Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 185262, 11 pages

http://dx.doi.org/10.1155/2015/185262

## On Modeling and Analysis of MIMO Wireless Mesh Networks with Triangular Overlay Topology

^{1}Department of Computer Science, South China Normal University, Guangzhou, Guangdong 510631, China^{2}Department of Computer Science, University of Memphis, Memphis, TN 38152, USA^{3}School of Computer Science & Education Software, Guangzhou University, Guangzhou, Guangdong 510006, China^{4}Department of Computer Science, Middle Tennessee State University, Murfreesboro, TN 37132, USA

Received 27 September 2014; Revised 17 January 2015; Accepted 27 January 2015

Academic Editor: Hsuan-Ling Kao

Copyright © 2015 Zhanmao Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Multiple input multiple output (MIMO) wireless mesh networks (WMNs) aim to provide the last-mile broadband wireless access to the Internet. Along with the algorithmic development for WMNs, some fundamental mathematical problems also emerge in various aspects such as routing, scheduling, and channel assignment, all of which require an effective mathematical model and rigorous analysis of network properties. In this paper, we propose to employ Cartesian product of graphs (CPG) as a multichannel modeling approach and explore a set of unique properties of triangular WMNs. In each layer of CPG with a single channel, we design a node coordinate scheme that retains the symmetric property of triangular meshes and develop a function for the assignment of node identity numbers based on their coordinates. We also derive a necessary-sufficient condition for interference-free links and combinatorial formulas to determine the number of the shortest paths for channel realization in triangular WMNs.

#### 1. Introduction

The WiMax group advocated the last-mile broadband services, IEEE 802.16 Standard [1], which defines broadband backbones as wireless mesh networks (WMNs). Such networks typically consist of two types of nodes, that is, base station (BS) and subscriber station (SS). BS is a wireless gateway connected to the Internet, while SS is a node that acts as a relay station. In multiple input multiple output (MIMO) WMNs, all nodes are equipped with multiple interfaces and support both multicast and mesh modes. Particularly, in a mesh mode, nodes can communicate with neighbors without the help of BS and the relay strategy provides an economical way to expand the mesh covering area. MIMO WMNs (in the rest of the paper, we use the term WMNs for conciseness) are well recognized as an efficient extension to the Internet backhaul [2].

WMNs possess some inherent characteristics that are different from ad hoc or wireless sensor networks. Since the nodes in WMNs are almost fixed and typically powered by electrical wires, the links or routing paths in WMNs generally last longer than those in mobile ad hoc networks. Also, every node in WMNs typically has nonzero traffic requests because it needs to route aggregated traffic from the terminal devices in its region for either upload or download. The topology of WMNs may be determined based on the predicted traffic requests or geographical environments. Since both BS and SS can be considered static, it is reasonable to view the mesh topology as a fixed graph.

The rapidly growing demand for ubiquitous Internet access requires an effective mathematical model for WMNs as it may simplify the tasks of routing, scheduling, and channel assignment. To achieve a maximum fair usage of multiple channels in WMNs, it is important to employ an efficient channel allocation scheme and an appropriate overlay graph topology for a given area [3]. In [4], the virtual topology is viewed as CPG to simplify the channel assignment problem through a graph. In addition, the CPG model also brings convenience for the analysis of routing and scheduling in WMNs. As nodes are static in WMNs, they can be identified geographically through their coordinates [5]. Therefore, the routing and scheduling problems can be analyzed using the node coordinates. Interference is another fundamental issue in either scheduling or routing, and the properties of interference under a given coordinate scheme, if defined properly and described rigorously, may bring benefits to resource utilization and interference avoidance.

In this work, we use CPG as a modeling approach and explore a set of unique properties of WMNs with a triangular topology. In each layer of CPG with a single channel, the network topology is a planar mesh. Our work makes several theoretical contributions to the analysis of WMN properties: (i) we design a coordinate scheme that retains the symmetric property of triangular meshes and develop a function to assign a unique identity number to a specific wireless node based on its coordinates. (ii) We derive a necessary-sufficient condition for interference-free links under the proposed node coordinate scheme. (iii) We derive combinatorial formulas in terms of the number of transceivers and channels to determine the number of the shortest paths for channel realization in triangular WMNs.

The rest of the paper is organized as follows. Section 2 surveys related work. Section 3 presents a channel-layered CPG model with emphasis on interference detection. In each layer of CPG, we propose a coordinate scheme, named parallel cluster coordinate, derive a necessary-sufficient condition for interference avoidance, and develop a function for node identity number assignment to support efficient WMN maintenance and administration. Section 4 derives formulas for determining the number of the shortest paths in WMNs.

#### 2. Related Work

We conduct a brief survey of work directly related to mathematical models for WMNs.

In the past decade, most efforts in WMN topology were focused on interference and performance in planar meshes [6–9]. With geographical information from satellite or control channel communications, it is relatively easy to acquire a planar topology since the nodes in WMNs are almost static. For example, in IEEE Standard 802.21 for media-independent handover, Media Independent Information Service (MIIS) stores the geographical information of all access network operators available in a particular region [10].

Square and hexagonal meshes have been proposed to act as wireless broadband backbones [11]. However, they are less competitive than triangular meshes, as the latter outperforms the former and other random meshes in terms of various performance metrics such as coverage area, link quality, per-user fair rate, and node density [12]. Hong and Hua conducted a comparative evaluation of the throughput performance between square, hexagonal, and triangular meshes. Their experiments showed that triangular meshes achieve higher throughput than others in several cases, and their total throughput does not vary significantly in response to topology changes in large wireless networks with a constant density [13]. Therefore, we also adopt a triangular mesh topology in our model.

A unified network model based on super graph may further facilitate the analysis of various aspects of WMNs such as interference, scheduling, routing, and channel assignment. However, research efforts along this line are still quite limited. Several researchers considered some of these aspects simultaneously [3, 11, 14], which motivates us to design a unified model for WMNs.

In a given network topology, a properly designed coordinate scheme may facilitate link interference detection and path finding. In hexagonal meshes, Chin et al. proposed a node coordinate scheme with three parallel line clusters [15], where a node is represented by a 3-tuple. In triangular meshes consisting of BS nodes with a node degree of six, Cao et al. proposed two BS-centered coordinate schemes [5] and explored interference and link groups in each of these schemes. Furthermore, in addition to coordinates, a router should also be assigned a unique identity number to support convenient simulation, administration, and maintenance. In [4, 5], Cao et al. also represented the coordinates of a node by a 3-tuple but did not tackle the identity number assignment problem.

The performance of WMNs is largely affected by link interference. Most research efforts on this subject have been made through generic methods or experimental studies, instead of conclusive results in the form of necessary-sufficient conditions [3, 16, 17]. In our work, we attempt to design a suitable node coordinate scheme and then model the interference in WMNs as a specific checking list based on set theory.

Routing in WMNs is a 2-step procedure, that is, path finding followed by channel assignment. One basic approach to find an alternative interference-free path is to count the number of shortest paths and the number of all possible channel assignment schemes. A tree-like path finding scheme is proposed in [6] without any node coordinate. Cao and Xiao proposed path counting formulas for a source-destination pair in square grids [18], while the path counting problem in triangular meshes is still left unexplored. In our work, based on the proposed CPG model and coordinate scheme, we tackle this problem in triangular meshes with multiple channels and interfaces. Channel assignment is another important problem involving several network layers in WMNs, which is essentially an NP-complete edge coloring problem [14, 19].

#### 3. A Channel-Layered Graph Model

As massive MIMO is on its way from theory to realistic deployment, one of the key problems is the interchannel cooperation, which calls for the development of sophisticated analytical channel models. Larsson et al. provide an overview of massive MIMO and motivate researchers to develop channel models capturing the essential channel behaviors despite their limitations [20]. For example, the Kronecker model, which is widely used to model channel correlation, is not an exact representation of reality but provides a useful model for certain types of analysis.

WMNs are conventionally modeled by a directed graph where a directed edge between two neighbor nodes represents a communication link over a specific channel. Since a node equipped with transceivers may have (at most) simultaneous links over orthogonal channels (assuming that more than channels are available), we can split a physical node into fully connected virtual nodes, each of which is equipped with a single transceiver. This way, we are able to represent the original WMN as identical layers of networks, each of which operates over a different channel.

##### 3.1. Cartesian Product of Graphs

The topology-based modeling approach has been commonly used in wireless networks for various purposes, but often in a planar view [8, 12, 14], and most of the discussions on scheduling, routing, and channel assignment are also based on a planar topology [6, 11, 13, 17]. The recent development of MIMO WMNs calls for a suitable model to describe MIMO-specific properties and understand the cooperative activities across different interfaces over multiple channels [20]. The planar topology can be used to determine the internode interference [8] but is insufficient to provide a visual representation for analyzing the cooperation between links or channels. On the other hand, modeling MIMO WMNs as a super graph still remains largely unexploited except the work in [4]. In this paper, our goal is to develop an effective model to facilitate the analysis of MIMO channel cooperation.

We propose to employ the CPG to model WMNs by combining a triangular mesh of physical nodes and a graph of fully connected virtual nodes. Together with the coordinates of triangular overlay nodes, the CPG model provides a convenient way to analyze the properties of interference avoidance, channel assignment, and routing path counting. This model retains the independence between orthogonal channels while providing a general approach to analyzing link behaviors over multiple channels.

*Cartesian Product of Graphs.* Given two graphs and , the Cartesian product is a graph such that(i)the graph has a vertex set ; that is, a vertex in is denoted by a pair , , and ;(ii)any two nodes and and and are adjacent in , if and only if one of the following holds: (a) and is adjacent to in , or (b) and is adjacent to in .

For illustration, Figure 1(a) shows a mesh network of five physical nodes (left side) and a graph of two connected virtual nodes (right side) corresponding to a physical node equipped with two transceivers, each operating on a different channel or . Figure 1(b) shows a channel-layered virtual topology of the original mesh network modeled by CPG.