient conditions in Theorem 1 are also the necessary conditions for a feasible contract.

Proof

Note that the sufficient conditions in Necessary Condition 2 are also necessary for a feasible contract. To be more explicit, Condition A is the same as the structure mentioned in equation (17), while Condition B pinpoints the IR constraint for the highest type $θ_{1}$ in a feasible contract. The IC constraints for type $θ_{s - 1}$ and type $θ_{s}$ correspondingly

B_{s - 1} - θ_{s - 1} R_{s - 1} \geq B_{s} - θ_{s - 1} R_{s}, B_{s} - θ_{s} R_{s} \geq B_{s - 1} - θ_{s} R_{s - 1}

prove Condition C ■

Optimal contract design

After obtaining the conditions for feasible contract, we look for the optimal contract in this section. The optimal contract has to enable the MBS to maximize its utility. In common scenes, the MBS merely knows the exact number of each type, that is, $N_{s}$ for the FBSs with type $θ_{s}$ , $\forall s \in S$ . Plus, since the MBS has no idea about $θ_{n}$ of $FB S_{n}$ , the MBS could not force a FBS to accept a given contract.

An ordinary way to deal with the problem is to find out the solution to equation (15), yet it appears to be difficult to a large extent because of complication. So we take the advantage of contract theory, especially the IR and IC constraints. Also notice that contract theory works well in such an asymmetric situation where $θ_{s}$ is reserved.

Specifically, we introduce a sequential optimization approach with three steps in the following:

Given a fixed feasible rate ${R_{s}, \forall s}$ , we obtain the best bandwidth ${B_{s}^{*} ({R_{s}, \forall s}), \forall s}$ ;

We then obtain the best rate ${R_{s}^{*}, \forall s}$ for the optimal contract;

We ultimately prove the optimality of ${(R_{s}^{*}, B_{s}^{*}), \forall s} .$

Following the line of section V in Duan et al.,³³ we derive the optimal contract below.

Proposition 1

Let $Ψ = {(R_{s}, B_{s}), \forall s}$ be a feasible contract with fixed rates $R_{1} \leq R_{2} \leq \dots \leq R_{S}$ , then the unique optimal bandwidths satisfy

\begin{matrix} B_{1}^{*} ({R_{s}, \forall s}) = θ_{1} R_{1} B_{s}^{*} ({R_{s}, \forall s}) = θ_{1} R_{1} \\ + \sum_{t = 2}^{s} θ_{t} (R_{t} - R_{t - 1}), \forall s = 2, 3, \dots, S \end{matrix}

(20)

Proof

To begin with, it is easy to find out that the bandwidths in equation (20) could form a feasible contract using Theorem 1. Thereby, we would concentrate on the optimality and uniqueness in what follows.

We first prove that, with fixed rates, the bandwidths defined in equation (20) would lead to the maximization of MBS’ utility in equation (16). Contradiction method is used here.

For the MBS, suppose another feasible bandwidths ${{\hat{B}}_{s}, \forall s}$ that is better than ${B_{s}^{*}, \forall s}$ . As u₀ in equation (16) is decreasing with respect of the sum of bandwidths, there would be $\sum_{s = 1}^{S} N_{s} {\hat{B}}_{s} < \sum_{s = 1}^{S} N_{s} B_{s}^{*}$ , and we have at least one bandwidth, for example, ${\hat{B}}_{s} < B_{s}^{*}$ for type $θ_{s}$ .

For s = 1, there would have ${\hat{B}}_{1} < B_{1}^{*}$ . Considering $B_{1}^{*} - θ_{1} R_{1} = 0$ , ${\hat{B}}_{1} - θ_{1} R_{1} < 0$ would be derived, which apparently violates the IR constraint.

Also, for $s > 1$ , we have ${\hat{B}}_{s - 1} + θ_{s} (R_{s} - R_{s - 1}) \leq {\hat{B}}_{s}$ according to Condition C in Theorem 1. Since we also have $θ_{s} (R_{s} - R_{s - 1}) = B_{s}^{*} - B_{s - 1}^{*}$ by equation (20), thus

{\hat{B}}_{s - 1} - B_{s - 1}^{*} \leq {\hat{B}}_{s} - B_{s}^{*} < 0

Continuing the above process, we could finally reach ${\hat{B}}_{1} - B_{1}^{*} < 0$ , which violates the IR constraint as just mentioned.

Consequently ${{\hat{B}}_{s}, \forall s}$ fails to belong to feasible contract, indicating that ${B_{s}^{*}, \forall s}$ is the best.

We then demonstrate its uniqueness. Using contradiction method again and supposing another feasible bandwidths ${{\hat{B}}_{s}, \forall s} \neq {B_{s}^{*}, \forall s}$ exists such that $\sum_{s = 1}^{S} N_{s} {\hat{B}}_{s} = \sum_{s = 1}^{S} N_{s} B_{s}^{*}$ , we have at least two bandwidths ${\hat{B}}_{s} \neq B_{s}^{*}$ . We pick out the one that ${\hat{B}}_{t} < B_{t}^{*}$ . Obviously we could recall the previous part of the proof of optimality and find out the violation of the IR constraint for type $θ_{1}$ FBS, so the uniqueness is also guaranteed. ■

With Proposition 1, equation (15) could be transformed to

\begin{matrix} \arg max_{{R_{s}, \forall s}} u_{0}, \\ subject to 0 \leq R_{1} \leq R_{2} \leq \dots \leq R_{S} \end{matrix}

(21)

where

\begin{matrix} u_{0} = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{s = 1}^{S} N_{s} R_{s}) - b_{0}]}} \\ - d_{0} \sum_{s = 1}^{S} N_{s} (θ_{1} R_{1} + \sum_{t = 2}^{s} θ_{t} (R_{t} - R_{t - 1})) \end{matrix}

(22)

Theorem 3

As an optimal contract, the contract items for FBSs with the lowest type are positive, while contract items for others are zero, that is

\begin{matrix} (R_{S}, B_{S}) > 0, \\ (R_{s}, B_{s}) = 0, \forall s < S \end{matrix}

Proof

Contradiction method is used here. Suppose there is an optimal contract in which $R_{s} > 0$ for $\forall s < S$ . Thereafter, the overall rate is $\hat{R} = \sum_{s = 1}^{S} N_{s} R_{s}$ and the utility of MBS becomes

\begin{matrix} {\hat{u}}_{0} = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \hat{R}) - b_{0}]}} \\ - d_{0} \sum_{s = 1}^{S} N_{s} (θ_{1} R_{1} + \sum_{t = 2}^{s} θ_{t} (R_{t} - R_{t - 1})) \end{matrix}

If such a constant overall rate $\hat{R}$ is only provided by type $θ_{S}$ FBSs, there would have $\hat{R} = N_{S} R_{S}$ and

\hat{u}'_{0} = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \hat{R}) - b_{0}]}} - d_{0} N_{S} θ_{S} R_{S}

It is obvious that

\sum_{s = 1}^{S} N_{s} (θ_{1} R_{1} + \sum_{t = 2}^{s} θ_{t} (R_{t} - R_{t - 1})) > N_{S} θ_{S} R_{S}

so $\hat{u}'_{0} > {\hat{u}}_{0}$ , violating the optimality of the contract and further proving the theorem. ■

Theorem 3

It facilitates us to simplify the optimization problem (21) as

\begin{matrix} R_{s}^{*} = 0, \forall s = 1, 2, \dots, S - 1 \\ R_{S}^{*} = \arg max_{R_{S} \geq 0} {\tilde{u}}_{0} \end{matrix}

(23)

where

\tilde{u} \sqrt{b^{2} - 4 {ac}_{0}} = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + N_{S} R_{S}) - b_{0}]}} - d_{0} N_{S} θ_{S} R_{S}

Because Sigmoid function contains both concave and convex segment, it is difficult to give a closed-form solution $R_{S}^{*}$ . However, one-dimensional exhaustive search could be applied here in order to obtain the global optimal solution.⁴¹

Based on the above-mentioned theorems and proposition, we could derive the theorem below.

Theorem 4

$Ψ = {R_{s}^{*}, B_{s}^{*}, \forall s \in S}$ is optimal, deducted from Proposition 1, Theorem 3, and equation (23).

Further discussions

Interaction protocol

In this part, we elaborate the interaction between the MBS and the FBSs. The algorithm consists of four steps:

The MBS broadcasts the possible contract $Ψ = {(R_{s}, B_{s}), \forall s \in S}$ to FBSs.

Each FBS chooses one contract and feeds its choice back to the MBS. To be competitive, every FBS would make efforts to make its type as small as possible.

According to the contract, the MBS pays some spectrum to each involved FBS, and the FBS purchases the residual part of the bandwidth.

After everything is set down, the MBS and the FBSs begin to give service.

Spectrum paid by the MBS

Compared with the idea that the MBS could reimburse some gratuity directly, it appears confusing to some extent that the MBS refunds the FBSs with spectrum. We will explain it in terms of social surplus.

In our model, social surplus is defined as the aggregate utility of MBS and all involved FBSs

\begin{matrix} SS = u_{0} + \sum_{s = 1}^{S} N_{s} u_{s} \\ = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{s = 1}^{S} N_{s} R_{s}) - b_{0}]}} - d_{0} \sum_{s = 1}^{S} N_{s} B_{s} \\ + \sum_{s = 1}^{S} N_{s} d_{s} (B_{s} - θ_{s} R_{s}) \\ = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{s = 1}^{S} N_{s} R_{s} - b_{0})]}} - \sum_{s = 1}^{S} N_{s} d_{s} θ_{s} R_{s} \\ + \sum_{s = 1}^{S} (d_{s} - d_{0}) N_{s} B_{s} \end{matrix}

In contrast, if the MBS directly pays refund $m_{s}$ in terms of money to type $θ_{s}$ FBS, the social surplus would be

\begin{matrix} SS' = u_{0} + \sum_{s = 1}^{S} N_{s} u_{s} \\ = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{s = 1}^{S} N_{s} R_{s}) - b_{0}]}} - \sum_{s = 1}^{S} N_{s} m_{s} \\ + \sum_{s = 1}^{S} N_{s} (m_{s} - d_{s} θ_{s} R_{s}) \\ = v_{0} \frac{1}{1 + e^{- a_{0} [(R_{0} + \sum_{s = 1}^{S} N_{s} R_{s} - b_{0})]}} - \sum_{s = 1}^{S} N_{s} d_{s} θ_{s} R_{s} \end{matrix}

Due to the low price of spectrum resulted from MBS’ preemption, that is, $d_{0} < d_{s}, \forall s \in S$ , $SS > SS'$ is established, indicating that the social surplus of the former one prevails.

Bandwidth constraint

In section “Utility of the FBS,” we assumed that

B_{n}^{actual} = B_{m, n} + B_{f, n} - B_{n} = \frac{R_{n}}{x_{n} \log (1 + α_{n})} - B_{n} \geq 0

(24)

which means that the bandwidth the MBS paid for $FB S_{n}$ would be fully utilized. At this moment, we validate this assumption.

In terms of Proposition 1 and Theorem 3, we have known that $(R_{s}, B_{s}) = 0, \forall s < S$ and $B_{S} = θ_{S} R_{S}$ . Therefore, all of them could be denoted as $B_{n} = θ_{n} R_{n}$ for all N FBSs.

Contradiction method is then used and suppose that

B_{n} > \frac{R_{n}}{x_{n} \log (1 + α_{n})}

(25)

Substituting

B_{n} = θ_{n} R_{n} = \frac{d_{n} - v_{n} (1 - x_{n}) \log (1 + β_{n})}{d_{n} x_{n} \log (1 + α_{n})} R_{n}

into equation (25), we would derive

\frac{d_{n} - v_{n} (1 - x_{n}) \log (1 + β_{n})}{d_{n} x_{n} \log (1 + α_{n})} > \frac{1}{x_{n} \log (1 + α_{n})}

and further

v_{n} (1 - x_{n}) \log (1 + β_{n}) < 0

(26)

Apparently, equation (26) fails to be satisfied, thereafter equation (24) is guaranteed, implying that the refunded bandwidth could always be employed.

Limited bandwidth in practice

In our previous parts, we assume that the MBS could purchase as much bandwidth as possible without any upper bound. However, in practice, there is always limited bandwidth available. Let $B^{up}$ denote the maximum bandwidth the MBS could buy, and we should consider the situation where $N_{S} B_{S}^{*} > B^{up}$ .

To cope with the problem, we randomly pick out $\bar{N_{S}}$ FBSs and

\bar{N_{S}} = ⌊ \frac{B^{up}}{B_{S}^{*}} ⌋

(27)

where $⌊ C ⌋$ denotes the biggest integer no more than C.

On this line, in case $N_{S} B_{S}^{*} > B^{up}$ , the MBS would give the optimal contracts to $\bar{N_{S}}$ randomly chosen FBSs instead of overall $N_{S}$ FBSs. Meanwhile, the MBS offers any contracts deviated from ${R_{S}^{*}, B_{S}^{*}}$ to those $N_{S} - \bar{N_{S}}$ FBSs failing to be chosen. Hence, we avoid the excessive requirement on bandwidth.

Incomplete information on contract design

In this section, we focus on the special scene where the MBS does not even know the number of each type of FBSs and it only knows the total number of FBSs N, the probability $q_{k}$ of any FBS belonging to type $θ_{S}$ , and the estimated maximum type, that is, $θ_{1}$ of FBSs. The difference from the above section is that here the MBS does not know which type is the lowest type among all FBSs in the current system and how many FBSs are in the lowest type, and thus the simple approach of only providing a positive contract item for type $θ_{S}$ as in Theorem 3 may not be optimal. If the MBS does that and it turns out that $N_{S} = 0$ in the current system, then there will be no FBSs participating in the cooperation with MBS.

To address the optimization in this scenario, we adopt the low computation complexity approximate algorithm, decompose-and-compare algorithm, to compute a close-to-optimal solution efficiently. Specifically, in this algorithm, we will compare S simple candidate contracts and pick the one that yields the largest utility for the MBS. The key ideas behind this heuristic algorithm are as follows:

One positive contract item per contract. In each of the S individually optimized candidate contracts, there is only one positive contract item (for one or more types). For the contract item in the sth candidate contract, this positive contract item is offered to FBSs with types equal to or smaller than type $θ_{s}$ . In other words, optimization of each candidate contract only involves a scalar optimization, instead of S variables.

A tradeoff between efficiency and uncertainty. Among S optimized candidate contracts, the best one that achieves the best tradeoff between efficiency and uncertainty will be picked. Under the incomplete information, it is not clear which type is the lowest among all FBSs existing in the current system. If a candidate contract offers the same positive contract item for types equal to or smaller than type $θ_{s}$ , then all FBSs in these types will choose to accept that contract item. The corresponding FBS’ payoff is decreasing in type, that is, a type $θ_{s}$ FBS receives zero payoff and a type $θ_{S}$ FBS receives the maximum positive payoff. Thus, choosing a candidate contract with a threshold type $θ_{s}$ too high will give too much payoffs to the FBSs (and thus reduce the MBS’ expected utility), but choosing a candidate contract with a threshold too low might lead to the undesirable case that no FBSs are willing to participate. This requires us to examine all possibilities (i.e. S candidate contracts) and pick the one with the best performance.

The decompose-and-compare algorithm works as follows:

Decomposition. Construct S candidate contracts, where for the sth contract

Offers the same contract item $(R_{s}, B_{s}) > 0$ to FBSs with a type equal to or larger than type $θ_{s}$ (called critical type), and zero for the FBSs above the critical type. That is, $(R_{1}, B_{1}) = (R_{2}, B_{2}) = \dots = (R_{s - 1}, B_{s - 1}) = 0$ , and $(R_{s}, B_{s}) = (R_{s + 1}, B_{s + 1}) = \dots = (R_{S}, B_{S})$ .

Computes the optimal $(R_{s}^{*}, B_{s}^{*})$ that maximizes MBS’ expected utility under the constraints of $(R_{1}, B_{1}) = (R_{2}, B_{2}) = \dots = (R_{s - 1}, B_{s - 1}) = 0$ and $(R_{s}, B_{s}) = (R_{s + 1}, B_{s + 1}) = \dots = (R_{S}, B_{S})$ .

Comparison. Choose the best contract out of S candidates to maximize the MBS’ expected utility.

Unlike the algorithm in common scene, here the computational complexity increases with the number of FBS types. This is because that the MBS may want to involve more than one type of FBSs in the contract to mitigate the uncertainty and avoid risk of having no relay in a particular network realization. However, the complexity of this approach will increase with the number of types. In addition, in section “Simulation results,” we show by numerical results that the proposed decompose-and-compare algorithm achieves a performance very close to the optimal solution in most cases.

Simulation results

In this section, we present some numerical results in order to evaluate how contract theory performs in our scenario. We consider an area of $1000 m \times 1000 m$ , which is covered by a MBS. The MBS is located in the center of this area. There are 600 FBSs randomly deployed in this region.

In our simulation setting, there are overall $N_{S} = 100$ FBSs with type $θ_{S}$ unless otherwise specified. Without loss of generality, the data rate per bandwidth that the FBSs serve MUs via hybrid access are identical to $R_{S} = 5$ , and each FBS serves its own FUs at a fixed data rate $R_{f, S} = 8$ . The utility gain per data rate is $v_{0} = 100$ for the MBS and $v_{n} = 4$ for FBSs. The MBS could provide $R_{0} = 1.6$ alone to serve MUs without any help from FBSs, while the reserved data rate requirement is $b_{0} = 1$ . Also, MUs’ sensitivity toward data rate increment is set to be $a_{0} = 0.5$ . The MBS could pay the spectrum at unit price $d_{0} = 15$ and the FBSs pay at $d_{n} = 35$ . All results are based on the above-mentioned parameters unless specified otherwise. In order to facilitate comparison, we add the circumstance where all FBSs choose closed access, and at this time the utility of MBS is calculated as $v_{0} / (1 + e^{- a_{0} (R_{0} - b_{0})})$ .

As our focus, the open proportion affects the type of each FBS and consequently the data rate of MUs served by the FBSs. It has been discussed before that a larger open proportion would make the type smaller and make the data rate of MUs higher. Figure 2 demonstrates such phenomena. When the open proportion rises from 0 to 1, in other words from closed access to hybrid access and ultimately to open access, the service data rate increases as well. The service rate from FBSs to MUs is another crucial factor since poor SNR counteracts the advantage of more bandwidth. We depict different situations where $R_{n}$ ranges from 4 to 8. With identical open proportion, it is apparent that better SNR is more beneficial to service rate.

Figure 2.

Date rate $R_{S}$ of MUs served by each type $θ_{S}$ FBS with different open proportion $x_{n}$ .

Thanks to the service data rate, the utility of MBS would be correspondingly enhanced. Figure 3 shows the contribution of FBSs’ open proportion to the MBSs utility. Notice here that the increase in utility is particularly remarkable when the FBSs step off closed access. Hence, merely a small part of spectrum is able to improve the utility of MBS to a large extent, meaning that each FBS could reserve abundant spectrum for its own FUs instead of sacrificing a lot.

Figure 3.

Utility of MBS u₀ with different open proportion $x_{n}$ .

The type of FBS is another important element. We have shown that smaller type is more competitive. Therefore, we investigate the relationship between $θ_{S}$ and data rate $R_{S}$ . From Figure 4, we could see that a smaller $θ_{S}$ results in a higher service data rate $R_{S}$ , which benefits MUs more. The number of type $θ_{S}$ FBS also plays an evident role. Each type $θ_{S}$ FBS would offer less data rate if $N_{S}$ increases. The underlying reason is that, $θ_{S}$ when is fixed, the total achievable data rate of MUs served by FBSs, that is, $N_{S} R_{S}$ is also fixed, and a larger $N_{S}$ leads to a smaller $R_{S}$ . Figure 4 illustrates the tendency ranging from 60 to 200 type $θ_{S}$ FBSs.

Figure 4.

Date rate $R_{S}$ of MUs served by each type $θ_{S}$ FBS with different smallest type $θ_{S}$ .

The analysis of remuneration to type $θ_{S}$ FBS is shown in Figure 5, revealing that the total bandwidth $N_{S} B_{S}$ refunded to the FBSs increases at first and decreases subsequently. Note that there is a turn of the curves. When $θ_{S}$ is small enough, the MBS could obtain a high utility even though reciprocating less bandwidth, yet it is less worthwhile to pay much bandwidth to FBSs with larger $θ_{S}$ considering their incapability.

Figure 5.

Total bandwidth refund $N_{S} B_{S}$ to each FBS with different smallest type $θ_{S}$ .

Intuitively, the price for the MBS to purchase bandwidth affects MBS’ utility as well as FBSs’ willingness to help. We thus manage to observe the utility of MBS by gradually increasing the unit price. Figure 6 checks the standpoint that higher price appears to be less attractive and leads to the decrease in utility. Hence, the difference between the price for the MBS and for the FBSs should be great enough as the motivation.

Figure 6.

Utility of MBS u₀ with different unit price of bandwidth for MBS d₀.

Moreover, we add a random algorithm for the comparison with the proposed contract theory–based algorithm. Specifically, the random algorithm picks out $N_{S}$ , which is the same as that obtained from the contract theory, FBSs at random to help serve the MUs. Then, the MBS pays some spectrum to these FBSs. Particularly, the amount of the spectrum paid to each selected FBS is calculated based on the sequential optimization approach in section “Optimal contract design.” With varying $N_{S} s$ , Figure 7 shows the utility of MBS with the FBSs selected based on contract design and the randomly selected FBSs, respectively. It is easily seen that, compared with the randomly selected FBSs, the FBSs selected based on contract design always achieves much better performance in improving the utility of MBS.

Figure 7.

Utility of MBS with different $N_{S} s$ .

Finally, to illustrate the performance of decompose-and-compare algorithm for incomplete information in section “Simulation results,” we consider only two types of FBSs: $θ_{1} > θ_{2}$ . The MBS only knows the total number of FBSs N and the probabilities q₁ and q₂ of two types, with $q_{1} + q_{2} = 1$ . Particularly, we consider two candidate contracts. The first candidate contract optimizes the same positive contract item $(R_{1}, B_{1}) = (R_{2}, B_{2}) > 0$ for both types. The MBS’ corresponding maximum expected utility is $E [u_{0}]^{1 - 2}$ . The second candidate contract sets $(R_{1}, B_{1}) = 0$ and optimizes the positive contract item $(R_{2}, B_{2}) > 0$ . The MBS’ corresponding maximum expected utility is $E [u_{0}]^{2}$ . Then we pick the candidate contract that leads to a larger MBS’ expected utility as the solution of the decompose-and-compare algorithm. In addition, as a benchmark, we compute the optimal solution to the MBS’ expected utility maximization problem via an K-dimensional exhaustive search and denote the corresponding optimal solution as $E [u_{0}]^{*}$ . Figures 8 and 9 show the comparison between MBS’ optimal expected utility using optimal exhaustive search method $(E [u_{0}]^{*})$ and the two candidate contracts of the decompose-and-compare algorithm, as a function of the open proportion $x_{n}$ . Specifically, two different parameter regimes, that is, large q₁ and small q₁, are considered. The large q₁ means that the probability that all FBSs belong to the high type $θ_{1}$ is large. From Figure 8, we can see that, with a large q₁, the candidate contract that offers the same positive contract items to both types (i.e. $E [u_{0}]^{1 - 2}$ ) achieves a close-to-optimal performance with all values of open proportion $x_{n}$ simulated here. This is because very often the MBS needs to rely on the high type $θ_{1}$ FBSs to help its transmission. Then, it is observed from Figure 9 that, with a small $q_{1}^{N}$ ( $E [u_{0}]^{2}$ ) is always better than ( $E [u_{0}]^{1 - 2}$ ) and achieves a close-to-optimal performance under all choices of $x_{n}$ This is because very often the MBS can find a low type $θ_{2}$ FBS for cooperation.

Figure 8.

Comparison between MBS’ optimal expected utility using optimal exhaustive search method and the two candidate contracts of the decompose-and-compare algorithm, as a function of the open proportion $x_{n}$ . We set $q_{1} = 0.9$ and N = 2.

Figure 9.

Conclusion

In this article, we investigate hybrid access for the femtocells with asymmetric information. To maximize the utility of MBS, the FBSs would open a portion of their spectrum to help serve MUs, while the MBS pays part of the spectrum to the FBSs as refund. Considering asymmetric information, we build a principal–agent model and use the rate–bandwidth combinations to describe the interaction between the MBS and the FBSs. We introduce contract theory to model the problem, incorporating the IR and IC constraints. We define the type of FBS at first and then derive the optimal contract after studying the necessary and sufficient conditions for feasible contract. In this way, the service quality of MUs could be improved thanks to the open spectrum of the FBSs. Also, several issues are discussed to corroborate our design.

Footnotes

Academic Editor: Sangheon Pack

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (61174127, 61573245).

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Hybrid access for femtocells based on contract theory

Abstract

Keywords

Introduction

Motivation

Related works

Contribution

System model

Problem formulation with contract theory

Utility of the MBS

Utility of the FBS

Contract formulation

Definition 1: IR

Definition 2: IC

Feasible contract

Necessary condition 1

Corollary 1

Necessary condition 2

Theorem 1

Proof

Proof of point 1

Proof of point 2

Theorem 2

Proof

Optimal contract design

Proposition 1

Proof

Theorem 3

Proof

Theorem 3

Theorem 4

Further discussions

Interaction protocol

Spectrum paid by the MBS

Bandwidth constraint

Limited bandwidth in practice

Incomplete information on contract design

Simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References