Minimum Cost Bandwidth Guaranteed Multicast Routing in Multi-channel Multi-radio Wireless Mesh Networks

Abstract

Multicast communication is an important service in wireless mesh networks (WMNs). It covers a broad range of applications, including data distribution, video conferencing, and distance learning. In this paper, we discuss the issue of bandwidth guaranteed multicast routing in multi-channel multi-radio WMNs. The problem of our concern is to construct a tree per multicast session such that the cost of the system, which is defined as the amount of total consumed bandwidth, is minimized. In order to solve the problem efficiently, we design Bandwidth Guaranteed Minimum Cost Tree construction (BGMCT) algorithm. Our algorithm yields cost-effective solutions as it exploits the wireless broadcast advantage (WBA) property of the wireless medium. In the proposed algorithm, we have developed two strategies for constructing minimum cost trees. Firstly, the number of the relay nodes in each tree is minimized. Secondly, the amount of overlapping between the shortest paths which connect different destinations of each session to its source node, is taken into account. The simulation results demonstrate that our algorithm outperforms existing solutions. Moreover, BGMCT provides near to optimal outcomes in a reasonable time.

Appendices

Appendix 1

Lemma 1

Constraint (9) guarantees that $T_m \left( {1\le m\le M} \right)$ is loop-free.

Proof

Assume there is a loop in $T_{m}$, and nodes $v_i \left( {1\le i\le q} \right)$ are in the loop. In this loop, $p_{v_{i+1} }^m =v_i \left( {1\le i<q} \right)$ and $p_{_{v_1 } }^m =v_q$. According to (9), $\varepsilon +h_{v_i }^m \le h_{v_{i+1} }^m \left( {1\le i<q} \right)$ and $\varepsilon +h_{v_q }^m \le h_{v_1 }^m$. Using these inequalities, it can be derived that $\varepsilon \le 0$, which is in contradiction to the initial assumption. $\square$

Lemma 2

In $T_m \left( {1\le m\le M} \right)$, the source node $s_{m}$ is connected to all destinations in $D_{m}$.

Proof

the existence of the path is shown in a bottom-up manner. According to (4), each destination $v\in D_m$ has a parent node. In addition, according to (8), $p_v^m$ must have a parent. Continuing this procedure, a chain of nodes is formed. The length of this chain is bounded by $\left| V \right|$. Since $\left| V \right|$ is finite, the chain has a tail. According to Lemma 1, $T_{m}$ is loop-free. Hence, the tail of the chain is a node that has no parent. The only node that preserves this constraint is $s_{m}$. $\square$

Appendix 2

Linearizing (11): To avoid the quadratic term on the right hand side of (11), it is replaced by the following constraints:

$$\begin{aligned}&nl_{uv}^{\textit{mk}} \le a_u^k\quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M\end{aligned}$$

(26)

$$\begin{aligned}&nl_{uv}^{\textit{mk}} \le a_v^k \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M\end{aligned}$$

(27)

$$\begin{aligned}&nl_{uv}^{\textit{mk}} \le l_{uv}^m \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M \end{aligned}$$

(28)

Linearizing (12): We formularize $\mathop {\min }\nolimits _{w\in \textit{RNH}_u^{\textit{mk}} } \left\{ {c_{uw} } \right\}$ at the first step. To do this, nodes of $N_{u}$ are sorted in ascending order according to the capacity of their common links with node $u$. If node $w_{i}$ is a receiver node of $\textit{NH}_u^{\textit{mk}}$, and its preceding nodes are not in the NH, $\mathop {\min }\nolimits _{w\in \textit{RNH}_u^{\textit{mk}} } \left\{ {c_{uw} } \right\}$ will be equal to $c_{uw_i }$. This condition is formally stated as:

$$\begin{aligned} \mathop {\min }\limits _{w\in \textit{RNH}_u^{\textit{mk}} } \left\{ {c_{uw} } \right\}&= c_{uw_1 } nl_{uw_1 }^{\textit{mk}} +c_{uw_2 } nl_{uw_2 }^{\textit{mk}} \left( {1-nl_{uw_1 }^{\textit{mk}} } \right) +\cdots + \nonumber \\&\quad c_{uw_{\left| {N_u } \right| } } nl_{uw_{_{\left| {N_u } \right| } } }^{\textit{mk}} \left( {1-nl_{uw_{\left| {N_u } \right| -1} }^{\textit{mk}} } \right) \ldots \left( {1-nl_{uw_2 }^{\textit{mk}} } \right) \left( {1-nl_{uw_1 }^{\textit{mk}} } \right) \nonumber \\&=\sum _{i=1}^{\left| {N_u } \right| } {c_{uw_i } nl_{uw_i }^{\textit{mk}} } -\sum _{i=2}^{\left| {N_u } \right| } {\sum _{j=1}^{i-1} {c_{uw_i } nl_{uw_i }^{\textit{mk}} nl_{uw_j }^{\textit{mk}} } } +\cdots + \left( {-1} \right) ^{\left| {N_u } \right| -1}c_{uw_{\left| {N_u } \right| } } nl_{uw_1 }^{\textit{mk}} \ldots nl_{uw_{\left| {N_u } \right| } }^{\textit{mk}}\nonumber \\ \end{aligned}$$

(29)

By combining (12) and (29), some binary products in the form of $nl_{uv}^{\textit{mk}} \prod \nolimits _{w_l \in p(N_v,j)} {nl_{uw_l }^{\textit{mk}} }$ $\left( {1<j\le \left| {N_u } \right| } \right)$ are appeared, where $p\left( {N_v, j} \right)$ presents the permutation of $j$ neighbors from the nodes of $N_{v}$. The term is linearized by replacing it with auxiliary variable $axh_{uvp}^{\textit{mk}}$ and including the following constraints in the model:

$$\begin{aligned}&nl_{uv}^{\textit{mk}} \le axh_{uvp}^{\textit{mk}} \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M,\quad \forall permutation\, p\end{aligned}$$

(30)

$$\begin{aligned}&nl_{uw_l }^{\textit{mk}} \le axh_{uvp}^{\textit{mk}} \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M, \quad \forall permutation \quad p, \forall w_l \in p\nonumber \\\end{aligned}$$

(31)

$$\begin{aligned}&axh_{uvp}^{\textit{mk}} \ge nl_{uv}^{\textit{mk}} \!+\!\sum _{w_l \in p(N_v, j)} {nl_{uw_l }^{\textit{mk}} } -j \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M,\quad \forall permutation\, p\nonumber \\ \end{aligned}$$

(32)

Doing so, the product term $t_u^{\textit{mk}} axh_{uvp}^{\textit{mk}}$ is appeared in (12), which is replaced by auxiliary variable $axht_{uvp}^{\textit{mk}}$. The mentioned variable is linearized as follows:

$$\begin{aligned}&axht_{uvp}^{\textit{mk}} \le bigM \quad axh_{uvp}^{\textit{mk}}\quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M, \quad \forall permutation \ p\nonumber \\\end{aligned}$$

(33)

$$\begin{aligned}&axht_{uvp}^{\textit{mk}} \le t_u^{\textit{mk}} \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M,\quad \forall permutation \ p\end{aligned}$$

(34)

$$\begin{aligned}&axht_{uvp}^{\textit{mk}} \ge t_u^{\textit{mk}} -bigM\left( {1-axh_{uvp}^{\textit{mk}} } \right) \nonumber \\&\quad \quad \quad \quad \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M, \quad \forall permutation \ p \end{aligned}$$

(35)

Linearizing $nl_{uv}^{\textit{mk}} t_u^{mk}$ : This quadratic term is appeared in (13), (16) and (17). To linearize the term, it is replaced by an auxiliary variable $axlt_{uv}^{\textit{mk}}$ and the following constraints are added to the model:

$$\begin{aligned}&axlt_{uv}^{\textit{mk}} \le bigM\quad nl_{uv}^{\textit{mk}} \qquad \qquad \qquad \quad \,\forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M\end{aligned}$$

(36)

$$\begin{aligned}&axlt_{uv}^{\textit{mk}} \le t_u^{\textit{mk}} \qquad \qquad \qquad \qquad \qquad \qquad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M\end{aligned}$$

(37)

$$\begin{aligned}&axlt_{uv}^{\textit{mk}} \ge t_u^{\textit{mk}} -bigM\left( {1-nl_{uv}^{\textit{mk}} } \right) \quad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M \end{aligned}$$

(38)

Linearizing $aux_v^k$ -variable: The $aux_v^k$ variable that appeared in (14), is equal to 1 if one of the $nl_{uv}^{\textit{mk}}$-variables $\left( {\forall \left( {u,v} \right) \in E, 1\le m\le M} \right)$ is set to 1. Therefore, it can be expressed linearly as follows:

$$\begin{aligned}&nl_{uv}^{\textit{mk}} \le aux_v^k\qquad \qquad \forall \left( {u,v} \right) \in E, 1\le k\le K, 1\le m\le M\end{aligned}$$

(39)

$$\begin{aligned}&aux_v^k \le \sum _{\left( {u,v} \right) \in E} {\sum _{m=1}^M {nl_{uv}^{\textit{mk}} } }\qquad \qquad \forall v\in V, 1\le k\le K \end{aligned}$$

(40)