Distributed resource allocation with fairness for cognitive radios in wireless mobile ad hoc networks

Abstract

Both spectrum sensing and power allocation have crucial effects on the performance of wireless cognitive ad hoc networks. In order to obtain the optimal available subcarrier sets and transmission powers, we propose in this paper a distributed resource allocation framework for cognitive ad hoc networks using the orthogonal frequency division multiple access (OFDMA) modulation. This framework integrates together the constraints of quality of service (QoS), maximum powers, and minimum rates. The fairness of resource allocation is guaranteed by introducing into the link capacity expression the probability that a subcarrier is occupied. An incremental subgradient approach is applied to solve the optimization problems that maximize the weighted sum capacities of all links without or with fairness constraints. Distributed subcarrier selection and power allocation algorithms are presented explicitly. Simulations confirm that the approach converges to the optimal solution faster than the ordinary subgradient method and demonstrate the effects of the key parameters on the system performance. It has been observed that the algorithms proposed in our paper outperform the existing ones in terms of the throughput and number of secondary links admitted and the fairness of resource allocation.

Appendices

Appendix 1

1.1 Convexity of the optimization problem (5)

The aim of this appendix is to prove that (i) the constraint set of the optimization problem P1 is convex; (ii) the objective function is concave.

Firstly, we will show the set of the constraints (3) and (4) is a nonempty convex set.

The constraint (3) can be rewritten as

$$ \sum\limits_{k = 1}^{{A_{i} }} { - R_{i}^{(k)} } \le - R_{i}^{\min } . $$

(50)

As proven above, $- R_{i}^{(k)}$ is convex. It is not difficult to obtain that the constraint (50) is a convex set.

Since $\mu_{i}^{(k)} = \frac{{\gamma_{ii}^{(k)} P_{i}^{(k)} }}{{N_{i}^{(k)} }}$, $\mu_{i}^{(k)}$is concave and $- \mu_{i}^{(k)}$ is convex. Thus, the constraint (2) is a convex set.

Since the constraint (4) is a linear combination of $P_{i}^{(k)}$, the summation of the constraints (3) and (4) is also a convex set.

Secondly, it follows from (5) that$R_{i}^{(k)}$is continuous and twice differentiable with respect to $P_{i}^{(k)}$.The twice derivation of $R_{i}^{(k)}$ with respect to $P_{i}^{(k)}$ can be obtained as follows

$$ \frac{{d^{2} R_{i}^{(k)} }}{{dP_{i}^{(k)2} }} = \frac{{ - B\left( {\kappa \gamma_{ii}^{(k)} } \right)^{2} }}{{\ln 2(N_{i}^{(k)} + \kappa \gamma_{ii}^{(k)} P_{i}^{(k)} )^{2} }}. $$

(51)

Since B is a positive real number, it is obvious that (51) is negative semidefinite. Thus, according to the Karush–Kuhn–Tucker condition, $R_{i}^{(k)}$ is concave. In addition, the constraints (3) and (4) is a convex set and the objective function in the problem P1 can be rewritten as a summation of a set of concave functions, thus the problem P1 is a convex optimization problem.

Appendix 2

2.1 Convexity of the optimization problem with fairness constraint

As proven in Appendix 1, the constraint set of the objective function (8) is a convex set.

The objective function with fairness constraint can be written as

$$ C^{'} = \mathop {\min }\limits_{{P_{i}^{(k)} \ge 0}} - \sum\limits_{i = 1}^{J} {\sum\limits_{{k = 1,k \in F_{i} }}^{{A_{i} }} {R_{i}^{(k)} } } = \mathop {\min }\limits_{{P_{i}^{(k)} \ge 0}} - \sum\limits_{i = 1}^{J} {\sum\limits_{{k = 1,k \in F_{i} }}^{{A_{i} }} {\rho_{i}^{(k)} B\log_{2} \left( {1 + \kappa \frac{{s_{i}^{(k)} }}{{\rho_{i}^{(k)} }}} \right)} } . $$

(52)

We can see from (52) that $R_{i}^{(k)}$ is continuous and twice differentiable with respect to $\rho_{i}^{(k)}$ and $P_{i}^{(k)}$, respectively. The Jacobian of $R_{i}^{(k)}$ is calculated as

$$ \nabla R_{i}^{(k)} (\rho_{i}^{(k)} ,s_{i}^{(k)} ) = B\left[ {\begin{array}{*{20}c} {\log_{2} \left( {1 + \kappa \frac{{s_{i}^{(k)} }}{{\rho_{i}^{(k)} }}} \right) - \frac{{\kappa s_{i}^{(k)} }}{{\ln 2\left( {\rho_{i}^{(k)} + \kappa s_{i}^{(k)} } \right)}}} \\ {\frac{{\kappa \rho_{i}^{(k)} }}{{\ln 2\left( {\rho_{i}^{(k)} + \kappa s_{i}^{(k)} } \right)}}} \\ \end{array} } \right]. $$

(53)

The Hessian of $R_{i}^{(k)}$ is calculated as

$$ \nabla^{2} R_{i}^{(k)} (\rho_{i}^{(k)} ,s_{i}^{(k)} ) = \frac{{B\kappa^{2} s_{i}^{(k)} }}{{\ln 2\left( {\rho_{i}^{(k)} + \kappa s_{i}^{(k)} } \right)^{2} }}\left[ {\begin{array}{*{20}c} { - \frac{{s_{i}^{(k)} }}{{\rho_{i}^{(k)} }}} & 1 \\ 1 & { - \frac{{\rho_{i}^{(k)} }}{{s_{i}^{(k)} }}} \\ \end{array} } \right]. $$

(54)

Since $\gamma_{i}^{(k)}$, $\gamma_{ii}^{(k)}$, $N_{i}^{(k)}$and $P_{i}^{(k)}$ are positive real numbers, it is obvious that the Hessian of $R_{i}^{(k)}$ is negative semidefinite and concave. Thus, the Hessian of $- R_{i}^{(k)}$ is positive semidefinite and convex. The objective function (52) with fairness constraint is convex. The optimization problem under consideration of fairness has optimal solutions.