An Offloading Mechanism Towards SE–EE Experiences Tradeoff in Heterogeneous Cellular Networks

Abstract

To achieve a tradeoff between spectral efficiency (SE) and energy efficiency (EE) experiences, we design an offloading mechanism to maximize the sum of weighted logarithmic utilities with respect to SE and EE for heterogeneous cellular networks, where the weighting parameters are used for adjusting the SE–EE experiences. Unlike the most existing offloading mechanisms that often try to find a tradeoff between SE and EC (energy consumption), we concentrate on a tradeoff between SE–EE experiences in our mechanism. That is to say, our mechanism directly optimizes EE and is weighted in favour of SE–EE fairness. To solve the finally formulated problem in a mixed-integer and nonlinear form, we firstly make some relaxation for the optimization objective to obtain its an upper bound. Then, we perform a decoupling operation to achieve a decomposable form of dual problem. Finally, we develop a feasible algorithm that can be well implemented in a distributed manner. Numerical results show that the designed mechanism can definitely reach a tradeoff between SE–EE experiences.

Appendix: Proof of Lemma 1

According to the principle of subgradient method, we have

$$\begin{aligned} \left\| {{\varvec{\lambda }}^{t+1}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}=&\left\| {{\varvec{\lambda }}^{t}}-\xi {\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{t} \right) }-{{\varvec{\lambda } }^{o}} \right\| _{2}^{2} \\ =&\left\| {{\varvec{\lambda }}^{t}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}-2\xi {{\left( {\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{t} \right) } \right) }^{T}}\left( {{\varvec{\lambda }}^{t}}-{{\varvec{\lambda }}^{o}} \right) +{{\xi }^{2}}\left\| {\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{t} \right) } \right\| _{2}^{2} \\ \overset{a}{\mathop \leqslant }&\left\| {{\varvec{\varvec{\lambda }}}^{t}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}-2\xi \left( {\mathcal {H}}\left( {{\varvec{\lambda }}^{t}} \right) -{{{\mathcal {H}}}^{o}} \right) +{{\xi }^{2}}\left\| {\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{t} \right) } \right\| _{2}^{2},\end{aligned}$$

(20)

where $\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{t} \right)$ represents the subgradient of ${\mathcal {H}}$ at tth iteration; a follows from the definition of subgradient.

By applying the inequality (20) recursively, we finally have

$$\left\| {{\varvec{\lambda }}^{t+1}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2} \leqslant \left\| {{\varvec{\lambda }}^{1}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}+{{\xi }^{2}}\sum \limits _{i=1}^{t}{\left\| {\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{i} \right) } \right\| _{2}^{2}}-2\xi \sum \limits _{i=1}^{t}{\left( {\mathcal {H}}\left( {{\varvec{\lambda }}^{i}} \right) -{{{\mathcal {H}}}^{o}} \right) }$$

(21)

Evidently, $\left\| {{\varvec{\lambda }}^{t+1}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}\geqslant 0$, and thus we have

$$2\xi \sum \limits _{i=1}^{t}{\left( {\mathcal {H}}\left( {{\varvec{\lambda }}^{i}} \right) -{{{\mathcal {H}}}^{o}} \right) }\leqslant \left\| {{\varvec{\lambda }}^{1}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}+{{\xi }^{2}}\sum \limits _{i=1}^{t}{\left\| {\nabla {\mathcal {H}}\left( {\varvec{\lambda }}^{i} \right) } \right\| _{2}^{2}}.$$

(22)

By combining it with

$$\begin{aligned} \sum \limits _{i=1}^{t}{\left( {\mathcal {H}}\left( {{\varvec{\lambda }}^{i}} \right) -{{{\mathcal {H}}}^{o}} \right) }&\geqslant t\underset{i\in \left\{ 1,2,\ldots ,t \right\} }{\mathop {\min }}\,\left( {\mathcal {H}}\left( {{\varvec{\lambda }}^{i}} \right) -{{{\mathcal {H}}}^{o}} \right) =t\left( {{\overline{{\mathcal {H}}}}_{t}}-{{{\mathcal {H}}}^{o}} \right) ,\end{aligned}$$

(23)

we have

$$\begin{aligned} {{\overline{{\mathcal {H}}}}_{t}}-{{{\mathcal {H}}}^{o}}=&\underset{i\in \left\{ 1,2,\ldots ,t \right\} }{\mathop {\min }}\,{\mathcal {H}}\left( {{\varvec{\lambda }}^{i}} \right) -{{{\mathcal {H}}}^{o}}\leqslant {{\beta }_{1}}{{\left\| {{\varvec{\lambda }}^{1}}-{{\varvec{\lambda }}^{o}} \right\| _{2}^{2}}}+{{\beta }_{2}}. \end{aligned}$$

(24)

Until now, we have established the proof of Lemma 1.