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Research Article

On Modeling and Analysis of MIMO Wireless Mesh Networks

with Triangular Overlay Topology

Zhanmao Cao,1,2 Chase Q. Wu,2Yuanping Zhang,3Sajjan G. Shiva,2and Yi Gu4

1Department of Computer Science, South China Normal University, Guangzhou, Guangdong 510631, China

2Department of Computer Science, University of Memphis, Memphis, TN 38152, USA

3School of Computer Science & Education Soware, Guangzhou University, Guangzhou, Guangdong 510006, China

4Department of Computer Science, Middle Tennessee State University, Murfreesboro, TN 37132, USA

Correspondence should be addressed to Chase Q. Wu; chase.wu@memphis.edu

Received September ; Revised January ; Accepted January

Academic Editor: Hsuan-Ling Kao

Copyright © Zhanmao Cao et al. is 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.

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 eective 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 andexplore a set of unique properties of triangular WMNs. In each layer ofCPG 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-sucient condition for interference-free links and

combinatorial formulas to determine the number of the shortest paths for channel realization in triangular WMNs.

1. Introduction

e WiMax group advocated the last-mile broadband servi-

ces, IEEE . Standard [], which denes broadband back-

bones 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 con-

nected 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 ecient extension to the Internet backhaul

[].

WMNs possess some inherent characteristics that are

dierent from ad hoc or wireless sensor networks. Since the

nodes in WMNs are almost xed and typically powered by

electrical wires, the links or routing paths in WMNs generally

lastlongerthanthoseinmobileadhocnetworks.Also,every

node in WMNs typically has nonzero trac requests because

it needs to route aggregated trac from the terminal devices

in its region for either upload or download. e topology

of WMNs may be determined based on the predicted trac

requests or geographical environments. Since both BS and SS

canbeconsideredstatic,itisreasonabletoviewthemesh

topology as a xed graph.

e rapidly growing demand for ubiquitous Internet

access requires an eective 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 ecient

channel allocation scheme and an appropriate overlay graph

topology for a given area []. In [], 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 identied

Hindawi Publishing Corporation

Mathematical Problems in Engineering

Volume 2015, Article ID 185262, 11 pages

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

Mathematical Problems in Engineering

geographically through their coordinates []. erefore, 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 interfer-

ence under a given coordinate scheme, if dened properly

and described rigorously, may bring benets to resource utili-

zation and interference avoidance.

In this work, we use CPG as a modeling approach and

explore a set of unique properties of WMNs with a triangu-

lar 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 specic wireless node

based on its coordinates. (ii) We derive a necessary-sucient

condition for interference-free links under the proposed

node coordinate scheme. (iii) We derive combinatorial for-

mulasintermsofthenumberoftransceiversandchannels

to determine the number of the shortest paths for channel

realization in triangular WMNs.

e rest of the paper is organized as follows. Section

surveys related work. Section 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-sucient con-

dition for interference avoidance, and develop a function for

node identity number assignment to support ecient WMN

maintenance and administration. Section 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 mathe-

matical models for WMNs.

In the past decade, most eorts in WMN topology were

focused on interference and performance in planar meshes

[–]. With geographical information from satellite or con-

trol channel communications, it is relatively easy to acquire a

planar topology since the nodes in WMNs are almost static.

For example, in IEEE Standard . for media-independent

handover, Media Independent Information Service (MIIS)

stores the geographical information of all access network

operators available in a particular region [].

Square and hexagonal meshes have been proposed to act

as wireless broadband backbones []. 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 []. Hong and Hua

conducted a comparative evaluation of the throughput per-

formance between square, hexagonal, and triangular meshes.

eir experiments showed that triangular meshes achieve

higher throughput than others in several cases, and their

total throughput does not vary signicantly in response to

topology changes in large wireless networks with a constant

density []. erefore, we also adopt a triangular mesh

topology in our model.

A unied network model based on super graph may

further facilitate the analysis of various aspects of WMNs

such as interference, scheduling, routing, and channel assign-

ment. However, research eorts along this line are still quite

limited. Several researchers considered some of these aspects

simultaneously [,,], which motivates us to design a

unied model for WMNs.

In a given network topology, a properly designed coor-

dinate scheme may facilitate link interference detection and

path nding. In hexagonal meshes, Chin et al. proposed a

node coordinate scheme with three parallel line clusters [],

where a node is represented by a -tuple. In triangular meshes

consisting of BS nodes with a node degree of six, Cao et al. pro-

posed two BS-centered coordinate schemes [] 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 [,], Cao

et al. also represented the coordinates of a node by a -tuple

but did not tackle the identity number assignment problem.

e performance of WMNs is largely aected by link

interference. Most research eorts on this subject have been

made through generic methods or experimental studies,

instead of conclusive results in the form of necessary-

sucient conditions [,,]. In our work, we attempt to

design a suitable node coordinate scheme and then model the

interference in WMNs as a specic checking list based on set

theory.

Routing in WMNs is a -step procedure, that is, path

nding followed by channel assignment. One basic approach

to nd 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 nding scheme

is proposed in []withoutanynodecoordinate.CaoandXiao

proposed path counting formulas for a source-destination

pair in square grids [], while the path counting problem in

triangular meshes is still le unexplored. In our work, based

on the proposed CPG model and coordinate scheme, we

tackle this problem in triangular meshes with multiple chan-

nels and interfaces. Channel assignment is another important

problem involving several network layers in WMNs, which is

essentially an NP-complete edge coloring problem [,].

3. A Channel-Layered Graph Model

AsmassiveMIMOisonitswayfromtheorytorealistic

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 []. 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 specic channel. Since a node

Mathematical Problems in Engineering

(a) Mesh and virtual nodes (b) Layered mesh

(c) Links in a channel (d) Maximal links

A

A

A

A

B

B

B

B

C

C

C

C

D

D

D

D

E

E

E

E

c1

c2

F : e CPG-modeled virtual topology of a physical network with ve physical nodes and two channels.

equipped with transceivers may have (at most) simul-

taneous 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. is way, we are able to

represent the original WMN as identical layers of networks,

each of which operates over a dierent channel.

3.1. Cartesian Product of Graphs. e topology-based model-

ing approach has been commonly used in wireless networks

for various purposes, but oen in a planar view [,,],

andmostofthediscussionsonscheduling,routing,and

channel assignment are also based on a planar topology [,

,,]. e recent development of MIMO WMNs calls

for a suitable model to describe MIMO-specic properties

and understand the cooperative activities across dierent

interfaces over multiple channels []. e planar topology

can be used to determine the internode interference []butis

insucient 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 []. In this paper, our

goal is to develop an eective 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.

is 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 ()×();thatis,

avertexin×is denoted by a pair (V,V),V∈ (),

and V∈ ();

(ii) any two nodes (,)and (V,V)and ,V∈ ()and

,V∈() are adjacent in ×, if and only if one

of the following holds: (a) =Vand is adjacent to

Vin ,or(b)=Vand is adjacent to Vin .

For illustration, Figure (a) shows a mesh network of ve

physical nodes (le side) and a graph of two connected virtual

nodes (right side) corresponding to a physical node equipped

with two transceivers, each operating on a dierent channel

1or 2.Figure (b) shows a channel-layered virtual topology

of the original mesh network modeled by CPG.

In this example, the CPG of two graphs in Figure (a)

results in a two-layered graph in Figure (b). A solid directed

Mathematical Problems in Engineering

edge in the top or bottom planar meshes in Figures (b),

(c), and (d) represents a communication link =(,)

𝑐,

transmitting data from node to node over a channel ,

while a dashed edge has a conict with some active links.

emaximumpossiblenumberofconcurrentactive

links on a given channel in a mesh is largely aected by

the selection of the senders. As shown in Figure (b), when

node in the top layer is sending data on channel 1,the

three neighbors of , that is, nodes ,,and,inthe

same layer (over channel 1)cannotsenddata.Furthermore,

node canbeselectedasasender,butneitherofits

neighbors, that is, nodes and ,canreceivedatafrom

without interference, illustrated as the dashed directed edges

in Figure (c). In this case, there is no other active link except

(,)𝑐1. However, if we choose the initial senders properly,

as shown in Figure (d), there could exist two concurrent

interference-free links on channel 𝑖,=1,2,thatis,(,)𝑐𝑖

and (,)𝑐𝑖.

Since all active links must be interference-free on the

same channel at the same time, links are generally sparsely

distributed in a planar mesh with respect to a certain

channel. Note that more links mean better service to trac

requests. e CPG model allows us to consider concurrent

paths in multiple layers over dierent channels through

radio cooperation among channel layers for a given trac

task, hence providing more capacity and higher throughput

comparedtothesituationwithasinglechannelandradio.

is is consistent with the experimental results presented

by Draves et al. []. e proposed CPG model is aligned

well with existing research in terms of interference relation,

link activity, and network throughput, as well as routing and

scheduling [,,,,] and enables us to conduct deeper

theoretical analysis of WMNs.

One advantage that CPG brings is to simplify the expres-

sion of multichannel links. Since dierent layers (operating

on dierent channels) are of the same topology, the schedul-

ing strategy derived in one layer is readily applicable to

another layer. For example, there exists a certain link distri-

bution pattern among concurrent links. In CPG, it is obvious

that such an interference-free link distribution pattern on one

channel also exists on others.

3.2. Coordinate Scheme. A well-designed coordinate scheme

may facilitate the analysis of WMNs. Since all channel

layers are of the same topology, we only need to design the

coordinate scheme for one layer or channel. e channel

information can be added to the node coordinates to uniquely

identify a specic layer.

Chin et al. proposed a coordinate scheme in hexagonal

cellular networks where each node has a degree of three

[]. Inspired by their work, we propose a parallel cluster

coordinate scheme in a local triangular mesh with one BS

node and a number of SS nodes with a degree of six. is

scheme can be readily extended to larger networks with

multiple BSs by inserting the BS information to the node

coordinates.

3.2.1. Parallel Cluster Coordinate Scheme. In a triangular

mesh, we rst dene three clusters of parallel lines along three

A

B

C

D

E

j

F

BS

i−2 i−1 i+1

j+2

j+1

j−1

j−2

k−2 k−1 k+1

k

i

F : e parallel cluster coordinate scheme.

dierent directions, that is, north-east, east, and south-east.

Sinceeachnodeisapointintersectedbythreelines,each

fromoneofthethreeclusters,weproposeaparallelcluster

coordinate scheme (PCCS), which uses a -tuple (,,) to

represent the coordinates of a node intersected by the th, th,

and th line in the corresponding clusters []. As illustrated in

Figure , the BS node with the coordinates (0,0,0)is located

at the center, and the SS nodes ,,,,,and,which

are one hop away from the BS node, have the coordinates

= (1,0,1), = (0,1,1), = (−1,−1,0), = (−1,0,−1),

= (0,−1,−1),and = (1,1,0).

Note that not every combination of three integers can

represent a node in a triangular mesh because some lines

dened by combinatorial -tuples do not intersect at a

common point. We summarize such lines as follows:

= −1,−1,,+1,+1,,,− 1,+1,

,+1,−1,+ 1,,− 1,− 1,,+ 1. ()

Under the proposed PCCS scheme, for a given sender

(,,), there are six possible receivers, which form its

neighbor set 𝑏(,,):

𝑏,,= − 1,+1,,+1,−1,,

,+1,+1,,−1,−1,

+1,,+1,−1,,−1.

()

PCCS retains the symmetric nature of a triangular mesh

and facilitates the calculation of the distance between a pair

of nodes.

3.2.2. Symmetric Property of PCCS. In PCCS, the coordinates

of any two nodes that are symmetric with respect to the

central BS node located at (0,0,0) are negated. For example,

the pairs of nodes and ,and ,andand in Figure

are symmetric, and their coordinates are negated from their

counterparts.

e symmetric properties of CPG and Cayley graph

have been well studied []. According to the vertex/edge

Mathematical Problems in Engineering

transitive properties, a link group can be transited to generate

another one in any layer.

Similarly, due to the symmetric property, we are able to

transform a link group to another one through rotation. For

example, link group of ((0,2,2),(0,1,1))𝑐and ((2,−2,0),

(1,−1,0))𝑐is interference-free. Aer a clockwise rotation of

/3, a new group of interference-free links is of ((2,0,2),

(1,0,1))𝑐and ((0,−2,−2),(0,−1,−1))𝑐,respectively.With

another clockwise rotation of /3, the derived interference-

free links are ((2,−2,0),(1,−1,0))𝑐and ((−2,0,−2),(−1,0,

−1))𝑐, respectively. Note that the rotation operation is edge

transitive, and it generates a new link group because of the

symmetric property.

3.2.3. Distance between Two Nodes. Asmosttracisupload/

download (to/from BS), we need to count the number of hops

fromarouternodetoBSinPCCS.Wehavethefollowing

properties.

Property 1. In triangular WMNs with PCCS, the minimum

number of hops from = (,,) to BS = (0,0,0)is

=||++||

2=max ||,,||()

which is consistent with the one in [].

Property 2. Suppose that BS is positioned at an arbitrary

location (0,0,0),insteadof(0,0,0).Node(,,) can be

translated to (,,)−(0,0,0)by a translation function:

:,,→ ,,−0,0,0.()

In general, suppose that (0,0,0)is the destination of a

trac path; we may virtually view (0,0,0)as the BS node

aer applying the translation of ().en,theformulain()

may facilitate further analysis.

Given two nodes =(

1,1,1)and =(

2,2,2)in

PCCS, the distance between and , denoted by (,),

is the minimum number of hops between them, which is

calculated as (,) = (|1−

2|+|

1−

2|+|

1−

2|)/2.

3.2.4. Mapping to 2D Points. In order to draw a planar mesh,

we need to map -tuple coordinates (,,)to -dimensional

(D) points (,).Todothis,werstoverlaparectangular

plane coordinate system to the PCCS triangular mesh. Let

the -axis overlap the axis =0while keeping the positive

rightward direction. Meanwhile, the -axispassestheBS

node and is vertical to the line =0with a positive upward

direction. en, we can determine (,)by projecting (,,)

onto axes and .ePCCSsupportsafunctionmapping

(,,)to (,)as follows.

Property 3. A one-to-one mapping function maps node

coordinates (,,)in PCCS to D coordinates (,):

,= ,, =

=+

2

=. ()

For example, node (0,1,1) is mapped to (1/2,1),and

(−1,3,2) is mapped to (1/2,3).Ifanodeisonthe-axis, =

−always holds. For example, (−1,2,1)is mapped to (0,2).If

node is located on =0in PCCS, =/2. For example,

(−2,2,0) is mapped to (−1,2).ispropertyfacilitatesthe

plotting of a triangular mesh in PCCS.

3.3. Interference-Free Conditions. To analyze the interference

between links, we need to consider node interference rela-

tions, which are critical to scheduling links in WMNs. As

interference is an inherent nature for radio media, the wire-

less communication performance may be severely degraded

if radios operate without a proper scheduling scheme [].

Minimizing interference has been extensively investi-

gated in the literature [,]. Subramanian et al. discussed

channel assignment in a multiradio situation []. Tan et al.

designed algorithms to set up a skeleton of minimum interfer-

ence for a single channel []. Scheduling links in a coopera-

tive way will improve the energy eciency and reduce colli-

sion. ese discussions assume variable transmitting power

or interface channel switching. However, the variation of

transmitting power may lead to the variation of network

topology, which may cause changes in the interference

relation.Xuetal.useasensingschemetoachievepowere-

ciency for convergence communication []. eir method

canhelpsettheinitialpowerinanalmost-staticWMN

topology while promising interference-free cognitive access

with link status as busy or idle.

When sender is sending over channel 0, neither can

receive data over channel 0, nor can its neighbors in its

eective radio coverage send data. If a valid neighbor node

receives data over channel 0,wehavealink(,)𝑐0.

Furthermore, the neighbors of receiver cannot send data

over channel 0at the same time. erefore, to be link

interference-free in multiradio multichannel environments

using PCCS, we need to consider three classes of node inter-

ferences: sender-to-sender, receiver-to-receiver, and sender-

to-receiver.

3.3.1. Sender-to-Sender. Givenacertainchannel,anytwo

sendersmustbeatleasttwohopsawayinatriangularmesh

to avoid mutual interference, which could help construct an

interference-free candidate set for possible senders.

Denition 1. Arouterset𝐶is called a sender candidate set if

every two nodes in 𝐶are at least two hops away.

Given two nodes 𝑖=(,,)and 𝑗=(,,)in 𝐶,the

distance between them must satisfy (𝑖,𝑗)≥2.

For a given node 𝑗∉

𝐶,wecanadd𝑗to 𝐶only if it

is at least two hops away to any node 𝑖in 𝐶.Wehavethe

following necessary condition on interference-free links:

𝑖,𝑗=|−|+|−|+−

2≥2. ()

3.3.2. Receiver-to-Receiver. Two nodes 𝑖and 𝑗can each act

as a receiver simultaneously if they have a distance larger than

. Similar to the sender candidate set, we use 𝐶to denote

Mathematical Problems in Engineering

T : Conditions for simultaneous links over one channel.

ree neighbor pairs Link interference-free conditions

Nodes Nodes Necessary Sucient

in 𝐶in 𝐶condition condition

𝑖,𝑗∀ =,(𝑖,𝑗)≥2()

𝑖,𝑗∀ =,(𝑖,𝑗)≥1() () ∧() ∧()

𝑖𝑗∀ =,(𝑖,𝑗)>1()

a receiver candidate set, in which any two nodes 𝑖=(,,)

and 𝑗=(,,)satisfythefollowingcondition:

𝑖,𝑗=|−|+−+−

2≥1. ()

Note that both 𝐶and 𝐶are candidate sets, and the

actual sender and receiver sets are a subset of 𝐶and 𝐶,

respectively. e relations between nodes largely depend on

the previous selected links.

Property 4. If links (1,1)𝑐and (2,2)𝑐can be scheduled at

the same time, then (1,2)must satisfy condition () and

(1,2)must satisfy condition ().

Note that Property is only a necessary condition for

interference-free links.

3.3.3. Sender-to-Receiver. Given a certain channel, a node

cannot receive data if it is in the eective radio range of

the sender of any other active link; otherwise, interference

occurs. If two links 𝑖and 𝑗( =) coexist, the nodes involved

in these two links must satisfy the following condition:

𝑖,𝑗=|−|+−+−

2>1, =, ()

where 𝑖is the sender of link 𝑖and 𝑗is the receiver of link

𝑗.

For example, in Figure (d), since ’s neig hb o r does not

conict with link (,)𝑐1,can be a receiver of another link

(,)𝑐1.

We summarize three necessary conditions for link coex-

istence in the aforementioned three classes in Tab l e .

We have the following theorem.

eorem 2. Two lin k s 𝑖and 𝑗can be simultaneously sched-

uled (coexist) on the same channel, if and only if they satisfy all

of the three necessary conditions in Table 1.

eorem is based on the PCCS scheme and the set

theory. As long as the PCCS node coordinates are given,

we are able to determine the interference between links.

ese known conditions are helpful to nd as many links as

possible, while contributing to concurrent central scheduling.

3.4. Transformations of Link Groups. We at t e m p t t o n d

as many coexisting links as possible in a given local area.

Since a link can be established only between a valid sender

candidate and a valid receiver candidate, the actual (nal)

k

j

(i−2,j+3,k+1) (i,j +3,k+3)

(i − 1, j + 3, k + 2)

(i−1,j+2,k+1) (i, j + 2, k + 2)

(i−1,j+1,k) (i+1,j +1,k+2)

(i,j+1,k+1)

(i, j, k) (i + 1, j, k + 1)

i

F : Coexisting links in a local area.

scheduled link set is a subset of 𝐶×𝐶. Generally, coexisting

links are sparsely distributed in the network. Based on a

known group of coexisting links around one triangle, we wish

to obtain a new group of coexisting links through certain

transformations.

Starting from a link 1=(

1,1)𝑐1,where1= (,,)and

1= (, +1,+ 1), we want to set up a dense link group in

alocalarea.esetof1’s neighbor nodes is 𝑏(1)={V|

(V,(,,)) ≤ 1,V∈}.esetof1’s neighbor nodes is

𝑏(1)={V|(V,(, + 1, + 1)) ≤ 1,∀V∈}.Asany

neighbor node of the sender or the receiver of an active link

should remain silent if it does not receive data from the sender

1, to avoid interference with 1, nodes in 𝑏(,,)∪𝑏(1)

cannot send data on 1when 1is active.

To expand the active link group containing 1=(

1,1)𝑐1,

two nodes 2= (, + 3, + 3) and 2=(,+2,+2),

which are not in 𝑏(1)and 𝑏(1),maynegotiateforanew

link. If successful, (2,2)𝑐1is added to the active link group.

According to the conditions in eorem ,link2=(

2,2)𝑐1

can coexist with 1.

Any node in 𝑏(1)∪

𝑏(2)cannot be a new sender,

except 1and 2. However, it is completely dierent on the

receiverside,whereonenodemaybethereceiverofanew

link,evenifitisin𝑏(1)∩𝑏(2). For example, in Figure ,

3=(−2,+3,+1)and 3= (−1,+2,+1)form a new

link 3=(

3,3)𝑐1, which can coexist with 1and 2.Around

the central triangle in Figure ,wendthreeinterference-free

links coexisting over one channel.

Although the three nodes of the central triangle are

involved in three active links as shown in Figure , neither of

these three nodes can act as a sender while the other two are

receiving data from their respective senders. To obtain the

same number of new coexisting links, we need to keep the

receivers unchanged, while considering a certain switching

to the three senders of active links 1,2,and3.

Mathematical Problems in Engineering

By passing the sending token from the current sender

toitsneighbornodeclockwisewhilekeepingtheoriginal

receiver, we are able to establish three possible new links:

1= −1,+1,,,+1,+1𝑐1,

2= +1,+2,+3,,+ 2, +2𝑐1,

3=−1,+3,+2,−1,+2,+1𝑐1.

()

Since the sender (−1,+3,+2)of

3is one hop

away from the receiver (,+2,+2)of

2,whichviolates

the third condition in eorem ,

2and

3cannot coexist on

channel 1,whichmeansthat

1,

2,and

3cannot coexist. In

other words, switching the senders of interference-free links

may incur new interferences. erefore, we must perform

interference check aer switching the sender of an active link.

Sender switching may generate a new dense link group in

a local mesh. For example, we obtain a new link ∗

1= (( +

1,,+1),(,+1,+1))𝑐1, by switching the sender of link 1

to its neighbor node anticlockwise. e three links ∗

1,2,and

3are interference-free. Similarly, link ∗

3=((−2,+2,),(−

1,+2,+1))𝑐1is also interference-free with links 1and 2.

Since triangular meshes possess symmetric properties,

transformations such as rotation and translation can retain

the interference-free features, which may save computing

time in nding new link groups [,].

3.5. Node Identity Number Assignment. e nodes in tri-

angular meshes can be viewed not only as wireless router

nodes, but also as resources or data sets. Assigning a unique

identication (ID) number to each node brings several

benets. For example, such IDs can help to locate or identify

nodes quickly for various administration or maintenance

purposes.

Given the coordinates (,,)of a node in PCCS, there

exists a general function that maps (,,) to a unique

integer. For example, in Figure ,wecanmapBS = (0,0,0)to

0, while =(1,0,−1)to 1,and = (−1,1,0)to 2.

Denition 3. In PCCS, -circle in a triangular mesh is a set

of nodes that have exact distance of hopstothenodeBS.

Suppose that (,,) is in -circle, where =max{||,

||,||} according to (). To construct a mapping function

(,,), we classify nodes according to their coordinates. For

any node = (,,), we consider the following seven cases

in Figure , which is a logical route extracted from Figure .

e rst and special case =0,(0,0,0) = 0.enodeIDs

in -circle increase along an anticlockwise direction. ese

cases are applicable to any -circle. e parallel solid line

segments are corresponding to identity counting piecewise

functions; for example, the nd case includes three parallel

segments of dierent -circle in Figure , in which nodes on

these segments get their ID number following the nd one in

().edashedlinemeansthattheinnernodeiscounted

already, and the outer node is the new start of the next circle.

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

F : e cases for node identity number assignment.

To assign an identity to each node on -circle, we need

to know the total number of nodes inside this circle:

=1+6(1+2+⋅⋅⋅+−1)=1+3(−1).()

Let = 3(−1);wehave=+1. For example, there

is one node inside circle =1,thereare+1=7nodes

inside circle =2,andthereare+1=18+1=19nodes

inside circle =3.

e number of nodes on -circle depends on the value

of . According to (),forany-circle, with the six solid

line segments in the order as shown by arrows in Figure ,

we assign each node a unique integer on -circle with an

ID number in {+1,+2,...,++1,...,+6}.

Aer overlapping Figure on Figure ,anodeisassigned

with its identity number in one of the seven cases in ()

corresponding to the line segment case in Figure :

,, =

0=0st

++1 =0∧= 2nd

+−+1 =0∧= rd

+2−+1 =0∧=− 4th

+3−+1 =0∧=− 5th

+4++1 =0∧=− 6th

+5++1 =∧ =,0 7th.

()

For example, in the th case, node (1,−4,−3)is assigned

with an ID number of 3×4×(4−1)+4×4+1+1=54.

Some nodes may satisfy two cases in () while getting the

same ID. For example, node (0,3,3),whichisonthe3-circle

and the line segment in parallel with -axis, is assigned with

an ID number of 3××(−1)++1=3×3×2+3+1=

22 in the nd case in (). Meanwhile, node (0,3,3),asitis

on one line segment of Case in Figure ,cangetitsIDof

3××(−1)+−+1=3×3×2+3+1=22by

followingtherdcasein().

We shall provide more explanations for the nd and

th cases to facilitate a better understanding of ().Inthe

nd case where =0∧=,thecoordinateof

node (,,)must be positive. Node (,,) on this -circle

(=) segment should be assigned with an ID number in

Mathematical Problems in Engineering

{+1,+2,...,++1,...,+}.Inthiscase,node(,,)

is located above the lines =0and >0,asshowninCase

in Figure . Hence, with the increasing , each node (,,)is

assigned with an ID number (,, ) = + +1. Generally,

on each -circle (>0), from the nd case to the th case,

each with nodes, the total number of nodes is 6.

In the th case, we need to avoid repeatedly counting the

node on line =0,asshowninFigure .enumberofhops

to the BS is =.elastnodeonthiscircleisonaline

segment of the -axis parallel cluster with =0according to

Figure . e condition =∧ =excludes the nodes

already counted in the previous Case in this circle, while

=∧ =0prevents the st case from being reconsidered.

Meanwhile, the condition =ensures that we count nodes

on the next circle following the nd case again where =.

In the th case, a node can be assigned with an ID number in

aniteset{+5+1,...,+5++1,...,+6}.With

the increasing ,where=0,1,...,−1,node(,,)on

thissegmentisassignedwithanIDnumberof+5++1.

Note that the mapping function (,,) is a segmented

linear function, which is invertible for any nite set.

4. Path Counting

Routing is one fundamental problem in WMNs. To develop

a good routing scheme, one needs to know the number of

alternative paths and the number of channel assignments

foragivenpairofsourceanddestination.Wediscusspath

ndingandrealizationbasedontheproposedCPGmodel

and coordinate scheme. e total number of shortest paths in

grid meshes was discussed in []. In this section, we tackle

the path counting problem in triangular meshes.

We use (,)to denote the distance between source =

(1,1,1)and destination =(

2,2,2).Firstly,(,) =

(|1−

2|+|

1−

2|+|

1−2|)/2. In order to transmit data

from to , we need to select one path from (,),which

is the set of all shortest paths from to .

4.1. Path Alternatives. Every step along the shortest path from

to is one hop forward in one of the directions ,,and.

e two smaller numbers of {|1−2|,|1−2|,|1−2|}indicate

the lines of parallel clusters that form a grid mesh for path

selection, as illustrated in the grid of dashed lines in Figure .

We refer to the two corresponding directions from to in

the grid mesh as the correct directions.

e correct directions ensure that the data is transmitted

through one of the shortest path, where every hop selection

makes one hop closer to the destination. rough the use

of correct directions, we are able to reduce the problem of

counting all shortest paths from to in triangular meshes

to a problem in grid meshes.

We provide an example in Figure to count the number

of paths, where |1−2|=2,|1−2|=3,and|1−2|=5.e

grid with dashed lines contains all the shortest paths from

to .

Inasimplesituationwhereoneof{|1−2|,|1−2|,|1−

2|}is , the dashed grid degrades to a line. erefore, there

is only one shortest path available.

D

X

W

U

V

S

F : e shortest path alternatives in triangular meshes.

For convenience, let ℘ = {|1−

2|,|1−

2|,|1−

2|}. e correct directions are consistent with min(℘) (the

minimum of three elements in ℘)andmid(℘) (the middle

of three elements in ℘). To select a shortest path, it is

necessary to remove the direction corresponding to max(℘)

(the maximum of three elements in ℘); otherwise, it would

lead to a longer path.

If the smallest in {|1−

2|,|1−

2|,|1−

2|} is 1or 2,

in the corresponding direction, the sender and receiver must

be on two neighbor parallel lines or two parallel lines with

one line between them. We can use the corresponding grid to

calculate the number (𝑆,𝐷) of paths from to as follows:

(𝑆,𝐷) =

1if min ℘ = 0,

mid ℘+1 if min ℘ = 1,

mid ℘+1mid ℘+2

2if min ℘ = 2.

()

For example, in Figure ,thenumberofpathsfromto

is determined by two directions and ,as|1−2|=2and |1−

2|=3are smaller than |1−2|=5.etotalnumberofpath

alternatives from to is.However,onestepalongthe

direction of (it becomes either +1or −1) obviously leads

to a longer path. We present two more lemmas on direction-

related properties as follows.

Lemma 4. On a shortest path, the coordinate displacements

between and along the three directions satisfy

min ℘+mid ℘=max ℘.()

Proof. e number of hops on a path dened by the correct

direction of min(℘)and mid(℘)is min(℘) +mid(℘),andthe

path traverses exactly min(℘)+mid(℘)+1dierent points. As

each node is intersected by three lines, each from one cluster,

one of the three lines must belong to the max(℘) cluster. It

follows that the path traverses min(℘) + mid(℘) + 1 lines

in the max(℘) cluster. Hence, the displacement of the two

line numbers (i.e., max(℘)) traversing and is equal to

theheightofthetree(min(℘) + mid(℘) + 1) − 1.Wehave

max(℘) =(min(℘)+mid(℘)+1)−1 = min(℘)+mid(℘).

Lemma 5. e distance from source to destination is

(,)=min ℘+mid ℘. ()

Mathematical Problems in Engineering

e proof of Lemma simply follows the denition of

(,)and Lemma .

In triangular WMNs, given a source-destination pair

and , the number of shortest paths satises the following

theorem.

eorem 6. When min(℘) > 0,thenumber(𝑆,𝐷) of paths is

determined in a grid of the two correct directions corresponding

to min(℘)and mid(℘)as follows:

(𝑆,𝐷) =(,)

min ℘, ()

which is the number of min(℘) combinations chosen from

(,)objects.

Proof. Without loss of generality, let and be the correct

directions. Note that is determined by min(℘), while is

determined by mid(℘). e coordinates of the next receiver

node would lead to one hop closer to the destination along

the direction or .

e number (𝑆,𝐷) of paths is equal to the number of

strings of ’s a n d ’s : is repeated min(℘) times, and is

repeated mid(℘)times in a permutation of min(℘)+ mid(℘)

elements, that is, (,) in Lemma .etotalnumberof

permutations is (,)! = (min(℘) + mid(℘))!,butwith

repetitions. Note that the same permutated strings can only

be counted once. e number of duplicated permutations is

min(℘)!and mid(℘)! in the direction of and ,respectively.

en, the total number of dierent paths is obtained by

dividing the total number of permutations by the number of

duplications in both directions:

(,)!

min ℘!×mid ℘! =(,)

min ℘=(,)

mid ℘. ()

For example, in Figure ,℘={2,3,5}with and being

the correct directions, where appears min(℘) =2 times and

appears mid(℘) = 3 times in every permutation. A valid

routing path is determined by the number of ’s and ’s as

well as their relative positions in the string. For example, ,

,andare all valid paths with two ’s and t h re e ’s . e

path 1=→→→→→,whichisinthe

set (,)of all shortest paths from to , can be expressed

as a constrained permutation of the two correct directions,

that is, .

Given the node coordinates, data packets are delivered

hop by hop along the correct directions, taking the receiver of

the current hop as the sender of the next hop, until max(℘)=

0.iscanbedonerecursivelyandmayhelpavoidthe

overhead of maintaining a routing table.

4.2.PathCountingwithChannelAssignment. For a given

path from to withaconstantnumberofavailableorthogo-

nal channels, we need to decide the number of feasible

channel assignment schemes for implementing this path by

using three channels.

D

XW

U

V

S

c0

c1

c2

c3

c4

F : Channel assignment to realize a path from to .

To illustrate this problem, we show a path with a channel

assigned to each hop in Figure , which is derived from

Figure :→

𝑐2→

𝑐1→

𝑐3→

𝑐2→

𝑐1,wherethe

label 𝑖between two neighbor nodes is the channel assigned

to the corresponding link.

e directed edges distributed in dierent layers form a

feasible orthogonal channel assignment to a path from to .

A valid path realization allows simultaneous transmission of

all the component links on the path. For example, in Figure ,

links (,)𝑐2,(,)𝑐2,(,)𝑐1,(,)𝑐1,and(,)𝑐3can

coexist at the same time.

Let the correct directions be and .Givenorthogonal

channels, the rst hop has 𝜛

1channel choices, the second

hop has 𝜛−1

1channel choices, and the third hop has 𝜛−2

1

channel choices. en, the three selected channels can be

used repeatedly without interference; that is, the fourth hop

selects the same channel as the rst hop, and the h hop

selects the same channel as the second hop and so on.

Channel 𝑖canbeselectedatmost(,)/3times.

In realizing a path using three channels at time ,therst

three hops determine the channel assignment order in every

three downstream hops. For example, if the rst three hops

are arranged in a channel order of 2,1,and3, then the

second three hops should be assigned channels in the same

order. Otherwise, a realization of the path from to would

requiremorechannels.Forexample,with(,) = 4,ifthe

rst three hops use channels in the order of 2,1,and3,the

fourth hop cannot use channels 1and 3,butonly2or a new

channel.

Note that using channels, the channel assignment

permutations of the rst three hops (i.e., 3!)areallpossible

schemes for downstream three-hop groups.

Lemma 7. When min(℘) =0,withorthogonal channels, the

number of valid channel assignment schemes for a given path

from source to destination using three channels at the same

time is

=

3×3!. ()

Mathematical Problems in Engineering

Based on eorem and Lemma ,wehavethefollowing

theorem.

eorem 8. When using three channels to realize a path, the

number of channel assignments for all shortest paths from to

is

(,)

min ℘⋅. ()

Proof. When min(℘) = 0, there is only one shortest path, and

the number of channel assignments for this path is obtained

by Lemma .

We focus on a gen e r a l c a s e wh er e m i n (℘) > 0.e

number of channel assignments can be counted in two

independent steps.

e rst step is to count all shortest paths from to in a

plane mesh. is step does not assign channels. For example,

Figure shows one such path: →→→→

→.

e second step assigns channels to the selected path

using three channels without interference. For example,

Figure shows the channel assignment for 1from to

in Figure :→

𝑐2→

𝑐1→

𝑐3→

𝑐2→

𝑐1.

Since the above two steps are independent, by the mul-

tiplication principle, the total number of path realization

schemes satises (). is result could be extended to

dierent numbers of channels assigned to one path.

5. Conclusion

We conducted a theoretical exploration on mathematical

models and combinatorial characteristics of MIMO WMNs.

For a single-channeled mesh, we designed a coordinate

scheme and a node identity assignment scheme and derived

the interference-free conditions. For multiradio multichannel

WMNs, we derived rigorous formulas to count the number of

shortest paths from source to destination.

It is of our future interest to nd some transformations

to generate new link groups from the known ones. Along

this direction, we plan to investigate the CPG vertex/edge

transitive properties for performance improvement.

Conflict of Interests

e authors declare that there is no conict of interests

regarding the publication of this paper.

Acknowledgments

e research is funded in part by Nature Science project of

Guangdong Province under Grant no. S and

China Scholarship Council no. . is research is

also partly sponsored by U.S. Department of Energy’s Oce

of Science under Grant no. DE-SC with University of

Memphis.

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