Energy efficient resource matching algorithm for multi-homing services in dynamic wireless environment

Abstract

This paper investigates the matching problem between radio resources and multi-homing services in a dynamic heterogeneous wireless environment. A mobile terminal with a multi-homing service simultaneously transmits data to multiple radio access networks using multiple air interfaces. This paper proposes an energy-efficient resource matching algorithm to statistically guarantee the service rate requirements of each user with the lowest power consumption. First, according to the concept of resource matching and the dynamic characteristics of wireless channels, this paper defines two matching probability metrics and subsequently selects the most appropriate metric to build a resource matching model to realize the maximum energy efficiency of each user while meeting the statistical guarantee constraints. In particular, the Rayleigh fading channel model is introduced in this paper to accurately reflect the dynamics of the wireless channel, and a complete expression of the matching probability metric is deduced. Second, according to the optimal solution characteristics of the optimization model, the original problem is decomposed into the inner power minimization problem and the outer matching probability maximization problem, and the convex optimization and the equality constrained optimization are separately used to solve the problem. Hence, the two-layer distributed solution algorithm is proposed. In particular, this paper conducts rigorous mathematical analyses on the convergence, optimality and complexity of the algorithm and constructs an improved resource matching model by relaxing the statistical guarantee parameters in the case of no solution to the model. Finally, the simulation results show that this algorithm has better performance than the three existing resource matching algorithms, can achieve lower user power consumption, and can realize the goal of the best match between the network resources and service requirements in a dynamic environment.

Appendices

Appendix 1: The concavity or convexity of $f\left( {p_{nms} ,b_{nms} ,r_{nms} } \right) = - \frac{{b_{nms} \sigma_{noise}^{2} }}{{p_{nms}^{{}} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}\left( {2^{{\frac{{r_{nms} }}{{b_{nms} }}}} - 1} \right)$

Proof

For f(p_nms, b_nms, r_nms), take the second order partial derivatives and second order mixed partial derivatives of b_nms, p_nms, and r_nms:

$$\frac{{\partial {}^{2}f}}{{\partial p_{nms}^{2} }} = - \frac{{2b_{nms} \sigma_{noise}^{2} }}{{p_{nms}^{3} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}\left( {2^{{\frac{{r_{nms} }}{{b_{nms} }}}} - 1} \right) \le 0$$

(21)

$$\frac{{\partial {}^{2}f}}{{\partial b_{nms}^{2} }} = - \frac{{r_{nms}^{2} \sigma_{noise}^{2} }}{{p_{nms}^{{}} b_{nms}^{3} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}2^{{\frac{{r_{nms} }}{{b_{nms} }}}} \left( {In 2} \right)^{ 2} \le 0$$

(22)

$$\frac{{\partial {}^{2}f}}{{\partial r_{nms}^{2} }} = - \frac{{\sigma_{noise}^{2} }}{{p_{nms}^{{}} b_{nms} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}2^{{\frac{{r_{nms} }}{{b_{nms} }}}} \left( {In 2} \right)^{ 2} \le 0$$

(23)

$$\frac{{\partial {}^{2}f}}{{\partial p_{nms} \partial r_{nms} }} = \frac{{\sigma_{noise}^{2} In2}}{{p_{nms}^{2} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}2^{{\frac{{r_{nms} }}{{b_{nms} }}}} \ge 0$$

(24)

$$\frac{{\partial {}^{2}f}}{{\partial p_{nms} \partial b_{nms} }} = \frac{{\sigma_{noise}^{2} }}{{p_{nms}^{2} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}\left[ {2^{{\frac{{r_{nms} }}{{b_{nms} }}}} \left( {1 - \frac{{r_{nms} }}{{b_{nms} }}In2} \right) - 1} \right]$$

(25)

Because b_nms ≥ 0, r_nms ≥ 0, and r_nms → 0 will cause b_nms → 0, then $\frac{{r_{nms} }}{{b_{nms} }} \ge 0$. $2^{{\frac{{r_{nms} }}{{b_{nms} }}}} \left( {1 - \frac{{r_{nms} }}{{b_{nms} }}In2} \right) - 1$ is the decreasing function of $\frac{{r_{nms} }}{{b_{nms} }}$. Therefore, we have:

$$\begin{aligned} \frac{{\partial {}^{2}f}}{{\partial p_{nms} \partial b_{nms} }} & = \frac{{\sigma_{noise}^{2} }}{{p_{nms}^{2} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}\left[ {2^{{\frac{{r_{nms} }}{{b_{nms} }}}} \left( {1 - \frac{{r_{nms} }}{{b_{nms} }}In2} \right) - 1} \right] \\ & \le \frac{{\sigma_{noise}^{2} }}{{p_{nms}^{2} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}\left[ {2^{0} \left( {1 - 0 \cdot In2} \right) - 1} \right] = 0 \\ \end{aligned}$$

(26)

Hence, given r_nms, f(p_nms, b_nms, r_nms) is the joint concave function of p_nmsand b_nms; given b_nms, f(p_nms, b_nms, r_nms) is the concave function of p_nms and r_nms, respectively.

Appendix 2: Optimality theorem and its proof

(I) Optimality theorem: Suppose $(\forall r_{nms}^{*} , \forall p_{nms}^{*} , \forall b_{nms}^{*} )$ is the global optimal solution of resource matching model based on formula (4), for any solution (∀r_nms, ∀p_nms, ∀b_nms) in feasible region, if for any user ∀m there has $\sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms} } } \le \sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms}^{*} } }$, then the matching probability of user ∀m with the allocations $\left( {r_{nms}^{*} , p_{nms}^{*} , b_{nms}^{*} } \right), \forall n, s$ is the largest.

(II) Proof: The logarithm function of matching probability of any user ∀m can be expressed as

$${\rm H} (r_{nms} ,p_{nms} ,b_{nms} ,\forall n,s ) { = }\sum\limits_{n = 1}^{{NS_{m} }} {\sum\limits_{s = 1}^{{SS_{mn} }} {\left( { - \frac{{b_{nms} \sigma_{noise}^{2} }}{{p_{nms}^{{}} \left( {\frac{\theta }{4\pi d}} \right)^{\vartheta } \cdot g_{s} g_{r} }}\left( {2^{{\frac{{r_{nms} }}{{b_{nms} }}}} - 1} \right)} \right)} } .$$

As can be seen, ${\rm H} (r_{nms} ,p_{nms} ,b_{nms} ,\forall n,s )$ is the increasing function of (p_nms, b_nms), ∀n, s and is the decreasing function of r_nms, ∀n, s. Then, reduction to absurdity is used to prove the Optimality Theorem.

Suppose there is a feasible solution (∀r_nms, ∀p_nms, ∀b_nms) that satisfies $\sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms} } } \le \sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms}^{*} } } ,\forall m$ and there is a certain user $m_{ 1}$ whose matching probability with the feasible solution $\left( {r_{{nm_{ 1} s}} ,p_{{nm_{ 1} s}} ,b_{{nm_{ 1} s}} } \right),\forall n,s$ is the largest. That is, ${\rm H}(r_{{nm_{1} s}} ,p_{{nm_{1} s}} ,b_{{nm_{1} s}} ,\forall n,s) > {\rm H}(r_{{nm_{1} s}}^{ * } ,p_{{nm_{1} s}}^{ * } ,b_{{nm_{1} s}}^{ * } ,\forall n,s)$. Because $\left( {r_{{nm_{ 1} s}}^{ * } ,p_{{nm_{ 1} s}}^{ * } ,b_{{nm_{ 1} s}}^{ * } } \right),\forall n,s$ is the global optimal solution for user $m_{ 1}$, ${\rm H}(r_{{nm_{1} s}}^{ * } ,p_{{nm_{1} s}}^{ * } ,b_{{nm_{1} s}}^{ * } ,\forall n,s) \ge In\left( {1 - \varepsilon } \right)$. Thus, let ${\rm H} (r_{{nm_{ 1} s}} ,p_{{nm_{ 1} s}} ,b_{{nm_{ 1} s}} ,\forall n,s ) { = }{\rm H} (r_{{nm_{ 1} s}}^{ * } ,p_{{nm_{ 1} s}}^{ * } ,b_{{nm_{ 1} s}}^{ * } ,\forall n,s ) { + }\Delta > In\left( {1 - \varepsilon } \right)$, where Δ > 0. Keeping the other quantities unchanged, ∀p_nms is replaced with $\forall p_{nms}^{{\prime }}$. Only when m = m₁, n = n₁ and s = s₁, let $p_{{n_{ 1} m_{ 1} s_{ 1} }}^{\prime } < p_{{n_{ 1} m_{ 1} s_{ 1} }}$; in other cases, let $p_{nms}^{{\prime }} { = }p_{nms}$. By reasonably selecting user $m_{ 1}$‘s sub-channel $n_{ 1} m_{ 1} s_{ 1}$ and carefully setting $p_{{n_{ 1} m_{ 1} s_{ 1} }}^{{\prime }}$, we can get: ${\rm H} (r_{{nm_{ 1} s}} ,p_{{nm_{ 1} s}}^{\prime } ,b_{{nm_{ 1} s}} ,\forall n,s )= {\rm H} (r_{{nm_{ 1} s}}^{ * } ,p_{{nm_{ 1} s}}^{ * } ,b_{{nm_{ 1} s}}^{ * } ,\forall n,s )\ge In\left( {1 - \varepsilon } \right)$. Hence $(\forall r_{nms} , \forall p_{nms}^{\prime } , \forall b_{nms} )$ is a feasible solution. Moreover, when ∀m ≠ m₁, there has $\mathop \sum \nolimits_{n = 1}^{{NS_{m} }} \sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms}^{\prime } } = \sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms} } } \le \sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {p_{nms}^{*} } }$; when $m{ = }m_{1}$, there has $\sum\nolimits_{n = 1}^{{NS_{{m_{ 1} }} }} {\sum\nolimits_{s = 1}^{{SS_{{nm_{ 1} }} }} {p_{{nm_{ 1} s}}^{\prime } } } < \sum\nolimits_{n = 1}^{{NS_{{m_{ 1} }} }} {\sum\nolimits_{s = 1}^{{SS_{{nm_{ 1} }} }} {p_{{nm_{ 1} s}} } } \le \sum\nolimits_{n = 1}^{{NS_{{m_{ 1} }} }} {\sum\nolimits_{s = 1}^{{SS_{{nm_{ 1} }} }} {p_{{nm_{ 1} s}}^{*} } }$. Therefore, $\sum\nolimits_{m = 1}^{MS} {\sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {\frac{{p_{nms}^{\prime } }}{{P_{m} }}} } } < \sum\nolimits_{m = 1}^{MS} {\sum\nolimits_{n = 1}^{{NS_{m} }} {\sum\nolimits_{s = 1}^{{SS_{nm} }} {\frac{{p_{nms}^{ *} }}{{P_{m} }}} } }$. From this, it can be concluded that the feasible solution $(\forall r_{nms} , \forall p_{nms}^{{\prime }} , \forall b_{nms} )$ is better than the optimal solution $(\forall r_{nms}^{*} , \forall p_{nms}^{*} , \forall b_{nms}^{*} )$, which contradicts the assumption that $(\forall r_{nms}^{*} , \forall p_{nms}^{*} , \forall b_{nms}^{*} )$ is the global optimal solution. Hence, the theorem is proved.