Optimizing link rate assignment and transmission scheduling in WMN through compatible set generation

Abstract

Radio links in wireless mesh networks (WMN) can select one of several modulation and coding schemes (MCS). A MCS assignment influences links data rates and their mutual interference, and therefore should be optimized. We consider joint optimization of link rate assignment and transmission scheduling in order to maximize the minimal flow in a WMN. One of the main difficulties stems from the requirement that each link has to use only one selected MCS for all its transmissions. This requirement leads to a complicated exact branch-and-price method, which is quite time-consuming for networks of practical size. Thus, we propose an original heuristic based on simulated annealing that utilizes specific characteristics of the problem. The method provides a balance between sub-optimality of the obtained solutions and the running time. The presented method is the main purpose and novelty of the paper. An extensive numerical study illustrates the effectiveness of the proposed approach.

A Appendix (proof of Proposition 1)

Consider an arbitrary optimal solution $\alpha , \; \zeta _e \; (e \in {\mathcal {E}}), \pi _e \; (e \in {\mathcal {E}}), \; \beta _e^m \; (e \in {\mathcal {E}}, m \in {\mathcal {M}})$ of the dual problem DP(${\mathcal {N}}$). Define:

• $\hat{\beta }_e = \text{ max } \; \{ \beta _e^m: \; m \in {\mathcal {M}} \}$, $e \in {\mathcal {E}}$

• $\hat{\beta }^i_e = \beta _e^m$ if $e \in \mathcal {C}_{i}$ and $r^m_{ei} = 1$, $e \in {\mathcal {E}}, i \in \mathcal {I}$ (recall that $r^m_{ei} = 1$ if MCS $m$ is used for link $e$ in compatible set $\mathcal {C}_{i}$)

• $\hat{\beta }^i_e = 0$ if $e \notin \mathcal {C}_{i}$ (then $r^m_{ei} = 0$ or all $m \in {\mathcal {M}}$)

• $\alpha = \text{ max } \;\left\{ \sum _{e \in {\mathcal {E}}} \pi _e B_{ei} - \sum _{e\in {\mathcal {E}}} \hat{\beta }_e^i : \; i \in \mathcal {I} \right\}$

• $i_0$ is the index of a compatible set with $\alpha = \sum _{e \in {\mathcal {E}}} \pi _e B_{ei_0} - \sum _{e \in {\mathcal {E}}} \hat{\beta }_e^{i_0})$

• $\hat{W}$ is the minimum of the dual function (6a).

With these definitions the following holds

$$\begin{aligned} \hat{W} = \left( \alpha + {\mathop {\sum }\nolimits _{e \in {\mathcal {E}}}} \hat{\beta }_e\right) \cdot T \end{aligned}$$

so that

$$\begin{aligned} \frac{\hat{W}}{T}&= \sum \nolimits _{e \in {\mathcal {E}}} \pi _e B_{ei_0} - \sum \nolimits _{e \in {\mathcal {E}}} \hat{\beta }_e^{i_0} + \sum \nolimits _{e \in {\mathcal {E}}} \hat{\beta }_e\\&= \sum \nolimits _{e \in {\mathcal {E}}} \pi _e B_{ei_0} + \sum \nolimits _{e \in {\mathcal {E}}} \left( \hat{\beta }_e - \hat{\beta }_e^{i_0}\right) . \end{aligned}$$

Since, by definition $\hat{\beta }_e \ge \hat{\beta }_e^{i_0}, \; e \in {\mathcal {E}}$, we have

$$\begin{aligned} \frac{\hat{W}}{T} \ge \sum \nolimits _{e \in {\mathcal {E}}} \pi _e B_{ei_0} \end{aligned}$$

which shows that the minimum of the dual function, $\hat{W}$, is attained also for the original values of $\alpha$, $\pi _e$ ($e \in {\mathcal {E}}$) and for $\beta _{e}^{m}=0$ ($e \in {\mathcal {E}}, m \in {\mathcal {M}}$), and hence for $\zeta _e = 0$ ($e \in {\mathcal {E}}$). $\square$