Energy efficient radio resource management for heterogeneous wireless network using CoMP

Abstract

The coordinated multi point (CoMP) transmission technique is considered a key feature in future wireless network to improve both cell edge users throughput by exploiting interference. However, to provide CoMP transmission several BSs need to be active, which eventually increases network energy consumption. The simultaneous active multiple BSs with different transmission characteristics in heterogeneous environment cause interferences on each other. In this paper, we study the energy efficient radio resource management (EE-RRM) scheme for heterogeneous wireless networks to reduce interference. In particular, our aim is to allocate subcarrier power by optimizing EE metric and minimize interference with knowledge of channel state information between BSs and user equipment. The EE-RRM problem is a fractional programming problem. In order to solve, we use Charnes–Cooper transformation technique and transform it into an equivalent concave optimization problem. The numerical results of our work present the effect of different interference, rate and power thresholds on the EE metric.

Appendix

Theorem

The optimal power allocation rule that maximizes the total energy efficiency in Eq. (30) is given by the following relation when the capacity is greater than the threshold in constraint C1:

$$ \frac{{y_{i,k}^{*} }}{{{\text{t}}^{*} }} = P_{i,k}^{*} = \left( {\frac{1}{{\beta_{i,k} }} - \frac{1}{{Q_{i,k} }}} \right) ^{ + } $$

where, $\beta_{i,k} = ln2(v - \alpha - \lambda_{i,k} + \mathop \sum \nolimits_{i = 1}^{M} ({\emptyset }_{i,k} + \eta_{i,k} )I_{CoMP} )$ , v, ϕ, λ, η, and α are Lagrange multipliers of Lagranging function L(y, t, ϕ, v, λ, u, η, ∊, α) and $Q_{i,k} = \frac{{R_{CoMP} }}{{\sigma^{2} + I_{CoMP} }},\, y_{i,k}^{*} = {\text{t}}^{ *} \left( {\frac{1}{{\beta_{i,k} }} - \frac{1}{{Q_{i,k} }}} \right) ^{ + } ,$ and ${\text{t}}^{ *} = \frac{1}{{\mathop \sum \nolimits_{i = 1}^{{N_{B} }} \left( {\frac{1}{{\beta_{i,k} }} - \frac{1}{{Q_{i,k} }}} \right) ^{ + } }}$

Proof of the Theorem

$$ {\text{Max}}\left( {{\text{y}},{\text{ t}}} \right) \quad t\mathop \sum \limits_{i = 1}^{{N_{B} }} a_{i,k} \log_{2} \left(1 + y_{i} Q_{C}^{ik}\right) - N_{B} {\text{t}} a_{i,k} log_{2 } t $$

(36)

$$\begin{aligned} &{\text{Subject to}}{:}\,\\ &C1{:}\, - \mathop \sum \limits_{i = 1}^{{N_{B} }} a_{i,k} \log_{2} (1 + y_{i} Q_{C}^{ik} ) + N_{B} {\text{t}} a_{i,k} log_{2 } t { + }R_{min} \le \, 0 \end{aligned}$$

(37)

$$ C2{:}\, y_{i,k} I_{CoMP} - I_{th} \le \, 0 $$

(38)

$$ C3{:}\,y_{i,k} - t P_{tot} \le 0 $$

(39)

$$ C4{:}\,- y_{i,k} \le \, 0 $$

(40)

$$ C5{:}\,y_{i,k} - 1 = \, 0 $$

(41)

$$ C5{:}\,- t < 0 $$

(42)

The Lagranging function of Eq. (36) can be given by:

$$ \begin{aligned} & {\text{L(y,t}},\upphi ,{\text{v}},\uplambda ,{\rm u},\upeta , \in ,\upalpha ) \, = \, {\rm t}\mathop \sum \limits_{i = 1}^{{N_{B} }} a_{i,k} \log_{2} \left(1 + y_{i} Q_{C}^{ik}\right ) - N_{B} {\text{t}} a_{i,k} log_{2 } t \\ & \quad + \;\mathop \sum \limits_{i = 1}^{M} \upphi_{i} y_{i,k} I_{CoMP} - I_{th} + \in \left( { - \mathop \sum \limits_{i = 1}^{{N_{B} }} a_{i,k} \log_{2} \left(1 + y_{i} Q_{C}^{ik}\right ) + N_{B} {\text{t}} a_{i,k} log_{2 } t + R_{min} } \right) \\ & \quad + \;\upalpha ( y_{i,k} - t P_{tot} ) - y_{i,k} \uplambda_{i,k} + {\text{v}}( y_{i,k} - 1) - {\text{ut}} \\ \end{aligned} $$

(43)

Suppose, y _i,k * and t* denote the optimal solution of the Lagrangian function (37), then the Karush–Kuhn–Tucker (KKT) condition can be written as follows:

$$ y_{i,k} I_{CoMP}-I_{th} \le \, 0, $$

(44)

$$ \upphi_{i} (y_{i,k} I_{CoMP} - I_{th} ) = 0 $$

(45)

$$ \upphi_{i} \ge 0 $$

(46)

$$ - \mathop \sum \limits_{i = 1}^{{N_{B} }} a_{i,k} \log_{2} \left(1 + y_{i} Q_{C}^{ik} \right) - N_{B} {\text{t}} a_{i,k} log_{2 } t + R_{min} \le \, 0 $$

(47)

$$ \in \left( { - \mathop \sum \limits_{i = 1}^{{N_{B} }} a_{i,k} \log_{2} \left(1 + y_{i} Q_{C}^{ik} \right) + N_{B} {\text{t}} a_{i,k} log_{2 } t + R_{min} } \right) = 0 $$

(48)

$$ \in\,\ge\,0 $$

(49)

where, $\upphi_{i}\,\,{\rm and}\,\,\in$ are Lagrange multipliers corresponding to interference and sub-carrier usage constraint with element ϕ _i  ≥ 0 and ∊ ≥ 0 respectively.

$$ y_{i,k} - t P_{tot} \le \, 0 $$

(50)

$$ \upalpha ( y_{i,k} - t P_{tot} ) = 0 $$

(51)

$$ \upalpha \, \ge \, 0 $$

(52)

α is Lagrange multiplier corresponding to the power constraints. The boundary constraints will be absorbed into KKT conditions when deriving optimal solution.

$$ - y_{i,k} \le 0 $$

(53)

$$ - y_{i,k} \uplambda_{i,k} = 0 $$

(54)

$$ \uplambda_{i,k} \ge 0 $$

(55)

In order to obtained optimal subcarrier allocation we take derivative of the L w.r.t ${\text{t*}}$ and y _i,k and equate this value to 0 for allocating the sub carrier i to the user k.

$$ \frac{\partial L}{\partial t^{*}} = 0 $$

(56)

$$ \frac{\partial L}{{\partial y_{i,k} }} = 0 $$

(57)

Substitute L from (43) into (57) and after some manipulation, we can write following relation:

$$ y_{i,k} Q_{c}^{i,k} \beta_{i,k} - {\text{t}}\left(Q_{c}^{i,k} - \beta_{i,k}\right) = Q_{c}^{i,k} \in $$

(58)

where, $\beta_{i,k} = ln2(v - \alpha - \uplambda_{i,k} + \mathop \sum \limits_{i = 1}^{M} \left( { {\emptyset }_{i,k} + \eta_{i,k} } \right)I_{CoMP} )$. Now, from (47–49), it can be seen that when ∊ = 0, (47) should have strictly inequalities. Therefore, when the rate constraint is satisfied with inequality, we can write Eq. (58) as follows:

$$ \frac{{y_{i,k}^{*} }}{\text{t* }} = P_{i,k}^{*} = \left( {\frac{1}{{\beta_{i,k} }} - \frac{1}{{Q_{i,k} }}} \right) ^{ + } $$

(59)

where ( · )⁺ is the non-negative function. Variables $\uplambda_{i,k}$ and η _i,k are the Lagrange multipliers chosen to satisfy the individual BS power constraint and data rate requirement respectively. I _CoMP represents the interference of the other BSs created by their power allocations on sub-carrier i. Therefore, the power allocation rule that maximizes the total energy efficiency in Eq. (24) is given by (59) when the capacity is greater than the threshold in constraint C1 has water-filling type policy. The optimal values of λ _i,k and η _i,k are obtained by using bisection method, due to the concavity of the transformed problem with respect to power allocation variables.