Conventionally, upstream SUs as players of the game are expected to act cooperatively and maximize their joint utility functions with fairness for players by constituting the collaborative coalition. As a result, the global optimization of transmit power and data rate will be attained through cooperation among players with group rationality, which has been recently reported in a cooperative bargaining game [41]. However, each upstream SU is unwilling to jointly adjust the power and rate because of the selfish behavior in forwarding data packets. This is a natural idea due to the fact that the transmissions lead to the consumption of network resources of upstream SUs, such as energy and spectrum. Therefore, the cross-layer optimization framework for congestion and power control will be restricted to noncooperation scenario. In the noncooperative differential game models *Γ*_{
PLPC
} and *Γ*_{
HHCC
}, the *i*th player competes to maximize the present value of its utility function derived over time interval [*t*_{0}, *T*]. For mathematical tractability, we define the starting time of the differential game models *Γ*_{
PLPC
} and *Γ*_{
HHCC
} as *t*_{0} = 0 hereinafter, but the results can be easily extended to more general cases.

### Optimal solution to *Γ*
_{
PLPC
}

For the noncooperation scenario, we formulate a dynamic optimization problem ℙ1 to derive the optimal solution to the noncooperative differential game model *Γ*_{
PLPC
} by taking into account the utility function maximization problem coupled with the linear differential equation constraint in (13):

$$ {\displaystyle \begin{array}{l}\mathrm{\mathbb{P}}1\kern0.5em \underset{p_i(t)}{\operatorname{Maximize}}:\kern1em {\int}_0^T\left(\frac{{\overline{p}}_i-{p}_i(t)}{C_{\left(i,b\right)}^{\ast}\left(\overline{\mathbf{P}}\right)}\left({\overline{p}}_i-{p}_i(t)\right)-\omega I(t)\right){e}^{- at}\mathrm{d}t\\ {}\kern2.00em \mathrm{Subject}\kern0.5em \mathrm{to}:\frac{\mathrm{d}I(t)}{\mathrm{d}t}=\sum \limits_{i\in \mathcal{N}}{p}_i(t)-\gamma I(t),\\ {}\kern13em I\left({t}_0=0\right)={I}_0.\end{array}} $$

(23)

We aim at deriving an optimal solution to ℙ1 by employing the theory of dynamic programming developed by Bellman [42]. Remark that the optimal solution is also viewed as a Nash equilibrium solution to ℙ1 if all the players play noncooperatively. Here, we relax the terminal time of *Γ*_{
PLPC
} to explore when *T* approaches ∞ (i.e., *T* → ∞) as an infinite time horizon. It is more realistic to obtain the long-term optimal power allocation for upstream SUs due to spectrum underlay strategy with cellular primary network. We use \( {p}_i^{\#}(t) \) to represent the optimal solution to ℙ1 and assume that there exists a continuously differentiable function *V*^{i}(*p*_{
i
}, *I*) satisfying the following partial differential equation:

$$ \kern1.5em {aV}^i\left({p}_i,I\right)=\underset{p_i(t)}{\operatorname{Maximize}}:\left\{\frac{{\left({\overline{p}}_i-{p}_i(t)\right)}^2}{C_{\left(i,b\right)}^{\ast}\left(\overline{\mathbf{P}}\right)}-\omega I(t)+\frac{\partial {V}^i\left({p}_i,I\right)}{\partial I}\left(\sum \limits_{j\in \mathcal{N}\backslash i}{p}_j^{\#}(t)+{p}_i(t)-\gamma I(t)\right)\right\}. $$

(24)

###
**Theorem 1**

*A vector of optimal transmit power* \( {\mathbf{P}}^{\#}=\left\{{p}_1^{\#}(t),{p}_2^{\#}(t),\cdots, {p}_N^{\#}(t)\right\} \) *of upstream SUs constitutes a Nash equilibrium solution to* ℙ1 *if and only if the optimal transmit power* \( {p}_i^{\#}(t) \) *of the ith player and the continuously differentiable function V*^{i}(*p*_{
i
}, *I*) *can be formulated as follows:*

$$ {p}_i^{\#}(t)={\overline{p}}_i-\frac{\omega {C}_{\left(i,b\right)}^{\ast}\left(\overline{\mathbf{P}}\right)}{2\left(a+\gamma \right)}, $$

(25)

$$ {V}^i\left({p}_i^{\#},I\right)=\frac{\omega }{a\left(a+\gamma \right)}\left(\frac{\omega {C}_{\left(i,b\right)}^{\ast}\left(\overline{\mathbf{P}}\right)\left(1+2N\right)}{4\left(a+\gamma \right)}- aI-\sum \limits_{i\in \mathcal{N}}{\overline{p}}_i\right). $$

(26)

*Proof*: See Appendix 1. ■.

From Theorem 1, we can observe that the existence and uniqueness of the Nash equilibrium point to ℙ1 are guaranteed under the constraint of analytical solution in (25) and (26). It is also revealed that the optimal transmit power \( {p}_i^{\#}(t) \) has been characterized by a fixed and unique value in (25). Evidently, Theorem 1 mathematically ensures the convergence of \( {p}_i^{\#}(t) \) to a Nash equilibrium point. The key point to derive the optimal solution to the differential game model *Γ*_{
PLPC
} is illustrated with a block diagram shown in Fig. 4a.

###
**Proposition 1**

*For the given large-scale slow-fading channel model, by letting G*_{1} = *ϖ*_{PBS}/(10^{6}*g*_{0})^{N}, *the optimal transmit power* \( {p}_i^{\#}(t) \) *of the ith player should follow the interference power constraint*:

$$ \prod \limits_{i\in \mathcal{N}}{p}_i^{\#}(t)\le {G}_1\prod \limits_{i\in \mathcal{N}}{\left\Vert {\vartheta}_i(t)-{\vartheta}_b(t)\right\Vert}^4. $$

(27)

*Proof*: See Appendix 2. ■.

Note that the optimal transmit power \( {p}_i^{\#}(t) \) of the *i*th player is fully constrained by the Euclidean distance between upstream SU *v*_{
i
} and bottleneck SU *v*_{
b
} under the given channel model. Substituting for \( {p}_i^{\#}(t) \) with its expression from Theorem 1 and taking into account the previous expression of shadow price *ℵ*_{
i
}, we can easily rewrite *ℵ*_{
i
} as:

$$ {\aleph}_i=\frac{\omega }{2\left(a+\gamma \right)}. $$

(28)

Apparently, shadow price *ℵ*_{
i
} tends to be a constant value for all the upstream SUs. Although (25) and (26) offer an analytical solution to ℙ1, it still remains to design an algorithm to ensure fast convergence of the update of optimal transmit power. Therefore, we devise a distributed optimal transmit power update (OTPU) strategy given in Algorithm 2 to update the optimal transmit power vector **P**^{#} for upstream SUs. Similar to [43], shadow price *ℵ*_{
i
} in Algorithm 2 needs to carefully be chosen to ensure fast convergence of the update of instant transmit power \( {p}_i^{\odot }(t) \). It is also noted that the update of instant transmit power \( {p}_i^{\odot }(t) \) for upstream SU *v*_{
i
} can be made locally according to its optimal transmit power \( {p}_i^{\#}(t) \) along with interference power constraint.

### Optimal solution to *Γ*
_{
HHCC
}

For notational simplicity, we begin by defining a notation *B*_{(i, b)} ≜ − *D*_{
i
}/log_{2}(1 + *χ* ⋅ SINR_{(i, b)}(**P**)). For the noncooperation scenario, we formulate a dynamic optimization problem ℙ2 to derive the optimal solution to the noncooperative differential game model *Γ*_{
HHCC
} by taking into account both the utility function maximization problem and the linear differential equation constraint in (19):

$$ {\displaystyle \begin{array}{l}\mathrm{\mathbb{P}}2\kern0.5em \underset{r_i(t)}{\operatorname{Maximize}}:\kern1em {\int}_0^T\left({A}_{\left(i,b\right)}x(t)+{B}_{\left(i,b\right)}\frac{r_i^2(t)}{x(t)}\right){e}^{-\tau t}\mathrm{d}t\\ {}\kern3.25em \mathrm{Subject}\kern0.5em \mathrm{to}:\frac{\mathrm{d}x(t)}{\mathrm{d}t}=x(t)-\sum \limits_{i\in \mathcal{N}}{r}_i(t),\\ {}\kern13em x\left({t}_0=0\right)={x}_0.\end{array}} $$

(29)

We turn to take advantage of the theory of maximum principle developed by Pontryagin [42] to derive an optimal solution or a Nash equilibrium solution to ℙ2. We further use \( {r}_i^{\#}(t) \) to represent the optimal solution to ℙ2 and assume that there exists a continuously differentiable function *W*^{i}(*r*_{
i
}, *x*) satisfying the partial differential equation as follows:

$$ -\frac{\partial {W}^i\left({r}_i,x\right)}{\partial s}=\underset{r_i(t)}{\operatorname{Maximize}}:\left\{\left({A}_{\left(i,b\right)}x(t)+{B}_{\left(i,b\right)}\frac{r_i^2(t)}{x(t)}\right){e}^{-\tau t}+\frac{\partial {W}^i\left({r}_i,x\right)}{\partial x}\left(x(t)-\sum \limits_{j\in \mathcal{N}\backslash i}{r}_j^{\#}(t)-{r}_i(t)\right)\right\}. $$

(30)

For tractability, we introduce two extra introduced auxiliary variables *Y*_{
i
}(*t*) and *J*_{
i
}(*t*) to characterize *W*^{i}(*r*_{
i
}, *x*). Specifically, we define *W*^{i}(*r*_{
i
}, *x*) ≜ (*Y*_{
i
}(*t*)*x*(*t*) + *J*_{
i
}(*t*))*e*^{−τt}.

###
**Theorem 2**

*A vector of optimal data rate* \( {\mathbf{R}}^{\#}=\left\{{r}_1^{\#}(t),{r}_2^{\#}(t),\cdots, {r}_N^{\#}(t)\right\} \) *of upstream SUs constitutes a Nash equilibrium solution to* ℙ2 *if and only if the optimal data rate* \( {r}_i^{\#}(t) \) *of the ith player can be expressed as:*

$$ {r}_i^{\#}(t)=\frac{Y_i(t)x(t)}{2{B}_{\left(i,b\right)}}, $$

(31)

*where Y*_{
i
}(*t*) *and J*_{
i
}(*t*) *satisfy the following differential equations:*

$$ \frac{\mathrm{d}{Y}_i(t)}{\mathrm{d}t}=\frac{1}{4{B}_{\left(i,b\right)}}{Y}_i^2(t)+\left(\tau -1\right){Y}_i(t)-{A}_{\left(i,b\right)}+\frac{Y_i(t)}{2}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{Y_j(t)}{B_{\left(j,b\right)}}, $$

(32)

$$ \frac{\mathrm{d}{J}_i(t)}{\mathrm{d}t}=\tau \cdot {J}_i(t). $$

(33)

*Proof*: See Appendix 3. ■.

For notational simplicity, we set \( {\varOmega}_{\left(i,b\right)}=1/\left(4{B}_{\left(i,b\right)}\right)+0.5{\sum}_{j\in \mathcal{N}\backslash i}1/{B}_{\left(j,b\right)} \) and \( \varepsilon =\sqrt{{\left(\tau -1\right)}^2+4{\varOmega}_{\left(i,b\right)}{A}_{\left(i,b\right)}} \), for 4*Ω*_{(i, b)}*A*_{(i, b)} + (*τ* − 1)^{2} > 0. We also denote *G*_{2} as a constant number. Substituting *Ω*_{(i, b)} into *ε*, we can rewrite *ε* as follows:

$$ \varepsilon =\sqrt{{\left(\tau -1\right)}^2+\frac{A_{\left(i,b\right)}}{B_{\left(i,b\right)}}+2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{1}{B_{\left(j,b\right)}}}. $$

(34)

###
**Proposition 2**

*The auxiliary variable Y*_{
i
}(*t*) *in the Nash equilibrium solution* \( {r}_i^{\#}(t) \) *to* ℙ2 *can be further given as*:

$$ {Y}_i(t)=\frac{\left(\left(\tau -1\right)+\varepsilon \right){e}^{\left(t-{G}_2\right)\varepsilon }+\varepsilon -\tau +1}{2{\varOmega}_{\left(i,b\right)}\left(1-{e}^{\left(t-{G}_2\right)\varepsilon}\right)}. $$

(35)

*Proof*: See Appendix 4. ■.

Combining *Y*_{
i
}(*t*) in (35) and *ε* in (34) yields the expression for *Y*_{
i
}(*t*) as:

$$ {Y}_i(t)=\frac{\left(\tau -1+\sqrt{{\left(\tau -1\right)}^2+\frac{A_{\left(i,b\right)}}{B_{\left(i,b\right)}}+2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{1}{B_{\left(j,b\right)}}}\right){e}^{\left(t-{G}_2\right)\sqrt{{\left(\tau -1\right)}^2+\frac{A_{\left(i,b\right)}}{B_{\left(i,b\right)}}+2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{1}{B_{\left(j,b\right)}}}}+\sqrt{{\left(\tau -1\right)}^2+\frac{A_{\left(i,b\right)}}{B_{\left(i,b\right)}}+2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{1}{B_{\left(j,b\right)}}}-\tau +1}{2\left(\frac{1}{4{B}_{\left(i,b\right)}}+0.5\sum \limits_{j\in \mathcal{N}\backslash i}\frac{1}{B_{\left(j,b\right)}}\right)\left(1-{e}^{\left(t-{G}_2\right)\sqrt{{\left(\tau -1\right)}^2+\frac{A_{\left(i,b\right)}}{B_{\left(i,b\right)}}+2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{1}{B_{\left(j,b\right)}}}}\right)}. $$

(36)

Substituting *B*_{(i, b)} ≜ − *D*_{
i
}/log_{2}(1 + *χ* ⋅ SINR_{(i, b)}(**P**)) into (36), we can rewrite *Y*_{
i
}(*t*) as follows:

$$ {\displaystyle \begin{array}{l}{Y}_i(t)=\frac{\left(\tau -1+\sqrt{{\left(\tau -1\right)}^2-\frac{A_{\left(i,b\right)}{\log}_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{D_i}-2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}}\right){e}^{\left(t-{G}_2\right)\sqrt{{\left(\tau -1\right)}^2-\frac{A_{\left(i,b\right)}{\log}_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{D_i}-2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}}}}{2\left(\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{4{D}_i}+0.5\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}\right)\left({e}^{\left(t-{G}_2\right)\sqrt{{\left(\tau -1\right)}^2-\frac{A_{\left(i,b\right)}{\log}_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{D_i}-2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}}}-1\right)}.\\ {}\kern5em +\frac{\sqrt{{\left(\tau -1\right)}^2-\frac{A_{\left(i,b\right)}{\log}_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{D_i}-2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}}-\tau +1}{2\left(\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{4{D}_i}+0.5\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}\right)\left({e}^{\left(t-{G}_2\right)\sqrt{{\left(\tau -1\right)}^2-\frac{A_{\left(i,b\right)}{\log}_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left(\mathbf{P}\right)\right)}{D_i}-2{A}_{\left(i,b\right)}\sum \limits_{j\in \mathcal{N}\backslash i}\frac{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(j,b\right)}\left(\mathbf{P}\right)\right)}{D_j}}}-1\right)}\end{array}} $$

(37)

Let *G*_{3} be a constant number. Based on (31) and (29), the optimal data rate \( {r}_i^{\#}(t) \) and the optimal state variable *x*^{#}(*t*) associated with \( {r}_i^{\#}(t) \) can be described as:

$$ {r}_i^{\#}(t)=\frac{Y_i(t)}{2{B}_{\left(i,b\right)}}\left({e}^{\left(1-0.5\sum \limits_{i=1}^N\frac{Y_i(t)}{B_{\left(i,b\right)}}\right)t}+{G}_3\right), $$

(38)

$$ {x}^{\#}(t)={e}^{\left(1-0.5\sum \limits_{i=1}^N\frac{Y_i(t)}{B_{\left(i,b\right)}}\right)t}+{G}_3. $$

(39)

From (38), we can see that the optimal data rate \( {r}_i^{\#}(t) \) is determined by both *B*_{(i, b)} and auxiliary variable *Y*_{
i
}(*t*) under the received SINR SINR_{(i, b)}(**P**) of link *l*_{(i, b)}. Unfortunately, it is difficult to directly obtain the relationship between SINR_{(i, b)}(**P**) and \( {r}_i^{\#}(t) \) through an analytical derivation because *Y*_{
i
}(*t*) cannot be further simplified into a concise structure. Thereby, given the channel gain *G*_{(i, b)} of link *l*_{(i, b)} under the large-scale slow-fading channel model, we use the numerical simulations to validate the effectiveness of the optimal data rate \( {r}_i^{\#}(t) \) of upstream SU *v*_{
i
}. We further remark that Theorem 2 and Proposition 2 characterize the existence of the Nash equilibrium point to ℙ2. It should be also admitted that the optimal data rate \( {r}_i^{\#}(t) \) has been formulated as a fixed and unique value in (38) by using auxiliary variable *Y*_{
i
}(*t*) in (37). Correspondingly, Theorem 2 mathematically ensures the convergence of \( {r}_i^{\#}(t) \) to a Nash equilibrium point. The key point to derive the optimal solution to the differential game model *Γ*_{
HHCC
} is also illustrated with a block diagram depicted in Fig. 4b. With the help of vectors **R**^{#} and **P**^{#}, shadow price *ℵ*_{
i
} can be calculated as:

$$ {\aleph}_i=\left(\frac{1}{\log_2\left(1+\chi \cdot {\mathrm{SINR}}_{\left(i,b\right)}\left({\mathbf{P}}^{\#}\right)\right)}\right)\frac{r_i^{\#}(t)}{x^{\#}(t)}. $$

(40)

### Proposition 3

*For the given vector* **R**^{#}* and vector* **P**^{#}, *a strict lower bound of shadow price ℵ*_{
i
}* can be approximately calculated as follows*:

$$ {\aleph}_i\ge {\left({\log}_2\left(\chi \cdot \frac{p_i^{\#}(t)}{\sum \limits_{j\in \mathcal{N}\backslash i}{p}_j^{\#}(t)}\right)\right)}^{-1}\frac{r_i^{\#}(t)}{x^{\#}(t)}. $$

(41)

*Proof*: See Appendix 5. ■.

Let \( {\mathrm{r}}_{\left(i,b\right)}^{\mathrm{base}} \) represent the baseline rate along link *l*_{(i, b)} from upstream SU *v*_{
i
} to bottleneck SU *v*_{
b
}. To guarantee the constraint of the saturation value of the buffer of bottleneck SU *v*_{
b
}, we design a distributed algorithm to obtain the optimal data rate update (ODRU) for upstream SUs as summarized in Algorithm 3, where **R** = {*r*_{1}(*t*), *r*_{2}(*t*), ⋯, *r*_{
N
}(*t*)} denotes the data rate vector of *N* upstream SUs at time *t*. A simple yet effective way to locally adjust the instant data rate \( {r}_i^{\odot }(t) \) of upstream SU *v*_{
i
} is to employ the optimal data rate \( {r}_i^{\#}(t) \) under the condition of buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \). Also, baseline rate \( {\mathrm{r}}_{\left(i,b\right)}^{\mathrm{base}} \) in Algorithm 3 should be carefully chosen to ensure the effectiveness of instant data rate.

### Distributed implementation

So far, we have devised Algorithms 2 and 3 to satisfy the interference power constraint along with buffer constraint by locally adjusting the optimal transmit power and the optimal data rate, respectively. In what follows, we would like to describe the distributed implementation strategy to realize the cross-layer optimization framework for congestion and power control by jointly optimizing PLPC-HHCC simultaneously. In conclusion, the cross-layer optimization scheme for joint PLPC-HHCC design is implemented in a distributed manner as follows:

#### Shadow price

Update shadow price *ℵ*_{
i
} using (28) and shadow price *ℵ*_{
i
} using (40), respectively.

#### Power controller in the physical layer

For each upstream SU, we initially assign optimal transmit power \( {p}_i^{\#}(t) \) using (25) to update instant transmit power \( {p}_i^{\odot }(t) \) at power controller. Due to the interference power constraint in (6) to protect PUs, \( {p}_i^{\odot }(t) \) should satisfy the following distributed power-update function when \( {\sum}_{i\in \mathcal{N}}{p}_i(t){G}_{\left(i,b\right)}>{\varpi}_{\mathrm{PBS}} \) via OTPU algorithm:

$$ {p}_i^{\odot }(t)=\left\{\begin{array}{l}\left|{p}_i^{\odot }(t)-{\aleph}_i\times {\overline{p}}_i\right|,\kern4.5em \mathrm{for}\kern1em 0<{\aleph}_i<1\\ {}\left|{p}_i^{\odot }(t)-{\left({\aleph}_i\right)}^{-1}\times {\overline{p}}_i\right|,\kern2em \mathrm{for}\kern1em {\aleph}_i\ge 1\end{array}\right.. $$

(42)

#### Rate controller in the transport layer

For each upstream SU, we also initially assign optimal data rate \( {r}_i^{\#}(t) \) using (38) to update instant data rate \( {r}_i^{\odot }(t) \) at rate controller. Owing to the buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \) to guarantee that bottleneck SU *v*_{
b
} will not become congested, \( {r}_i^{\odot }(t) \) should be subject to the following distributed rate update function when \( {\varphi}_b(t)>{\widehat{L}}_b \) via ODRU algorithm:

$$ {r}_i^{\odot }(t)\leftarrow {r}_i^{\odot }(t)-{\aleph}_i\times {\mathrm{r}}_{\left(i,b\right)}^{\mathrm{base}}. $$

(43)

#### Cross-layer coordination mechanism

With the aid of the updated power \( {p}_i^{\odot }(t) \), the link capacity supply *C*_{(i, b)}(**P**^{⊙}) with respect to each upstream SU is regulated by power controller by using (4) as shown in Fig. 3. The rate demand depends on instant data rate \( {r}_i^{\odot }(t) \) regulated by rate controller, which is nonlinear function of instant transmit power vector **P**^{⊙} according to (37) and (38).