Selective greedy routing: exploring the path diversity in backbone mesh networks

Abstract

This paper proposes a new routing protocol for IEEE 802.11s wireless mesh network to overcome the problems associated with the conventional proactive and reactive routing approaches. The proposed ‘Selective Greedy’ (SelG) routing protocol operates in two phases. In the first phase it exploits the proactive mode of Hybrid Wireless Mesh Protocol, the standard routing protocol for IEEE 802.11s mesh, and constructs a set of potential forwarders for every mesh point, that can act as the next-hop relay. In the second phase, during the actual data communication, a candidate is selected from the set of potential forwarders based on a local optimization. The optimization procedure considers the statistical effect of local link quality fluctuation and interference over the global routing path selection. This way the SelG protocol reduces the control packet flooding in the network (a major drawback for reactive protocols). At the same time, the optimization procedure captures the network dynamics, and thus avoids the possibility of routing based on stale information (a drawback for proactive protocols). The routing properties and the correctness of the SelG protocol is established theoretically, and the performance of the protocol is analyzed through simulation results. The proposed protocol is implemented in an indoor wireless mesh testbed, and the performance is evaluated and compared with other traditional approaches.

Appendices

Appendix 1: Proof for theorem 1

Fig. 25

Interference characterization

The proof extends the concept provided in Lemma 1, given in [7], for protocol interference model in multi-interface mesh network. According to the protocol interference model, two interfaces interfere with each other, if they are at-most $q \times R_c$ distance apart. Consider the following two cases,

• Case I. Assume $I_i^k$ transmits. In this case, none of the interfaces in $H(I_i^k)$ can transmit. Therefore,

  $$\begin{aligned} Z(I^k_i)&= 1 \\ \sum \limits _{I^m_j \in H(I^k_i)}{Z(I^m_j)}&= 0 \end{aligned}$$

  This satisfies the inequality given in 2.

• Case II. Assume $I_i^k$ does not transmit. In this case, two interfaces from $H(I_i^k)$ can transmit, provided that they are at-least $q \times R_c$ distance apart. Let us assume that there exists ${\mathfrak {S}}$ number of interfaces in $H(I_i^k)$, who are at least $q \times R_c$ distance apart. Considering the fact that all interfaces in $H(I_i^k)$ must be within the disk centered at $I_i^k$ and with radius $q \times R_c$ (based of the protocol model consideration, interference range is at-most $q \times R_c$), the problem of finding ${\mathfrak {S}}$ can be reduced to the circle packing problem discussed in [32]. Let us consider Fig. 25. If $MP_i$ does not use interface $I_i^k$, then the interfaces of $MP_u$ and $MP_v$ can transmit simultaneously if they are at-least $q \times R_c$ distance apart. Without considering the directionality of the interfaces, the problem of “finding maximum number of interfaces that can simultaneously transmit”, is similar to the problem of “finding maximum number of non-overlapping circles of radius $(q \times R_c)/2$ that can be packed within a circle of radius $1.5 \times q \times R_c$”. From [32], the value is $7$. Therefore, for uniform distribution of the directional beams of the interfaces,

  $$\begin{aligned} {\mathfrak {S}} \le \left( \frac{\alpha }{2\pi } \times 7 \approx 1.11364 \alpha \right) \end{aligned}$$

  where $\alpha$ is the beam-width of the interface $I_i^k$, expressed in radians. As a result,

  $$\begin{aligned} Z(I^k_i)&= 0 \\ \sum \limits _{I^m_j \in H(I^k_i)}{Z(I^m_j)}&\le 1.11364\alpha \end{aligned}$$

This follows the theorem.

Appendix 2: Proof for theorem 2

Let $\mathcal {Y}_i = \langle MP_j,I_i^k\rangle$ represent the optimal solution of Problem 1 for $MP_i$. Instead of allowing $\mathcal {Y}_i$ to take specific (MP, interface) value, let us define $\mathcal {Y}_i$ to be a vector in $\{\fancyscript{O}_1,\fancyscript{O}_2,...,\fancyscript{O}_k\}$, where each $\fancyscript{O}_j$ takes a possible solution, and is represented by an integer value from $[0...k]$. Let this integer is chosen as follows: consider an equilateral simplex $\Sigma _k$ in $\mathbb {R}^{k-1}$ with vertices $b_1,b_2,...,b_k$. Let $c_k = \frac{b_1+b_2+...+b_k}{k}$ be the centroid of $\Sigma _k$, and let $\fancyscript{O}_j = b_j - c_k$ for $1 \le j \le k$. Then problem 1 is represented as an integer semidefinite program with finite solution bound. From [41], the problem can be solved within $O(k)$ time complexity by exploring all the elements in the vector $\{\fancyscript{O}_1,\fancyscript{O}_2,...,\fancyscript{O}_k\}$. In the present scenario, $k = |\mathbb {F}_i|\times |\Gamma _i|$. This follows the theorem.

Appendix 3: Proof for theorem 3

The airtime link metric value is additive in nature, and therefore it supports isotonicity. According to Algorithm 2, before broadcasting a PPREQ message, an MP, $MP_i$, updates the path metric value as follows,

$$\begin{aligned} {{\mathfrak {P}}_{ij}^{k}}^* = \min \{{\mathfrak {P}}_{ij}^{k} + {\mathfrak {C}}_{ij}^{k} | \langle MP_j, I^{k}_i \rangle \in \mathbb {F}_i\} \end{aligned}$$

(5)

where ${{\mathfrak {P}}_{ij}^{k}}^*$ is the updated path metric value. Considering two sets $\mathbb {A} \subseteq \mathbb {R}$ and $\mathbb {B} \subseteq \mathbb {R}$, where $\mathbb {R}$ is the set of real numbers. if $\min \{x| x \in \mathbb {A}\} \le \min \{y| y \in \mathbb {B}\}$, then for any $\omega \in \mathbb {R}$, following inequality always holds true.

$$\begin{aligned} \min \{x| x \in \mathbb {A}\} + \omega \le \min \{y| y \in \mathbb {B}\} + \omega \end{aligned}$$

Based upon this inequality, it can be easily seen that ${{\mathfrak {P}}_{ij}^k}^*$ is isotonic. During greedy selection, let us assume the link dispersion factor be $\rho$. Considering ${{\mathfrak {P}}_{ij}^k}^* \le {{\mathfrak {P}}_{uv}^w}^*$ for two candidate forwarders $\langle MP_j, I_j^k \rangle$ and $\langle MP_v, I_u^w \rangle$, following inequalities always hold true.

$$\begin{aligned} \left( 1 + \rho \right) \times {{\mathfrak {P}}_{ij}^k}^* \le \left( 1 + \rho \right) \times {{\mathfrak {P}}_{uv}^w}^* \end{aligned}$$

(6)

$$\begin{aligned} \left( 1 - \rho \right) \times {{\mathfrak {P}}_{ij}^k}^* \le \left( 1 - \rho \right) \times {{\mathfrak {P}}_{uv}^w}^* \end{aligned}$$

(7)

Therefore the utility function, as given in Eq. (3), also follows the isotonicity property.

Appendix: Proof for theorem 4

The method of contradiction is used to proof the theorem. For the notational shorthand, let us use only the next-hop MP as the candidate from the set of potential forwarders, and imply that the corresponding interface is used. Let ${\mathfrak {P}}_i$ and ${\mathfrak {C}}_i$ denote the path metric and the link metric for the next hop $MP_i$ respectively, with the implication of corresponding interfaces.

Let us assume that the greedy selection introduces a routing loop at $MP_i$. Assume the routing loop be $\{MP_i, MP_j, ..., MP_k, MP_i\}$. Because of the additive and non-zero properties of the airtime metric value,

$$\begin{aligned}&\min \{{\mathfrak {P}}_i + {\mathfrak {C}}_i\} < \min \{{\mathfrak {P}}_j + {\mathfrak {C}}_j\} < ... \nonumber \\&\qquad {} < \min \{{\mathfrak {P}}_k + {\mathfrak {C}}_k\} < \min \{{\mathfrak {P}}_i + {\mathfrak {C}}_i\} \end{aligned}$$

(8)

$\forall {i};\;\min \{{\mathfrak {P}}_i + {\mathfrak {C}}_i\} \in \mathbb {R}^+$, where $\mathbb {R}^+$ is the set of positive real numbers. Therefore, the inequality given in Eq. (8) is never possible. This contradicts with our assumption.

Hence, the greedy selection never introduces a routing loop.