Distributed algorithms for resource allocation of physical and transport layers in wireless cognitive ad hoc networks

Abstract

In this paper, by integrating together congestion control, power control and spectrum allocation, a distributed algorithm is developed to maximize the aggregate source utility and increase end-to-end throughput. Despite the inherent difficulties of non-convexity and non-separability of variables in the original optimization problem, we are still able to obtain a decoupled and dual-decomposable convex formulation by applying an appropriate transformation and introducing some new variables. The objective is accomplished by the interaction and coordination among three sub-algorithms of the algorithm through the congestion prices. The convergence properties of the three sub-algorithms are also proved. Simulation results illustrate several other desirable properties of the proposed algorithm, including the impacts of node mobility and path and packet losses on convergence and robustness. This work is a preliminary attempt towards a systematic approach to jointly designing a congestion control sub-algorithm and a power control sub-algorithm coupled with a spectrum allocation sub-algorithm.

Appendix A: Proof of Theorem 3

Frequently, in optimization problems to be solved with gradient method, the gradient is not computed exactly. It is easy to show that if there exists an error e ⁽ⁱ⁾ _l in the gradient $\nabla_{l}^{(i)} V_{power} \left( {p_{l}^{(i)} } \right),$ the gradient direction becomes $\nabla_{l}^{(i)} V_{power} \left( {p_{l}^{(i)} } \right) + e_{l} .$

Substituting $p_{l}^{(i)} = e^{{\hat{p}_{l}^{(i)} }}$ into (25), we can get

$$ \nabla_{l}^{(i)} V_{power} (p_{l}^{(i)} ) = B_{l}^{(i)} \left[ {\lambda_{l}^{(i)} - p_{l}^{(i)} \sum\limits_{j \ne l} {{\frac{{\lambda_{j}^{(i)} G_{jl}^{(i)} }}{{\sum\nolimits_{m \ne j} {G_{jm}^{(i)} p_{m}^{(i)} } + \sigma_{j}^{(i)} }}}} } \right]. $$

(40)

It follows from (3) that

$$ \sum\nolimits_{m \ne j} {G_{jm}^{(i)} p_{m}^{(i)} } + \sigma_{j}^{(i)} = {\frac{{G_{jj}^{(i)} p_{j}^{(i)} }}{{\gamma_{j}^{(i)} \left( {p_{j}^{(i)} } \right)}}} $$

(41)

Substituting (41) into (40), we have

$$ \nabla_{l}^{(i)} V_{power} (p_{l}^{(i)} ) = B_{l}^{(i)} \left[ {\lambda_{l}^{(i)} - p_{l}^{(i)} \sum\limits_{j \ne l} {{\frac{{\lambda_{j}^{(i)} G_{jl}^{(i)} \gamma_{j}^{(i)} (p_{j}^{(i)} )}}{{G_{jj}^{(i)} p_{j}^{(i)} }}}} } \right] . $$

(42)

In our optimization problem, an error in G ⁽ⁱ⁾ _jl leads to an error e ⁽ⁱ⁾ _l in the gradient vector, from (43), this can be formulated as follows:

$$ e_{l}^{(i)} = B_{l}^{(i)} p_{l}^{(i)} \sum\limits_{j \ne 1} {{\frac{{\Updelta G_{jl}^{(i)} \lambda_{j}^{(i)} \gamma_{j}^{(i)} (p_{j}^{(i)} )}}{{G_{jj}^{i} p_{j}^{(i)} }}}} . $$

(43)

Let us consider the case that e ⁽ⁱ⁾ _l is small relative to the gradient, that is,

$$ \left\| e \right\|^{2} < \left\| {\nabla V_{power} (P)} \right\|^{2} . $$

(44)

By taking the square in both sides of (43) and adding, it follows that

$$ \begin{gathered} \left\| {\nabla V_{power} (P)} \right\|^{2} = \sum\limits_{l} {\left[ {B_{l}^{(i)} (t)p_{l}^{(i)} (t)} \right]^{2} } + \sum\limits_{l} {\left[ {B_{l}^{(i)} (t)p_{l}^{(i)} (t)\sum\limits_{j} {{\frac{{\lambda_{j}^{(i)} (t)G_{jl}^{(i)} (t)\gamma_{j}^{(i)} (t)}}{{p_{j}^{(i)} (t)G_{jj}^{(i)} (t)}}}} } \right]}^{2} \hfill \\ \, - 2\sum\limits_{l} {B_{l}^{(i)} (t)p_{l}^{(i)} (t)\sum\limits_{j \ne l} {{\frac{{\lambda_{l}^{(i)} (t)\lambda_{j}^{(i)} (t)G_{jl}^{(i)} (t)\gamma_{j}^{(i)} (t)}}{{p_{j}^{(i)} (t)G_{jj}^{(i)} (t)}}}} } , \hfill \\ \end{gathered} $$

(45)

Similarly, from (44), we can obtain

$$ \left\| e \right\|^{2} = \sum\limits_{l} {\left[ {B_{l}^{(i)} p_{l}^{(i)} \sum\limits_{j \ne l} {{\frac{{\Updelta G_{jl}^{(i)} \lambda_{j}^{(i)} \gamma_{j}^{(i)} (p_{j}^{(i)} )}}{{G_{jj}^{(i)} p_{j}^{(i)} }}}} } \right]}^{2} . $$

(46)

Substituting (45) and (46) into (44) and simplifying, one can get the result of Theorem 3. This completes the proof of Theorem 3.