Multi-Channel Assignment in Wireless Sensor Networks: A ...

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Multi-Channel Assignment in WirelessSensor Networks: A Game TheoreticApproach

Qing Yu, Jiming Chen, Yanfei Fan, Xuemin Sherman Shen,Youxian SunDate Submitted: 23 November 2009Date Published: 24 November 2009

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Subject Classification Vehicular Technology

Keywords Channel assignment and reservation;;

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this paper has been accepted by IEEE INFOCOM 2010

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Copyright Copyright © Date Submitted: 23 November 2009 Qing Yu et al.This is an open access article distributed under the CreativeCommons Attribution 2.5 Canada License, which permitsunrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

1

Multi-Channel Assignment in Wireless Sensor

Networks: A Game Theoretic Approach

Qing Yu†, Jiming Chen†‡, Yanfei Fan‡, Xuemin(Sherman) Shen‡

and Youxian Sun†

† State Key Lab of Industrial Control Technology, Zhejiang University,

Hangzhou, 310027, P.R.China

Email: [email protected]; jmchen,[email protected]

‡ Department of Electrical and Computer Engineering, University of Waterloo,

Waterloo, ON, N2L 3G1, Canada

Email:yfan, [email protected]

Abstract

In this paper, we formulate multi-channel assignment in Wireless Sensor Networks (WSNs) as an

optimization problem and show it is NP-hard. We then propose a distributed Game Based Channel

Assignment algorithm (GBCA) to solve the problem. The GBCA takes into account both the network

topology information and the transmission routing information. We prove there exists at least one

Nash Equilibrium in the channel assignment game. Furthermore, we analyze the suboptimality of Nash

Equilibrium and the convergence of the Best Response in the game. Simulation results show that GBCA

can reduce interference significantly and achieve satisfactory network performance in terms of delivery

ratio, throughput, packet transfer delay and energy consumption.

I. INTRODUCTION

Wireless Sensor Networks (WSNs) with low-cost, low-power and multi-functionality, make

their applications with large scale and dense deployments possible. However, traditional ways

to use a single channel among the whole WSNs in the same geographical region, can not work

Downloaded from engine.lib.uwaterloo.ca on 19 February 2022 Page 1 of 21

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properly when the WSNs become larger and denser since frequent interference among the WSNs

will decrease the network performance dramatically. The half-duplex radio transceiver in current

available sensor node is already able to operate on multi-channel. Thus, using the available

multiple channels efficiently to exploit parallel transmission in WSNs becomes very attractive.

There already exist a considerable number of multi-channel protocols for wireless networks

[1], [2], [3], [4], [5]. However, most of them have strong assumptions that the transceivers

either use the frequency hopping spread spectrum wireless cards or can operate on multiple

channels simultaneously. Unfortunately, these protocols are not applicable to WSNs, because

current available sensor node with only one simple half-duplex radio transceiver can not satisfy

these strong assumptions. Besides, the extra overhead of some protocols caused by dynamic

channel negotiations or RTS/CTS packets, poses major challenge on both constrained energy

and limited bandwidth in WSNs.

Recently, some multi-channel protocols have been proposed specially for WSNs. Most of

them dynamically assign channels to links according to the transmitting flows in the network

[6], [7], [8], [9], [10]. Although these protocols can reduce interference to some degrees, the

requirements of frequent negotiations and highly accurate time synchronization cause extra large

overhead. Alternatively, static channel assignment protocols are based on network topology [11],

[12]. They work with less overhead than the dynamical protocols, but can not reduce interference

efficiently because they neither take full advantage of the routing information nor dynamically

track the transmitting flows. For example, MMSN [11] uses the network topology information

to balance the channel usage among two-hop neighbors. Without routing information, it may

make the channel usage in the links which are not involved in the routing excess but the channel

usage in the links which are involved in the routing tight. In this paper, observing that the

interference in the network depends to a large extent on the routing, we take into account both

the Topology Information and the Routing Information (TIRI) to reduce the interference more

efficiently. However, in general, it is difficult to fully exploit the TIRI in WSNs. For example,

PMIT [12] statically divides the whole network into mutually exclusive single-channel subtrees,

which takes advantage of the inter-tree routing information but does not exploit the intra-tree

routing information. In order to fully exploit TIRI, all the autonomous sensor nodes must be

involved in the channel assignment and share information dynamically.

Game theory is the study of the interaction of autonomous agents. The features of WSNs—


3

decentralized operation, self configuration, and energy awareness—result in that each sensor

node in WSNs behaves as an autonomous agent, and thus make game theory as a promising

way to solve the competitive problems in WSNs, such as coverage [13], routing [14], sensor

activation [15], etc. To the best of our knowledge, this paper is the first attempt to use game

theory to cope with the channel assignment problem in WSNs. Based on the observations that

the interference suffered by the sender is positively related to the number of links heard by the

receiver, and the routing in applications of WSNs is generally static and of tree/forest structure,

the whole network can be considered to be composed of lots of parent-children sets which are

constructed from TIRI. In order to fully exploit TIRI, we model the channel assignment among

these sets as a game with the total interference of the whole network as the social objective.

Specifically, parent in each set is modelled as player, and its payoff is the sum of the interference

suffered by the set and the interference caused by the set. The channel assignment game is evolved

according to the Best Response (BR) dynamic, i.e., each player chooses the channel to minimize

its payoff according to the channels chosen by the other players. Furthermore, we prove that

BR converges into a Nash Equilibrium (NE), and analyze the social efficiency of NE. Based on

BR, we propose a Game Based Channel Assignment algorithm (GBCA) to handle the NP-hard

channel assignment problem in WSNs in a distributed way. The main contributions of this paper

are summarized as follows:

• We analyze the relationship between the number of links heard by the receiver and the

interference suffered by the sender, and prove that this relationship is linear. We formulate

the channel assignment in WSNs as an optimization problem and prove that it is NP-hard.

• The channel assignment in WSNs is modelled as a channel assignment game to fully exploit

TIRI. We prove that the game is an exact potential game and there exists at least one NE

in the game. We analyze the price of anarchy of the game and bound it by c−1c

where c is

the number of available channels, which means any NE is suboptimal. We adopt BR as the

evolution rule of the channel assignment game and prove that BR converges into a NE in

finite iterations.

• We propose the GBCA, which is a distributed algorithm, based on BR. GBCA can provide

a suboptimal solution to the channel assignment problem in polynomial time, and it is com-

patible with both scheduling based and contention based medium access control schemes.


4

Simulation results show that GBCA reduces interference significantly and achieves better

network performance in terms of delivery ratio, throughput, packet transfer delay and energy

consumption than MMSN [11] does.

The remainder of the paper is organized as follows. Section II analyzes the interference

and formulates the optimization problem for channel assignment. We propose the game based

algorithm to solve this problem and analyzes the convergence and suboptimality of the algorithm

in Section III. The performance of the algorithm is evaluated by simulations in Section IV.

Finally, we conclude our paper in Section V.

II. PRELIMINARIES AND PROBLEM FORMULATION

A. Interference Analysis

In wireless communication, whether a message gets corrupted or not is closely related to

whether other links, which are heard by the receiver, are transmitting message. In general, as

long as there is another message being transmitted simultaneously through some of these links,

there is a collision at the receiver and the message may be corrupted. We analyze the relationship

between the number of links heard by the receiver and the interference suffered by the message

transferred to the receiver in a similar way as in [16] in the following.

Assume that there are I links heard by the receiver and the probability that a message

is transmitted through a link at a given time is q. Therefore, the probability of a successful

transmission with one try is

p = (1− q)I−1. (1)

If there is no maximum number of retries constraint, the average number of retries before a

successful transmission is

R =+∞∑k=1

kp(1− p)k−1 − 1 =1

p− 1. (2)

Apparently, the equation above usually holds because the MAC layer allows relatively large

number of retries. Substituting Equation (1) into Equation (2), we have

R = (1 +q

1− q)I−1 − 1. (3)


5

Since q is very small for the light load in WSNs, an approximate R can be obtained as follows.

R.=

q

1− q(I − 1). (4)

Equation (4) reveals that the more links the receiver hears, the more retries a message needs

to be successfully received. Furthermore, the more times the sender retries, the more energy it

consumes and the longer delay the transmission suffers, which in turn results in both throughout

and delivery ratio decrease. Thus, the average number of retries, i.e., R, is a key metric to

characterize the interference suffered by the sender. Furthermore, according to Equation (4), I is

approximately linear with R, so we adopt I as the interference metric and call it the interference.

B. Problem Formulation

The links towards a receiver can be divided into two categories: intersecting links and interfer-

ing links [17]. The transmissions in intersecting links aim to the receiver, while the transmissions

in interfering links do not aim to the receiver but can be overheard by the receiver. If only

single channel is used, then both intersecting links and interfering links will interfere with

the transmission aiming to the receiver. Though we can not reduce the interference caused by

intersecting links because there is only one half-duplex radio transceiver in sensor node and thus

the receiver can not receive multiple messages simultaneously, we can reduce the interfering

links by using multiple channels. In this paper, we assign channel in a receiver-centric way,

i.e., each sensor node is assigned one channel to receive message. If providing enough non-

overlapping channels, we can make all interfering links use different channels from the ones

their receivers use. In this case, the receiver can not overhear the transmissions in interfering

links, and the interference in the network can be remarkably reduced. However, the number

of non-overlapping channels is usually fixed and limited in practice [12]. Hence, the problem

becomes to optimally assign the limited channels to minimize interference. We first define some

notations in Table I and make some useful assumptions as follows: 1) The network routing is

of tree/forest structure, i.e., the children deliver their messages to their parents in the network;

2) Channels are all non-overlapping and do not interfere with each other; 3) The links between

two sensor nodes are symmetric, i.e., given that sensor nodes i and j are connected, if i uses

j’s channel to transmit then i can be heard by j, and vice versa.


6

TABLE I

NOTATION DEFINITIONS

Symbol Definition

| · | | · | represents the cardinality of a set.

C the set of non-overlapping channels, i.e.,

C=1,2,...,c.

G(V,E) the topology graph of sensor network. It is a

directed graph and includes routing information.

V the set of sensor nodes including both normal

sensor nodes which know their parents and sink

sensor nodes which only collect messages.

parent(i) the parent of sensor node i in the tree/forest

structure.

Child(i) the set of children of sensor node i in the

tree/forest structure.

P the set of all non-leaf sensor nodes which

have children in the tree/forest structure, i.e.,

P=1,2,...,m.

f the channel assignment that assigns each non-

leaf sensor node one channel to receive mes-

sage, i.e., f : P → C.

E the set of links including both intersecting links

and interfering links. The links among sensor

nodes always appear in pairs and with different

directions.

s(e),r(e) the sender and receiver of link e.

Es the set of intersecting links, i.e., Es=e : e ∈ E

and s(e)∈ Child(r(e)).

Ef the set of interfering links, i.e., Ef=e : e ∈ E

and s(e) /∈ Child(r(e)).

ch(e) the channel of link e, i.e.,

ch(e)=f(parent(s(e))).

J(e) the potential interference that link e may yield

to the network, i.e., J(e)=|Child(r(e))|.

I(i, f) the interference suffered by sensor node i in

assignment f , i.e., I(i, f)=|e : e ∈ E, r(e) =

parent(i) and ch(e) = f(parent(i))|. Since

the sink sensor nodes just collect messages in

the network, they suffer no interference.

Lr(f), Lu(f) for a given f , Lr is the set of all the interfering

links that can not be heard by their receivers,

and Lu is the set of all the interfering links

that still can be heard by their receivers, i.e.,

Lr(f)=e : ch(e) = f(r(e)), e ∈ Ef and

Lu(f)=e : ch(e) = f(r(e)), e ∈ Ef.


7

We use the total interference suffered by all sensor nodes as the optimization objective, since

it is corresponding to the average number of retries over the network and in turn reflects the

network performance.

The primal optimization problem (PP): Given G(V,E) and C, the primal optimization problem

is to find a channel assignment f to minimize the total interference∑i∈V

I(i, f).

Furthermore, for a given G(V,E) and an assignment f , we have∑i∈V

I(i, f) =∑e∈Es

J(e) +∑

e∈Lu(f)

J(e). (5)

Since both the total potential interference that all the intersecting links may yield and the total

potential interference that all the interfering links may yield are constants in a given G(V,E),

we have ∑e∈Es

J(e) = A,∑

e∈Lu(f)

J(e) +∑

e∈Lr(f)

J(e) = B, (6)

where A and B are constants regardless of f . Substituting Equation (6) into Equation (5), we

have ∑i∈V

I(i, f) = A+B −∑

e∈Lr(f)

J(e). (7)

Hence PP can be equivalently transformed to a dual problem.

The dual optimization problem (DP): Given G(V,E) and C, the interference reduction problem

is to find a channel assignment f to maximize the total removed interference U(f) =∑

e∈Lr(f)

J(e).

III. GAME BASED CHANNEL ASSIGNMENT

In this section, we first analyze the hardness of the channel assignment problem in WSNs, then

model the problem as a channel assignment game, and finally provide a distributed algorithm

based on the game to handle the problem.

A. Problem Assessment

The minimum same-color edges coloring problem (MSCP): Given undirected graph G(V,E)

and integers 0< K ≤ |V|, 0≤ M < |E|, find a coloring (i.e., assign each vertex one of the K

colors) such that the number of same-color edges (i.e., the colors of its two vertexes are the

same) is not more than M .


8

Lemma 1: MSCP is NP-complete.

Proof: we set M=0 in MSCP, thus MSCP is restricted to GRAPH K-COLORABILITY

problem [18] which is a typical NP-complete problem. Hence, we prove that MSCP is NP-

complete.

The relative decision problem (RP): Given G(V,E), C and a non-negative integer W , deter-

mine whether there exists a channel assignment f such that∑i∈V

I(i, f) ≤ W .

Theorem 1: PP is NP-hard.

Proof: To prove this theorem, it is sufficient to prove RP is NP-complete. It is easy to see

that RP is in NP. Then, we transform MSCP into RP. For an arbitrary G, we create a G in the

following way. For each node i in G, we create two normal sensor nodes in G, i.e., pi and si, and

pi is the parent of si. Two directed edges are added: one is from si to pi and the other is from

pi to si. For each edge eij in G, we create two directed edges in G: one is from si to pj and the

other is from pj to si. Finally, we create a sink sensor node r and |V| edges each of which is from

pi to r. Thus, we get a new graph G(V,E) where |V |=2|V|+1 and |E|=3|V|+2|E|. Apparently,

this transformation is polynomial time. Let the channel assigned to pi in G correspond to the

color assigned to i in G and the interfering links heard by the non-leaf sensor nodes to the

same-color edges. Thus, MSCP with K, M and G is transformed, in polynomial time, into RP

with |C|=K, W=|V|2+|V|+M and G. According to Lemma 1, MSCP is NP-complete, hence RP

is also NP-complete. Therefore, PP is NP-hard.

Obviously, DP is also NP-hard as DP is completely equivalent to PP.

B. Channel Assignment Game

Since DP is NP-hard, it is hard to find a solution with both polynomial execution time and

optimal result. Instead, we model the channel assignment game to cope with DP with polynomial

execution time and suboptimal result in this subsection.

Recalling the assumption of tree/forest structure, we construct parent-children sets from the

network to exploit its TIRI, i.e., each non-leaf sensor node in the network and its children

constitute a Parent-Children Set (PCS). We define interfering PCSs as a pair of PCSs such that

the message sent by some child in one PCS may be heard by the parent in the other, e.g., PCS3

and PCS4 in Fig.1, and PCS1 and PCS4. Correspondingly, the parents in the pair of PCSs are

called interfering parents, e.g., player3 and player4 in Fig.1. All the interfering parents of parent


9

i is denoted by a set Θ(i). In fact, the interference suffered by any child in a PCS is determined

by the channel usage among the PCS and its interfering PCSs, i.e., the more parents in Θ(i) use

the same channel as parent i uses, the more interference the children of parent i suffer. Thus,

in order to minimize the interference suffered by their children, the PCSs should autonomously

compete and interact with each other for the channel usage. In this case, the channel assignment

among the PCSs can be naturally modelled as a channel assignment game.

In the channel assignment game, the players are all parents of the PCSs, i.e., P . Each player i

chooses a channel si ∈ C as its strategy. The strategies of all players make a channel assignment

s=(s1, s2, ..., sm), and we denote the strategies of all players except for player i by s−i. The

payoff of player i is a function of s and denoted by ui(s). To minimize the total interference

of the network, we consider both the interference suffered by the children of player i and the

interference caused by its children when constructing ui(s). When choosing a channel, player

i must try its best to choose a channel different from its interfering players to minimize the

interference suffered by its children; On the other hand, player i should bring as less interference

as possible to the children of its interfering players because excessive selfishness may result in

extra interference to each other’s children. Therefore, we define ui(s) in two parts as follows:

ui(s) =∑

e∈X(i,s)

J(e) +∑

e∈Y (i,s)

J(e), (8)

X(i, s) = e : e ∈ Lu(s), s(e) ∈ Child(i),

Y (i, s) = e : e ∈ Lu(s), r(e) = i.

For example, in Fig.1, the payoffs of player1, player2, player3 and player4 are 3, 4, 4 and 3,

respectively.

The channel assignment game is designed to be a repeated game, and players negotiate the

channel usage according to the Best Response (BR) dynamic. In each iteration of the game,

each player chooses the channel to minimize its payoff based on the channel assignment in

last iteration, but the interfering players can not change their channels simultaneously. However,

BR does not guarantee convergence in all kinds of games and the stable state is not always

with optimal social utility. Hence, we study the characteristics of our channel assignment game

in terms of convergence and suboptimality in the remaining subsection. We begin with the

definition of the famous stable state in game theory—Nash Equilibrium (NE), and then discuss


10

c2 c1 c2

PCS1 PCS2 PCS3

player1 player2 player3

c1

player4

PCS4

Fig. 1. An example of channel assignment among interfering PCSs, where player2 and player3 use channel c1 to receive

message, and player1 and player4 use channel c2 to receive message. The solid arrows represent intersecting links, while the

dashed arrows represent interfering links.

200 300 400 5000

0.2

0.4

0.6

0.8

1

number of nodes

(a)

2 4 6 80

0.2

0.4

0.6

0.8

1

number of channels

resi

dual

inte

rfer

ence

rat

io

(b)

200 300 400 50010

20

30

40

50

60

number of nodes

itera

tions

to c

onve

rge

(c)

2 4 6 8

20

30

40

50

number of channels

itera

tions

to c

onve

rge

(d)

GBCAMMSNbound

GBCAMMSNbound

GBCA GBCA

Fig. 2. Numerical results of GBCA in terms of convergence and suboptimality

the convergence of BR to NE and the suboptimality of NE.

Nash Equilibrium: In a game, a strategies profile s∗ is called NE iff. for each player i and an

arbitrary strategy si in its strategy space, the following inequality is always satisfied,

ui(s∗) ≤ ui(si, s

∗−i). (9)

To exploit the convergence of BR, we have the following theorem.

Theorem 2: The channel assignment game is an exact potential game [19].

Proof: We construct the potential function as follows:

φ(s) =1

2

∑i∈P

ui(s). (10)


11

20 40 600.8

0.85

0.9

0.95

1

number of flows

(a)

20 40 60200

250

300

350

400

450

500

number of flowsth

roug

hput

(kbp

s)

(b)

20 40 600.008

0.01

0.012

0.014

0.016

0.018

0.02

number of flows

pack

et tr

ansf

er d

elay

(s)

(c)

20 40 602

2.2

2.4

2.6

2.8x 10

−6

number of flows

ener

gy c

onsu

mpt

ion

per

byte

(mW

hr) (d)

MMSNGBCA

MMSNGBCA

MMSNGBCA

MMSNGBCA

Fig. 3. Performance comparison between GBCA and MMSN with number of flows increasing

When player i unilaterally changes its strategy from s1i to s2i , we have

φ(s2i , s−i)− φ(s1i , s−i)

=1

2

∑j∈P

[uj(s2i , s−i)− uj(s

1i , s−i)]

=1

2

∑j∈sj=s2i

∩Θ(i)


1i , s−i)]

−∑

j∈sj=s1i ∩

Θ(i)


2i , s−i)]

+ui(s2i , s−i)− ui(s

1i , s−i)

. (11)

Let Z(a, b)=e : s(e) ∈ Child(b), r(e) = a, we have∑j∈sj=s2i

∩Θ(i)


1i , s−i)]

=∑

j∈sj=s2i ∩

Θ(i)

∑e∈Z(j,i)

J(e) +∑

e∈Z(i,j)

J(e)

=∑

e∈X(i,(s2i ,s−i))

J(e) +∑

e∈Y (i,(s2i ,s−i))

J(e)

= ui(s2i , s−i). (12)

Similarly, the second sum in Equation (11) is equal to ui(s1i , s−i). Hence, Equation (11) becomes

φ(s2i , s−i)− φ(s1i , s−i) = ui(s2i , s−i)− ui(s

1i , s−i). (13)


12

Thus, we prove that the channel assignment game is an exact potential game and one of its

potential functions is φ(s).

Since φ(s) is bounded and only gets integral values, Equation (13) indicates that φ(s) will

continue decreasing in BR until it reaches a local minimum point. The number of iterations to

converge is finite. From this finite improvement property [19], we have the following conclusions.

Corollary 1: The channel assignment game always has NE, and the assignment minimizing

φ(s) must be a NE.

Theorem 3: In the channel assignment game, BR always converges to a NE, the number of

iterations to converge is less than (|V | − 1)2.

Proof: The potential function, as defined in Equation (10), can be calculated as follows.

φ(s) =1

2

∑i∈P

∑e∈X(i,s)

J(e) +∑

e∈Y (i,s)

J(e)

=∑

e∈Lu(s)

J(e) =∑i∈P

∑e∈Y (i,s)

J(e) (14)

For each i ∈ P , |Y (i, s)| ≤ |V | − 1− |Child(i)| and∑i∈P

|Child(i)| ≤ |V | − 1. Thus,

φ(s) ≤∑i∈P

|Child(i)|(|V | − 1− |Child(i)|) (15)

= (|V | − 1)∑i∈P

|Child(i)| −∑i∈P

|Child(i)|2

< (|V | − 1)2 (16)

Hence, we have 0 ≤ φ(s) < (|V | − 1)2.

According to BR, each player tries to choose the channel to minimize its own payoff in each

iteration. Thus, in iteration k, the situation is always one of the following two cases:

Case 1: none of the players changes channel, i.e., for each player i and an arbitrary si,

ui(sk) ≤ ui(si, s

k−i). According to the definition of NE, sk is a NE.

Case 2: at least one player changes channel. Suppose that there are m ≥ 1 such players and

denote them by a set Ω(k). According to BR, these players must not be interfering players,

hence,

φ(sk)− φ(sk−1) =∑

i∈Ω(k)

[ui(sk)− ui(s

k−1)]. (17)

Moreover, each term in the sum of Equation (17) is no more than -1. Thus,

φ(sk)− φ(sk−1) ≤ −m ≤ −1. (18)


13

Therefore, before satisfying Case 1, each iteration will make φ decrease at least one. Since

0 ≤ φ < (|V | − 1)2, BR will take at most (|V | − 1)2 iterations to converge to a NE.

Remark 1: According to Theorem 3, the number of iterations to converge to a NE is O(|V |2).

In Section III-C, it will be shown that each iteration can be executed in polynomial time. Thus,

the channel assignment can be handled in polynomial time.

Remark 2: In Inequalities (15) and (16), the upper bound of φ(s) is very relaxed. The number

of links towards a sensor node is usually bounded by a constant K, which is related to the node

density. The children of a non-leaf sensor node are usually comparable in number to its interfering

links. Let 0 < δ < 1 be the upper bound of the ratio of interfering links to all links towards a

non-leaf sensor node. Then φ(s) ≤ Kδ(|V | − 1). Moreover, φ(s) usually decreases more than

one in one iteration because many players could change their channels simultaneously if they

are not interfering players. Hence, the number of iterations to converge is usually much less

than (|V | − 1)2.

From Equation (14), φ(s) is equal to the total interference caused by the unremoved interfering

links. Thus, from Equation (6), φ(s) also reflects the social objective in DP. According to

Corollary 1, the optimal assignment must be a NE. Based on these observations, we study the

social efficiency of NE as follows.

Corollary 2: Any NE of the channel assignment game is locally optimal, i.e., in a NE, all

non-leaf sensor nodes can not reduce the interference unilaterally and thus DP reaches a local

maximum point.

Theorem 4: The Price of Anarchy [20]: let so be the optimal solution of DP and s∗ be an

arbitrary NE, then we have c−1c

≤ U(s∗)U(so)

≤ 1.

Proof: According to the definition of NE, each player chooses a channel from C to make

its current payoff as less as possible in the Nash Equilibrium s∗. Thus, we have the following

inequality.

ui(s∗) =

∑e∈X(i,s∗)

J(e) +∑

e∈Y (i,s∗)

J(e)

≤ 1

c

∑e∈s(e)∈Child(i)

∩Ef

J(e)

+∑

e∈r(e)=i∩

Ef

J(e). (19)


14

Therefore, the potential function can be bounded as:

φ(s∗) ≤ 1

2c

∑i∈P

∑e∈s(e)∈Child(i)

∩Ef

J(e)

+∑i∈P

∑e∈r(e)=i

∩Ef

J(e)

(20)

=1

2c× 2

∑e∈Ef

J(e) =B

c(21)

Thus, according to Equations (6) and (14), we have

U(s∗) = B −∑

e∈Lu(s∗)

J(e) = B − φ(s∗) (22)

≥ B − B

c≥ c− 1

cU(so) (23)

Apparently, U(s∗) ≤ U(so), thus,

c− 1

c≤ U(s∗)

U(so)≤ 1. (24)

Remark 3: According to Theorem 4, any NE is a suboptimal solution of DP. Taking Micaz

for example, there are 8 non-overlapping channels available [12]. According to Inequality (23),

in a NE, at least 87.5% of the interference caused by interfering links is reduced. According to

Inequality (24), the NE is at most 12.5% worse than the optimal assignment.

Remark 4: Since φ(s) can only get integral values, and it is not always true that all the

interfering players of a player use different channels and have the same potential interference to

it, then the relaxations in Inequalities (19) and (20) are usually a little excessive. Thus, NE is

usually much closer to the optimal assignment than the given bound.

Since the suboptimality of NE is closely related to c, we have an encouraging corollary as

follows when c is large enough.

Corollary 3: When c > maxi∈V

|Θ(i)|, any NE is optimal and removes all the interference caused

by interfering links.

Proof: Since c > maxi∈V

|Θ(i)|, in an arbitrary NE, any player i can and have to choose a

channel s∗i which is different from its interfering players’s. Hence, ui(s∗i , s

∗−i) = 0. Furthermore,

φ(s∗i , s∗−i) = 0. According to Equation (22), we have U(s∗) = B. Moreover, U(s∗) ≤ U(so) ≤ B.

Therefore, U(s∗) = U(so) = B.


15

Algorithm 1 Game Based Channel Assignment Algorithm1: Input: the initial channel assignment s0

2: Output: the stable channel assignment NE s∗

3: for each iteration k=1,2,... do

4: for any non-leaf sensor node i ∈ P do

5: //window RTC:

6: compute the channel to minimize the non-leaf sensor node i’s payoff: ch = arg minsi∈C

ui(si, sk−1−i );

7: if ch is not equal to sk−1i then

8: broadcast a REQ message with i’s id;

9: else

10: ski = ch;

11: end if

12: collect messages in window RTC;

13: //window PTC:

14: if node i has received REQs in window RTC then

15: find the max id in REQs and reply a PER message to it;

16: end if

17: collect messages in window PTC;

18: //window STC:

19: if node i has broadcasted REQ in window RTC then

20: if node i has received PERs from all its neighbors in window PTC then

21: set both its rcv ch and ski to ch;

22: broadcast a CHA message with ch and its id;

23: else

24: set both its rcv ch and ski to sk−1i ;

25: end if

26: end if

27: collect messages in window STC;

28: //window RCC:

29: if node i has received CHAs in window STC then

30: if the id of one CHA is its parent then

31: set its snd ch to the ch in this CHA;

32: end if

33: rebroadcast the CHAs with i’s id added;

34: end if

35: collect messages in window RCC;

36: if node i receives CHAs in window RCC and the CHAs are not from itself then

37: update channel information in its RNNT according to the ch and id of the CHAs;

38: end if

39: end for

40: if sk is equal to sk−1 then

41: The NE s∗ = sk is reached;

42: break;

43: end if

44: end for


16

In summary, in the channel assignment game, BR will converge to a NE after at most (|V |−1)2

iterations, and the suboptimality of NE is guaranteed by c−1c

≤ U(s∗)U(so)

≤ 1.

C. Game Based Channel Assignment Algorithm

The payoff of a player depends only upon the channels chosen by its interfering players,

so players only need to exchange information with their interfering players to implement BR.

Hence, based on BR, we propose a distributed Game Based Channel Assignment algorithm

(GBCA) to cope with DP. For the channel assignment game, the most important elements are

the payoff functions of players, which reflect the benefit of players and further determine the

NE of the game and its performance, and the BR dynamic of players, which determines the

dynamic evolution of the game. The payoff functions and BR also constitute GBCA. Thus,

according to Section III-B, the existence of stable state, the convergence of BR to stable state

and the suboptimality of stable state are guaranteed in GBCA. We state the detail of GBCA as

follows:

Assume that each sensor node knows its parent, its children and the sensor nodes which have

interfering links towards it. Each sensor node has a structure (rcv ch, snd ch, id) (i.e., the

channel it uses to receive, the channel it uses to send and its ID number) and a Related Non-leaf

Nodes Table (RNNT) which is used to record information about its interfering parents: their

channels, their number of children and the corresponding intermediate sensor nodes. GBCA is

divided into two phases: the interfering parents discovery phase and the channel negotiation

phase.

In the first phase, each sensor node broadcasts twice. In the first broadcast, each sensor node

broadcasts a message with its id and its number of children; In the second broadcast, each sensor

node rebroadcasts the messages, which it received in the first broadcast, with its id added. After

these broadcasts, each non-leaf sensor node knows its interfering parents, their number of children

and the intermediate sensor nodes. All these information can be used to calculate payoff in the

next phase.

In the second phase, BR is implemented and the channel negotiation is conducted in iterations.

Each iteration occupies a time slot and is divided into four time windows: the RTC window,

the PTC window, the STC window and the RCC window. This phase is the core of GBCA, we

present its pseudo-code in Algorithm 1.


17

2 4 6 80.8

0.82

0.84

0.86

0.88

0.9

0.92

number of channels

(a)

2 4 6 8320

330

340

350

360

370

number of channelsth

roug

hput

(kbp

s)

(b)

2 4 6 80.01

0.012

0.014

0.016

0.018

0.02

number of channels

pack

et tr

ansf

er d

elay

(s)

(c)

2 4 6 82

2.5

3

3.5

4

4.5x 10

−6

number of channels

ener

gy c

onsu

mpt

ion

per

byte

(mW

hr) (d)

MMSNGBCA

MMSNGBCA

MMSNGBCA

MMSNGBCA

Fig. 4. Performance comparison between GBCA and MMSN with number of channels increasing

Apparently, each iteration in the second phase could be completed in polynomial time. Addi-

tionally, (|V | − 1)2 iterations are sufficient to converge to a NE. Therefore, this algorithm will

converge in polynomial time.

In addition, the communications in these two phases use a designated channel. After the two

phases, the channel of each non-leaf node is assigned, then the network starts to use this channel

assignment to communicate. Time synchronization is only needed in the second phase but not

required when transmitting data. Conducting data transmission task, GBCA is compatible with

lots of medium access control schemes, including both scheduling based ones and contention

based ones. So we do not intend to design a medium access control scheme specially for GBCA.

Currently, IEEE 802.15.4 is very popular in WSNs, so we use its medium access control scheme

in the simulations of this study with just one modification: when performing CCA, the sensor

node should check both its channel to send and its parent’s channel to send to avoid colliding

with its siblings and parent.

IV. PERFORMANCE EVALUATION

A. Convergence and Suboptimality Evaluation

In Section III-B, we analyze the convergence and suboptimality of GBCA theoretically. In

this subsection, we evaluate these characteristics of GBCA numerically. We set the field to

200m×200m, and communication radius to 30m. Furthermore, the number of non-overlapping


18

20 30 40 500.86

0.87

0.88

0.89

0.9

0.91

0.92

communication radius(m)

(a)

20 30 40 50345

350

355

360

365

370

communication radius(m)th

roug

hput

(kbp

s)

(b)

20 30 40 500.01

0.012

0.014

0.016

0.018

0.02


pack

et tr

ansf

er d

elay

(s)

(c)

20 30 40 502

2.5

3

3.5

4x 10

−6


ener

gy c

onsu

mpt

ion

per

byte

(mW

hr) (d)

MMSNGBCA

MMSNGBCA

MMSNGBCA

MMSNGBCA

Fig. 5. Performance comparison between GBCA and MMSN with communication radius increasing

channels is 2∼8 and the number of sensor nodes is 200∼500. According to Inequality (23), we

use 1/c as the upper bound of residual interference ratio (i.e., the ratio of the residual interference

to the total potential interference). We depict the numerical results in Fig.2, where each point is

the average result of 50 repeated computations.

From Fig.2.(a) and (b), we can observe: 1) GBCA always reduces more interference than

MMSN, and its residual interference is far less than the upper bound, which means NE is usually

much better than its lower bound and closer to the optimal solution in Inequality (24); 2) With

the number of sensor nodes increasing, the residual interference ratio of GBCA increases. That

is because the increasing number of sensor nodes in the same area exacerbates interference;

3) With the number of channels increasing, the residual interference ratio of GBCA decreases

quickly to zero since the increasing number of channels gives more choices to each non-leaf

sensor node and removes more interference.

From Fig.2.(c) and (d), we can see: 1) GBCA converges very fast and usually less than 50

iterations, which is far less than the bound (|V |−1)2; 2) Approximately, the number of iterations

needed for convergence increases linearly with the number of sensor nodes since more nodes

in the network cause more channel changes; 3) With the number of channels increasing, the

number of iterations first increases and then decreases since the number of sensor nodes to

change channel increases with the increase of the number of channels, when it is relatively

small, and the probability that sensor nodes become stable at early iteration increases with the

increase of the number of channels, when it is relatively large.


19

B. Network Performance Evaluation

GBCA uses game theory as a tool to exploit TIRI as much as possible to reduce the interference

suffered by the network. In this subsection, we evaluate its network performance and compare it

with MMSN (this study implements the even-selection, which can be used when non-overlapping

channels are not sufficient and leads to less interference [11]) by simulations with OMNET++.

We conduct three groups of simulations. In all these simulations, the field is set to 200m×200m,

and 320 sensor nodes are uniformly distributed in the field. They form a forest with 16 sink

sensor nodes. The physical model of sensor nodes is in accordance with CC2430 [21], and the

modified CSMA mentioned in Section III-C is used to control medium access.

In the first group of simulations, we set the number of non-overlapping channels to 6 and

communication radius to 30m, and vary the number of flows from 25 to 55. The rate of each

flow is 25 packets per second. These flows are randomly generated in space but the number of

flows transmitting simultaneously is fixed according to the setting. The simulation results are

shown in Fig.3 where each point is the average result of 45 repeated simulations and so is for

the remaining two groups. From Fig.3, we can see: 1) GBCA outperforms MMSN in all these

metrics, i.e., it has larger delivery ratio and throughput but yields smaller packet transfer delay

(i.e., the time a sensor node spends competing for medium access and transmitting a packet) and

energy consumption; 2) These advantages get more remarkable with the increase of the number

of flows since GBCA takes full advantage of TIRI but MMSN just tries to balance the channel

usage in two-hop neighborhood. And when the load of the network gets heavier, the effect of

TIRI becomes more obvious.

In the second group of simulations, we set the number of flows to 40 and communication

radius to 30m, and vary the number of non-overlapping channels from 2 to 8. The way to

generate flows is the same as the first group and so is for the third group. The simulations results

are shown in Fig.4. We can see: 1) GBCA outperforms MMSN in all these metrics. 2) GBCA

achieves the near-best performance with fewer non-overlapping channels than MMSN does, e.g.,

it almost yields the largest delivery ratio and throughput when the number of channels is 5, while

MMSN may achieve this when the number of channels is 7 or more. That is because GBCA can

combine TIRI with the limited channels rationally to reduce the interference in the network. This

characteristic of GBCA makes it more practical than MMSN, as the non-overlapping channels


20

are very limited in practice [12].

In the third group of simulations, we set the number of non-overlapping channels to 6 and

the number of flows to 40, and vary the communication radius of sensor nodes from 20m

to 50m, which means the number of neighbors of sensor node increases and generates more

interference. We depict the simulation results in Fig.5. From Fig.5, we can see: 1) GBCA

outperforms MMSN in all these metrics; 2) These advantages become more significant with the

communication radius increasing; 3)When the radius is small, the delivery ratio and throughput

of GBCA increase faster than MMSN with the communication radius increasing, and when the

radius is large, the delivery ratio and throughput of GBCA decrease slower than MMSN with

the communication radius increasing. It is also because GBCA exploits TIRI more fully than

MMSN does. The increment in delivery ratio and throughput with the increase of radius when

the radius is relatively small results from the following reason: the sensor nodes get shorter path

to deliver message when the simulation builds the forest.

V. CONCLUSIONS

In this paper, we have studied how to assign channels in WSNs to reduce interference

and improve network performance. Unlike previous static assignment protocols, we take full

advantage of both the network topology information and the routing information to assign

channels in WSNs. We formulate channel assignment in WSNs as an optimization problem,

and propose a distributed game based algorithm GBCA, which inspires the dynamic interactions

between non-leaf sensor nodes and fully exploits both the network topology information and

the routing information, to solve this problem. We prove that GBCA can converge to a NE in

finite iterations, and provide bounds on the suboptimality of NE theoretically. The convergence

and suboptimality of GBCA are further testified numerically. Network performance simulations

show that GBCA achieves better network performance than MMSN in terms of delivery ratio,

throughput, packet transfer delay and energy consumption.

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