FEVER Mechanism

Election Rule (VER). First, CIH requires all MSes to report their CQI θ_i ={Li,f, L_i,m}.

Then, CBH uses the channel information harvested by CIH to calculate MSes' most pre-ferred downlink power p^∗_i. These values are derived by H({θi|Mi ∈ S}) = H(¯θ) = {p^∗i|Mi ∈ S}, where H(·) is in accordance with the results given by (2.7). Finally, VER collects{p^∗i|Mi ∈ S} derived from CBH and applies Virtual Election rule to choose the femtocell downlink power P_f. Then, in Cell Selection stage there are two levels: BEst response Dynamic (BED) and Max-min Allocation Rule (MAR). According to the chosen P_f, MSes select the cell they want to be served by, which can be predicted by Theorem 1. Thus S_f and S_m is decided in this level. Since we have all CQI from CIH, we induce a cheat-proof procedure here: if an MS M_j which is not in S_f(P_f) calculated by CBH selects the femtocell in this stage, we surely know this MS is cheating in Cell-Breathing stage and thus we can block this MS from using the femtocell service. Finally, the femto-cell allocates the backhaul data rate according to the max-min allocation rule A_{f ever}(·).

Then, we give the following theorem to show the truthfulness of FEVER mechanism.

Theorem 3 (Truthfulness of FEVER mechanism). All M_ireport their CQI θ_i^∗ ={^∗_i,f,^∗_i,m} truthfully in FEVER mechanism B_{f ever}(·). That is, ∀Mi ∈ S, ∀¯θ ∈ Θⁿ= (L× L)ⁿ,

u_i(A_i(B_{f ever}( ¯θ^∗))) ≥ ui(A_i(B_{f ever}(¯θ))).

Proof. Due to page limitation, we sketch the outline of the proof here. We discuss three cases: M_i's vote v_iderived in CIH is equal to, larger than, or smaller than the selected vote v_κ. For the first case, surely M_iwill just report its CQI truthfully. For the second case, the vote chosen by FEVER is larger than its most preferred downlink power v_i. Therefore, γ_i,f = Γ_f(v_κ) according to Assumption 1 and Theorem 2. If M_ichooses to report L^′_i,f <

L^∗_i,f, the modified vote v_i^′ will be larger than v_i, and the vote chosen by VER v_κ^′ will be no smaller than v_κ according to VER. So, γ_i,f^′ = Γ_f(v^′_κ) ≤ Γf(v_κ). Since it eventually gets a lower downlink data rate, it has no incentive to report L^′_i,f < L^∗_i,f. However, if M_i chooses to report L^′_i,f > L^∗_i,f, since the modified vote v^′_i will be smaller than vi, and the original selected vote v_κ is larger than v_i, reporting L^′_i,f makes no difference to v_κ.

Additionally, since γ_i,f is already not constrained by its wireless data rate Γ_i,f(v_κ), its downlink data rate remains the same γ_i,f^′ = Γ_f(v_κ) = γ_i,f. Therefore, it has no incentive to report L^′_i,f > L^∗_i,f. Similarly, in the third case M_i still chooses to report L^∗_i,f. For L_i,m, given Mi ∈ Sj if vκ is selected and L^∗_i,m is reported, it makes no difference to vκ unless given the reported L^′_i,m, Mi will make a choice of S_j^′ ̸= Sj|j={f,m}. However, since Mi's true CQI is L^∗_i,m, it will choose S_jin the Cell Selection stage, and thus the cheating will be revealed by the femtocell. Hence, FEVER mechanism is a truthful mechanism. Note that this proof requires the capacity assumption is satisfied or Γ_f(p) may not be a decreasing function.

If the capacity assumption, that is,∀Mi ∈ S, ^C_N^f ≥ γi,m, is not satisfied, the prefer-ence on power is no longer single-peaked. However, FEVER mechanism is still truthful when the selected vote order κ is equal to 1. We show this characteristic in the following theorem.

Theorem 4 (Truthfulness without capacity assumption). When∃Mi ∈ S, ^C_N^f < γ_i,m, all M_i still report their CQI θ^∗_i = {^∗i,f,^∗_i,m} truthfully in FEVER mechanism Bf ever(·) when selected vote order κ = 1.

Proof. Since κ = 1,∀Mi ∈ Sf, γ_i,f = Γ_i,f(v₁). For a given M_j ∈ Sf, if it chooses to re-port^′_i,f <^∗_i,f, its allocated data rate Γ^′_i,f is strictly lower than Γ_i,f because it claims a lower wireless data rate. In contrast, if it chooses to report^′_i,f > ^∗_i,f, it claims a higher wireless data rate and the CBH will decide a lower vote v₁^′ < v₁, and thus its real downlink data rate γi,f = Γ(v^′₁^∗_i,f) < Γ(v^′₁^∗_i,f). Thus, it will truthfully reporti,f. For ^∗_i,m, a misreported

′i,m does not affect the vote v₁ unless it is large enough that the MS will not be counted in S_f in CBH. According to the cheat-proof procedure in BED, this MS cannot return to the femtocell and its downlink data rate is γ_i,m < Γ_i,f(v₁). Hence,∀Mi ∈ Sf, they all report CQI θ^∗_i. Similarly,∀Mj ∈ Sm, they also report CQI θ^∗_j. Thus FEVER mechanism is truthful even when∃Mi ∈ S, ^C_N^f < γ_i,m.

The choice of κ = 1 not only relaxes the capacity assumption but also makes FEVER mechanism to choose the most capacity efficient operation point in most network scenario,

which will be shown in the following section.

2.5.1 Performance Analysis of FEVER mechanism

In FEVER mechanism, the selected vote order κ is predefined in VER to derive the κth votes as the choice of P_f. Now we discuss the influence of κ on the (throughput) efficiency and (allocation) fairness of the resulting expected throughput{γi,f|Mi ∈ Sf} and{γi,m|Mi ∈ Sm}. We will show that there is a tradeoff between efficiency and fairness when different κ is chosen.

Price of Anarchy

First we define the price of anarchy (P OA): the ratio of the maximum social utility to the social utility at NE. P OA is used to measure the efficiency loss due to the selfish behavior of MSes in NE. If P OA is equal to one, there is no efficiency loss at NE, and NE leads to the same performance as the social-utility maximized system. In the femtocell cell-breathing control framework, we define the social utility as Us = ^∑_Mi∈Sui. The price of anarchy of FEVER mechanism is described as below:

Theorem 5. The price of anarchy of FEVER mechanism P OA(κ) is one if the selected vote order κ = 1 and the social-utility maximized power P_f^∗can be described as

∑j i=1

Γ(P_f^∗L_i,f)≤ Cf and

j+1∑

i=1

Γ(P_f^∗L_i,f)≥ Cf.

In addition, P OA(κ) is increasing with κ and bounded by 2.

Proof. First, we shortly describe how to find the optimal downlink power in the fem-tocell: we first sort MSes in increasing order of ^L_L^1,m

1,f. (This guarantees that MSes join the femtocell sequentially when the femtocell BS increases P_f.) Intuitively, the optimal power that maximizes social utility is the one that just allocates all the available back-bone capacity to MSes. If the solution does not exist and the macrocell's data rate is rel-atively large, the optimal power will be the one that minimizes the unallocated backbone data rate before an additional MS joins S_f and uses all the capacity. Thus, the optimal

power P_f^∗is given by^∑j_i=1Γ(P_f^∗L_i,f)≤ Cf and^∑j+1_i=1Γ(P_f^∗L_i,f) > C_f, where j satisfies P_f^∗L_i,f ≥ PmL_i,m, ∀i ≤ j and Pf^∗L_i,f < P_mL_i,m ∀i ≥ j. If^∑ji=1Γ(P_f^∗L_i,f) = C_f, the social utility U_s^∗ =^∑_Mi∈Su_i = C_f+^∑N_i=j+1Γ_i,m. We observe that P_f, which is the peak value of M1, is equal to PF according to Theorem 2. Thus, if κ = 1, FEVER mechanism always chooses P_f^∗, and thus P OA(1) = 1.

The reason that P OA(κ) is an increasing function is intuitive. When P_f increases, there are two possible cases: one is that a new MS M_j joins S_f, and the other is that no MS changes its selection. For the former case, the social utility will decrease by γ_j,msince M_jchooses to join S_f and gives up the throughput offered by the macrocell. For the latter case, there is no change in the resulting social utility. In addition, we know v_κ increases with κ, so P_f = B_{f ever}(·, κ) is an increasing function of κ. Thus, P OA(κ) is an increasing function of κ.

Finally, since the worst case in FEVER mechanism will be that all users choose to join the femtocell and all macrocell's offered services are wasted under selfish behavior, we have γ_i,f|κ=N > γ_i,m ∀Mi ∈ S. In addition, we know^∑M_i ∈ Sγi,f|κ=N = C_f. Thus, P OA(κ)≤ P OA(N) = ^Cf+

∑

Mi∈Sm(P ∗f)γi,m

Cf < ^Cf_C^+Cf

f = 2.

Theorem 5 tells us that when κ = 1, FEVER mechanism always chooses the down-link power that maximizes the social utility. In addition, according to Theorem 4, even the capacity assumption is relaxed, FEVER mechanism can still maximize the social utility because all MSes truthfully report their CQI when κ = 1. For κ > 1, the social util-ity decreases with κ since more MSes joins S_j and waste the throughput offered by the macrocell. Thus the output becomes inefficient.

Noted that in some scenarios, the optimal power P_f^∗ may not be the one described in Theorem 5. This happens when γ_j+1,m < C_f −^∑ji=1Γ(P_f^∗L_i,f). In this case, allowing M_j+1to join the femtocell is beneficial to the system since the unallocated backhaul ca-pacity is significantly large. In this case P OA(1) will be slightly larger than one. The degree of efficiency loss depends on the macrocell downlink data rate of γ_j,m. Given γ_j,m, P OA(1) < ^Cf+γ_C ^j,m

f = 1 + ^γ_C^j,m

f . If capacity assumption is satisfied, P OA(1) < 1 +_N¹.

Fairness

Then, we discuss the fairness among M_i ∈ S when κ changes. Since the allocation rule A_{f ever}(·) follows Assumption 1, the resource allocation (backhaul capacity) among all MSes in S_f surely satisfy max-min fairness. In addition, given a predetermined κ, all MSes' utilities satisfy the max-min fairness under FEVER mechanism.

Theorem 6. Given κ, all MSes' utilities{ui|Mi ∈ S} under FEVER mechanism satisfies max-min fairness, that is,

∃{u^′_i} ̸= {ui}, u^′_k > u_k for some M_k ∈ S

⇒ ∃Mi ∈ S, u^′i < u_i ≤ uk.

Here we would like to further discuss the fairness efficiency among all MSes in the overlay system. To discuss the fairness efficiency, we apply a common fairness index: Jian fairness index I({ui}) = ⁽

∑

1≤i≤Nui)²

N∑

1≤i≤Nu²_i [24] here. Jian fairness index is bounded between

0 and 1. A higher fairness index means the resource allocation is fairer. The following theorem describes the fairness efficiency of FEVER mechanism.

Theorem 7. The fairness index I({ui}) of output of FEVER mechanism is an increasing function of κ.

Proof. We use Property 1 in [24] to prove this. Before that, we define the reallocation sequence{(Mi¹, M_j¹, du₁), (M_i², M_j², du₂), ...}, where (Mi^l, M_j^l, du^l) means M_i^l's utility (downlink data rate) minus du^l while M_j^l's utility plus du¹. We also recall the notations S_f^u(p) and S_f^c(p) in the proof of Theorem 1 in Appendix.

According to Theorem 1, for a given p and p^′ = dp + p > p, we have S_f(p) ⊂ Sf(p^′) and γ_f(p) > γ_f(p^′). Without losing generality, we assume dp is small enough that only one of following cases happens when power is increased to p^′: Sf(p^′) = Sf(p) or Sf(p^′) = S_f(p)∪ {MSj}.

Since the reallocation only happens in MSes in Sf(p^′), we focus on the effect of these users and ignore users served by the macrocell. For the first case, no new MS joins S_f(p^′).

Since^∑_Mi∈Sf(p′)u_i = ^∑_Mi∈Sf(p)u_i = C_f and γ_f(p) > γ_f(p^′), there exists at least one reallocation sequence{(Mi^l, M_j^l, du^l)}, where Mi^l ∈ Sf^c(p) and M_j^l ∈ Sf^u(p^′), can realize the reallocation from{γi,f(p)} to {γi,f(p^′)}.

According to the proof of Theorem 1 in Appendix,∀Mi ∈ Sf^c(p), γ_i,f(p^′) = γ_f(p^′) <

γf(p). And ∀Mj ∈ Sf^u(p^′), γj,f(p) ≤ γj,f(p^′) ≤ γf(p^′). Thus, for all reallocation (M_i^l, M_j^l, du^l),du^l ≤ γi^l,f(p) − γi^l,f(p^′) = γ_il,f(p) − γ(p^′) ≤ γi^l,f(p) − γj^l,f(p) = u^l_i(p)− u^lj(p).

According to Property 1 in [24], Jian fairness index increases when du≤ ui−uj. This means after every reallocation (M_i^l, M_j^l, du^l), Jian fairness index increases. Thus, the final output I({ui(p^′)}) is larger than I({ui(p^′)}) when p^′ > p in the first case. Similarly, In the second case we can reach the same conclusion. Thus, the fairness index of the output of FEVER mechanism is an increasing function of femtocell downlink power p. Finally, since v_κ is increasing with κ, I(κ) = I({ui(v_κ)}) is increasing with κ.

Notice that when κ = N and p^∗_N ≤ Pf^max, the output of FEVER mechanism has a fairness index of one.

Observed in Theorem 5 and 7, there is a tradeoff between throughput efficiency and allocation fairness. When a larger κ is chosen, the output of FEVER mechanism becomes less capacity efficient since more users give up the macrocell services. Doing so increases the allocation fairness since these users now are offered with higher expected throughput, in exchange with lower expected throughput to those users already selected by the femto-cell. The choice of κ should be application-oriented. On one hand, if the service provider wants to make use of the additional backhaul data rate offered by the femtocell efficiently, he can choose a smaller κ. On the other hand, if he wants to make users share the backhaul fairly, he may choose a larger κ.

2.6 Subscriber Group Modes

We now investigate the compatibility of FEVER mechanism to subscriber group modes in femtocell system. In implementing the subscriber group modes in femtocells, two

is-sues are of concerns: access control on low-priority MSes and resource reservation for high-priority MSes. These two concerns change the access policy of femtocells and may brings some incompatibility to existing access policies for typical cell base stations.

We first apply necessary extensions into our downlink cell-breathing framework and two-stage game model. We add a new phase: Access Control Phase between Cell Breath-ing and Cell Selection phases into the framework. In this phase, the femtocell should decide and broadcast the allowed mobile user set S^a. Only users in S^a are allowed to access the femtocell. Notice that the choice of S^adepends on the subscriber group mode and the available resource.

To model the user behavior in the new framework, we extend the two-stage game model in Section 2.3. The Access Control Phase is included in the first stage (Cell-Breathing Stage) of our two-stage game model. In the new game model, the Cell-(Cell-Breathing rule is extended to output not only the femtocell downlink power Pf but also the allowed mobile user set S_a. In addition, in the Cell-Selection stage we impose a new game rule:

only users belonging to S_a have the right to access the femtocell. Mathematically, the player set in the second stage is now restricted to S^a instead of S. Other users M_i ̸∈ S^a can only select the macrocell. In the femtocell, a subscriber group S^g ⊂ S is predeter-mined. Each MS M_i ∈ S^ghas a desired expected throughput γ_i^req. We assume their utility is maximized whenever the desired rate is satisfied. Thus, the utility function of M_i ∈ S^g is ui(γ) = min(γ, γ_i^req). Lastly, we assume the backhaul data rate can support at least the demand of these MSes, that is, C_f ≥ ^∑Mi∈S^gγ_i^req. With this assumption, we define the reserved capacity function C^g(p):

C^g(p) = ^∑

Mi∈S^g,a(p)

(min(Γ_i,f(p), γ_i^req)), (2.10)

where M_i ∈ S^g,a(p) if and only if M_i ∈ S^g and Γ_i,f(p) ≥ Γi,m. C^g(p) is the backhaul data rate reserved for users in S^g. Notice that the assumption and reserved data rate imply that when all users in S^g select the femtocell, they all can still have a reasonable service quality. For other users, they may have an unacceptable (lower than the one provided by macrocell) expected throughput if users in S^g have higher priority.

2.6.1 FEVER Mechanism in Subscriber Group modes

We now propose Subscriber Group FEVER (SG-FEVER) mechanism, a voting-based truthful mechanism that is compatible to three subscriber group modes. Similar to FEVER mechanism, SG-FEVER is composed of three levels in Cell-Breathing stage: Channel Information Harvester (CIH), Subscriber Group Cell-Breathing Helper (SG-CBH), and Subscriber Group Virtual Election Rule (SG-VER).

In SG-CBH, instead of directly applying max-min allocation rule to derive the peak of each user, we first define S^g,a(p) and C^g(p) according to (2.10). Then, the max-min allo-cation rule A_{f ever}(·) is applied to these two groups independently: the reserved capacity C^g(p) is used for users in S^g,a(p), while the remaining capcity C^−g(p) = C_f − C^g(p) is used for users in S^−g(p), which is given by

S^−g(p) ={Mi|Γi,f(p)≥ Γi,m and Mi ̸∈ S^g}.

The peak p^∗_i and the wireless data rate Γ_i,f(p) of each user M_i ∈ S are derived in this process.

Finally, SG-VER collects{p^∗i|Mi ∈ S} derived from CBH and applies Virtual Election rule to choose the femtocell downlink power P_f with a predetermined selected vote order κ. In addition, SG-VER broadcasts the allowed user set S^a, which depends on the choice of subscriber group mode.

Then, in Cell Selection stage there are still two levels: BEst response Dynamic (BED) and Subscriber Group Max-min Allocation Rule (SG-MAR). In SG-FEVER mechanism, MSes select the BS they want to be served by based on their best responses given in Theorem 1 if they are in the allowed user set S^a. If not, the users have no choice but to choose the macrocell BS as their serving BS . Thus, S_f and S_mare decided in this level.

The cheat-proof procedure in SG-FEVER is enhanced: if an MS is not in S^g(P_f) selects the femtocell in this stage, femtocell BS blocks the MS form using femtocell services.

Finally, the femtocell allocates the backhaul data rate according to the decision in SG-CBH, and the expected throughput is then determined. Since the implementation of OSG

is straightforward, we discuss the implementation of other subscriber group modes.

CSG Mode

In this mode, only users in S^gare allowed to access the femtocell. Thus S^a={Mi|Γi,f(P_f)≥ γ_i,m} ∩ S^g. Note that SG-FEVER promises that users in S^g will be allocated at least the wireless data rate Γ_i,f(P_f) from the femtocell BS, and at most their required throughput γ_i^req. When κ = |S^g|, ∀Mi ∈ S^g, γ_i,f(v_κ) = γ_i^req. That is, all users derive their required throughput from the femtocell.

Corollary 2. [Truthfulness under CSG Mode] SG-FEVER mechanism is a truthful mech-anism when S^g is non-empty and S^a={Mi|Γi,f(P_f)≥ γi,m} ∩ S^g.

Proof. We first discuss the case that M_i ̸∈ S^g. If so, the MS cannot choose the femtocell whatever it reports in SG-CBH since M_i ∈ S/ ^a ⊂ S^g. Thus, any θ_i ∈ Θ, including θi^∗, is M_i's best response. Then, if M_i ∈ S^g, since SG-CBH always provides the user at least its raw wireless data rate Γi,f(Pf), its preference on Pf is single-peaked with peaks p^∗_i at Γi,f(p^∗_i) = γ_i^req. So, Mi will report θ_i^∗ in SG-FEVER mechanism. We conclude that reporting θ_i^∗ is the dominant strategy of any M_i ∈ S in SG-FEVER mechanism under CSG mode, that is, SG-FEVER is a truthful mechanism.

Hybrid Mode

For Hybrid mode, the subscriber group S^gis also predefined, but other users are still al-lowed to access the femtocell if there are remaining resources (backhaul capacity). Given a predetermined non-empty set S^g, we define S^a = {Mi|Γi,f(P_f) ≥ γi,m} in this mode.

Since we remove the restriction that all users not belonging to S^g are excluded from the femtocell, there is a chance that a user M_i ̸∈ S^g will be included in S^a. We prove that SG-FEVER mechanism is a truthful mechanism in Hybrid mode.

Corollary 3 (Truthfulness under Hybrid Mode). SG-FEVER mechanism is a truthful mech-anism when S^g is non-empty and S^a={Mi|Γi,f(P_f)≥ γi,m}.

Proof. It has been shown in Corollary 2 that∀Mi ∈ S^g, reporting θ_i^∗ is their dominant strategy under SG-CBH. Then, ∀Mi ̸∈ S^g, the remaining capacity C^−g(p) is allocated according to A_{f ever}(·) in SG-CBH. Thus, according to Theorem 3, their dominant strategy is reporting θ_i^∗ either. So, SG-FEVER mechanism is a truthful mechanism under Hybrid mode.