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Negotiation-Based TDMA Scheme for Ad HocNetworks from Game Theoretical PerspectiveHui Leifang, Li Jiandong*, Li Hongyan, Ma YinghongBroadband Wireless Communications Laboratory, Information Science Institute, State Key Laboratory of Integrated Service

Networks, Xidian University, Xian 710071, Shaanxi Province, P. R. China

Abstract: A negotiation-based TDMA scheme for ad

hoc networks, which was modeled as an asynchronousmyopic repeated game and self-adjusted to choose

proper time slots is proposed. During the simulation,

the game theory has been utilized to model the

negotiation procedure as a potential game. Comparedto the traditional centralized TDMA schemes, our

scheme operates in a decentralized manner and is

scalable to topology changes. Simulation results showthat, with respect to the coloring quality, the

performance of our scheme is close to that of the

classical centralized algorithms with much lowercomplexity. Moreover, there is a fairness benefit on itcompared to CSMA/CA.

Key words: TDMA scheme; asynchronous myopic

repeated game; coloring quality; classical centralizedalgorithms

I. INTRODUCTION

Due to the multi-hop nature of ad hoc networks,

different users can reuse the common channel in

appropriate manners as long as they do not interfere

with each other. TDMA is one of the multiplexingmethods. It is especially effective for networks with

high load or deadline-sensitive traffic. Slot assignmentis the core component of TDMA protocols. Assigning

different time slots to conflicting users is the objective

of the slot assignment issue and is the subject of this

paper.Since the optimal static time slot scheduling is NP-

hard[1-4], various heuristic methods have been

developed. Ramanathan[2] models this problem as agraph coloring problem and designs three efficient

greedy algorithms RAND, MNF and PMNF. But the

centralized characteristic is not suitable for ad hoc

networks. The first proposed TDMA protocol for adhoc networks is FPRP[1,3]. Nodes select slots

randomly by using a five-phase algorithm. But a node

may not be assigned a slot and requires many runs toincrease the chance to get a slot. Ref.[4] proposes a

distributed TDMA slot assignment algorithm based on

a distance-2 coloring scheme. It requires each node to

maintain state within its three-hop neighborhood,which could be quite difficult and resource intensive.

In NB-TDMA[5], TDMA scheduling is done on

demand. A node wishing to transmit data toward asink dispatches a mobile agent. This creates a

coupling between the routing and MAC operation.

Moscibroda[6] et al. proposes a graph coloring

scheme, which performs distance-1 coloring, in whichonly adjacent nodes have different colors. This

scheme does not prevent hidden terminal collisions.

DRAND[7] is the latest proposed TDMA scheme for

ad hoc networks, which is the distributed version ofRAND[2]. It gives the same maximum slot number as

RAND, but the exchange procedure is complicated.

Owing to these drawbacks, a negotiation-basedTDMA scheme for ad hoc networks is proposed in this

paper, which is executed in a distributed manner and

is self-adjusted to choose proper time slots. Withrespect to the number of time slots required (It is themeasure of coloring/scheduling quality[3].), our

scheme has similar performance to the classical

PMNF[2] with lower computational burden and isbetter than RAND [2]/DRAND[7].

The rest of this paper is organized as follows.

Section II outlines system model and expressions,

followed by the problem formulation in Section III.Section IV gives the detailed procedure of our

scheme. The scalability is introduced in Section V and

simulation results are provided in Section VI. Finally,

Section VII concludes the paper.

II. SYSTEM MODEL ANDEXPRESSION

We consider an ad hoc network with several

homogeneous users randomly deployed in a square

area. Similar to other literatures, users around one userare its one-hop neighbors or two-hop neighbors in

terms of the hops. Time is split into frames which are

subdivided into time slots with equal length.

All users share a common channel. We assume that

each user has a half-duplex transceiver. Capture effectis not considered and the interference occurs when one

user is transmitting and receiving simultaneously orreceiving packets from different flows at the same

time. Signal propagation delay is ignored, thus packets

can be received by destinations immediately.

Furthermore, we assume that users always havepackets to transmit all the time.

As mentioned earlier, a decentralized scheme is

preferred for ad hoc networks. Under this mode, evenif the stable state has been achieved through

negotiation, it is still hard to make all users to know

the state within a short time. Therefore, the latest

system state is necessary. Besides, in TDMA system,


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all users should be synchronized. In a word, one user

needs to act as a coordinator to implement thesefunctions, and all users are synchronized with the

coordinator.

We assume that there are N users in the network.

Each user has a unique ID, labeled as i, 1 i N .

iNC and _ 2iNC denote the one-hop and two-hop

neighbor set of user i respectively. In each frame,there are TS time slots. We use A to represent the

available time slots matrix. Its element 1ija =

indicates time slotj (1 )j TS is available for user

i. Similarly, O represents the time slot occupancy

matrix. Its element 1ijO = indicates user i occupies

time slotj (1 )j TS to transmit with rate iR .

III. PROBLEM FOR MULATIONOur TDMA procedure is divided into two phases: thenegotiation phase and the collision-free transmission

phase. The latter one is also known as TDMA phase1.

During the negotiation phase, each user continuouslyjudges its time slot choice based on the information

collected before, makes change if necessary and then

broadcasts relevant information. Its choice will

influence the upcoming decisions and vice versa.Game theory[8-12] is a natural modeling technique.

From the game theoretic perspective, users are

decision makers, i.e. players, and time slots set { }j

are the action space.

After getting proper time slots in the negotiationphase, the collision-free transmission phase get

started.

3.1 Game theoretical model for the negotiation

phase

Since all users are synchronized, at the beginning of

each time slot, all users try to access it with a certainprobability (the selection of this probability will be

discussed later). It means that there is always a

random combination of users accessing the channel in

each time slot, forming a subset of the user set. Thisprocess repeats until the collision-free phase has been

achieved.

This procedure is well matched with the repeatedgame[8-9]. Firstly, the problem is a repeated game

with asynchronous timing. Besides, each user can only

get its neighbor information by listening to thechannel. Using the game theoretical term, it is myopic.

Therefore, users actions in every time slot form a

normal game, and the negotiation phase is an

asynchronous myopic repeated game.For the normal game, the utility function of a user,

say i, can be defined as

( ) (1 ) ( )

_ 2 _ 2

u d R o R d kd k iii i ik NC NC k NC NC i i i i

= +

%

U U

(1)

It expresses the local throughput of user i with

current choice id . If it selects time slot j, then id j= .

In the first term, ikdo denotes whether user k occupies

time slot j, thus the product of 1ikd

o reflects the

collision. It means that as long as the same decision

has been made within is two-hop area, user i mightget a throughput of 0. The second term denotes the

estimated throughput of its one-hop and two-hop

neighbors, which can be got by the recorded

information.From (1), we know that user decisions within two-

hop area are needed. In each time slot, users who

catched channel successfully will broadcast its one-

hop neighbors choices and its own decision. Theirone-hop neighbors who receive the information will

update the record.

Accordingly, we define a potential function as

_ 21

( , ) ( (1 ))k

k k

N

i i k md

k m NC NC

P d d R o=

= U

(2)

It is the sum of each user's throughput, in which id is

the decision set for users except i.

Theorem 1. The stage game { ,{ },{ }}iN j u , with

iu defined as (1) and potential function defined as (2),

is a finite exact potential game.Proof. The proof of Theorem 1 follows similar lines

of the proof in Ref. [13].So far, the stage game has already been designed

and proved to be an exact potential game. With this

property, we will discuss the performance of the game

model.

3.2 Model analysis

The utility function should be selected to have theparticular meaning of the local throughput. But as

mentioned in Ref.[8], it must also have appealing

mathematical properties that guarantee the equilibrium

convergence. Nash Equilibrium (NE)[8-10] is acommonly used stable solution. Then how is the

performance of the modeled myopic repeated game?

Will it converge to the expected steady state?

3.2.1 Existence

Intuitively speaking, the steady state here means that

all the potential collided users transmit in different

time slots. It is obvious that the steady states do exist.From the perspective of game theory, all the finite

potential games have at least one NE (Theorem 4.24

mentioned in Ref.[9]). Therefore, there are multipleNEs for this model.

3.2.2 Optimality


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Optimality here means that the number of time slots is

minimized. Ref.[2] has pointed out that the optimalsolution of the NP-hard problem can only be got

through exhaustive search. Our model can get efficient

scheduling but not the optimal one.

3.2.3 Convergence

It makes no sense to speak of convergence for anormal form game as it is defined as having only a

single iteration. Convergence is much frequentlydiscussed in the context of repeated games.

Convergence has close relationship with the decision

rules[9] and decision timings[9] in every stage game.Potential game is a special game model. It is

guaranteed to converge to the NE with different

combinations of decision timings and decision rules. It

has been shown in Ref.[9] that finite potential gamesconverge in round-robin, random and asynchronous

decision timings, no matter which decision rule it is

using.These features shed lights on our algorithm design.

As long as the learning procedure is designed based

on the convergence condition, our myopic repeated

game model must arrive to the corresponding steadystate.

IV. THE NEGOTIATION-BASEDTDMA MAC SCHEME

4.1 Frame format

A frame consists of three parts as shown in Figure 1.

Fig.1 Frame format

In the frame head, SI(Synchronization Information)

contains timing information to provide accurate

synchronization in each frame. CP (Current Phase)indicates either it is in negotiation phase or collision-

free phase. All users determine their actions according

to the CPvalue. During the negotiation phase, CP=0;

otherwise, CP=1. TS (Time Slot) represents thenumber of time slots in this frame. All the information

in the frame head is sent by the coordinator with

proper power to notify all users the current systemstate.

Time slots section is used to compete and negotiate

in the negotiation phase and transmit packets in

TDMA phase.It should be emphasized that we assume there is a

user acting as a coordinator, but the selection of it is

beyond the scope of this paper.

4.2 Packet types

Several types of packets are involved in the

negotiation phase.

-- RTSP(RTS Packet): It is used for similar purposeasRTSin CSMA/CA; however, the required length

is much shorter than that ofRTS. It is used to

reserve the channel;

-- CTSP (CTS Packet): It is used for similar

purposes as CTS in CSMA/CA; however, therequired length is much shorter than that of CTS. It

is used to reply to RTSPand also to silence its ownone-hop neighbors (i.e. the two-hop neighbors of

the user who sentRTSP);

-- NOTIFP (Notification Packet): It is used tobroadcast its one-hop neighbor occupancy

information recorded and its current choice;

-- FBP(Feed Back Packet): It is used to feedback

the current occupancy information to thecoordinator. It can only be sent in frame tail.

4.3 Backoff issue

The backoff scheme we discussed here is similar tothat of IEEE 802.11.

In the negotiation phase, for users who try to access

channel in the current time slot, backoff is executedfirst. The user with the shortest backoff in its

neighborhood broadcasts RTSP after his backoff,

while those who have longer backoffs will cancel their

attempts once RTSPs are received. This strategy caneffectively alleviate concurrent attempts among

adjacent users.

In CSMA/CA, RTS and CTS exchanges betweensource and destination pair, and other users who hear

one of them will keep silent in the designated time.

Different from that, our CTSPs are replied by all the

one-hop neighbors of the source, collisions may occurat common neighbors. Figure 2 is an example.

UserB, CandD receiveRTSPfrom UserA. If they

reply CTSPat the same time, collisions occur at bothUserA andE. For UserE, the collided CTSPmakes it

unaware of the reservation from UserA. In order to

avoid this case, the random backoff is also adopted

when users reply CTSP.

Fig.2 Topology example

The handshake and backoff procedure is depicted in

Figure 3.


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Fig.3 Handshake and backoff procedure

At the beginning of time slot i, suppose UserA andC try to participate in the negotiation. They backoff

first (UserA has a shorter counter than Cs). Then

UserA sends RTSP after his backoff, while User C

cancels backoff and gives up the attempt. For the one-

hop neighbor of UserA, UserB, C, and D reply withCTSP to RTSP after a random backoff and also set

NAV to keep silent. UserEhears CTSPfrom UserBand Cand set NAV. Waiting for a maximum backoff

time after RTSP, User A broadcasts NOTIFP

containing its choice.

4.4 Scheme description

1) Negotiation phase

We assume that in the initial state, TS is set to N,which is the number of users in the system. In order to

minimize the number of time slots needed, each user

chooses TS1 as their initial choice. Each user

maintains three variables iTS , indicating the current

choice; iA , the set of current available time slots and

ip , the current access probability, which is defined as

the reciprocal of the access index.Details of the negotiation phase are described as

follows:

- Initialization:

TS is set to N by the coordinator. For each user,

01, { },

i i iTS A j p p= = = .

- Repeat of the frame:

Frame Head: SI, CP=0 and TS=N is sent by thecoordinator;

Time Slots: in time slotj, each user decides whether

to access channel according to ip . The user, who tries

to negotiate, backoffs a random time.

If nothing has received during the backoff period,the user, say i, sendsRTSPand then waits;

IfRTSPis received by the backoffing user, sayk, it cancels the current backoff, starts a new

backoff, sends CTSP after backoff and thenkeeps silent in this slot;

IfRTSPis received by the non-backoffing user,

say m, it sends CTSP after a random backoff

and keeps silent in this slot; If one user receives CTSPbut it is not the user

sentRTSPin current slot, the user, say n, keeps

silent in this slot; After a maximum waiting time is run out, user i

selects the time slot with the least index in iA

as its current choice and broadcastsNOTIFP;

k and m update kA and mA respectively once

NOTIFPis received.

Users who participate in the current negotiation

decrease their ip , while other users increase ip

. If ip exceeds the given range, set 0ip p= .

Frame Tail: each user returnsFBPcontaining iTS to

the coordinator in a round-robin mode.

- Until: No one changes its decision in a frame.

It should be pointed out that the adjustment of ip is

to make sure that all the users have chance to

participate in the negotiation.2) Collision-free phase/TDMA phase

Once there is no user changing their decisions in a

frame, the negotiation phase is terminated and the

collision-free phase is started, which is indicated byCP=1 in the frame head. The coordinator decides TS

value based on the collected information in frame tail.

Till now, a TDMA MAC scheme for ad hocnetwork is designed. Its advantages are multiple folds.

Firstly, the negotiation procedure is also a process of

neighbor discovery. Compared to the traditional

centralized TDMA schemes and the latest DRAND [7],a priori topology information is not required.

Secondly, the decentralized negotiation process only

collects local knowledge thus lightens thecomputational burden compared to the centralized

solution. Finally, fairness is taken into consideration.

During the negotiation phase, ip is changeable to

ensure the competition chance. Once the collision-free

phase is achieved, each user transmits once in a frame.

While in CSMA/CA, the user who has alreadytransmitted successfully is prone to transmit more,

which causes unfairness.

V. SCALABILITY

The collision-free transmission under static topology

has been achieved. But how to make it be scalable tochanges? Now we give the basic mechanisms.

5.1 Coordinator alternation

The coordinator broadcasts the basic information inthe whole procedure. In order to avoid one user

consuming too much energy, an alternation is required

among all users.

Current coordinator can include the alternation


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expectation in SIto indicate that a new coordinator is

wanted. The user who wants to be the coordinatorcontains the application in the FBP. Then current

coordinator selects one by a predefined rule and

conveys this information in SIof next frame. Thus, allusers will be aware of the new coordinator.

The following time benchmark can barely followthe current one or be redefined by the new

coordinator.

5.2 Topology change

If one user is going to leave, it informs its neighbors

in advance. Even though there is no advancenotification, its one-hop neighbors will clear this

record if the corresponding time slot has been idle for

several frames.For a new user joining in, if the system is in the

negotiation phase, it just follows the time benchmark

and participates in the negotiation. While in collision-free phase, since the time slots have already beennegotiated, the new senses in each time slot. If there

are still available time slots (due to the release of

leaving users), it tries to capture the idle time slot bybackoffing first. Backoff is used to avoid the collision

of simultaneous attempts in that idle time slot.

Therefore, the user with the fastest backoff will

capture that time slot. Even if two users attempt at thesame time after the same backoff time and then

collision happens, this information can still be

validated in the following time slots or frames.If all the time slots have been occupied, new users

send applications in the frame tail. Similarly, backoff

is also performed before application to stagger

multiple new users. Once several applications havebeen received by the coordinator, it simply increases

the number of time slot to accommodate new users

whereas the current users will not be influenced. It isquite straightforward, but may cause redundant time

slots. Here is another method. The coordinator

initiates the negotiation process again by setting

CP=0. Due to the re-negotiation, the latter methodwill get an appropriate TS value but requires longer

time to achieve steady states compared to the former

one.

VI. SIMULATION RESULTS

We first consider a small ad hoc network with 10

users, i.e. N=10. Their locations are generated

randomly within an 80-by-80 area, with a uniform

distribution for its X and Y coordinates. The regulartransmission range is set to 40 for each user, making

every link bidirectional. The initial access index is set

to 5, thus 0 1/ 5p = . And the index window is [2, 15].

If one user negotiates in a time slot, its access index

will be increased by 2 for the next time slot, and those

users whose attempts have been terminated by

RTSP/CTSPor those who do not access at all in thecurrent time slot will have their indices decreased by

1, as long as they are still in the range of the index

window; otherwise, the initial value is reset.

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

(TS1)(TS1)

(TS1)

(TS1)

(TS1)

(TS1)

(TS1)

(TS1)

(TS1) 10

9

8

7

6

5

4

3

2

Y

Coordinates

X Coordinates

1(TS1)

Fig.4 Topology example and the initial state

Figure 4 is the initial state of a random generated

topology. We can see that each user selects TS1 in the

initial state. When the collision-free phase is reached,each user selects a time slot which is different from

the choices of its one-hop and two-hop neighbors, as

shown in Figure 5. Take User 1 as an example, sinceTS1 to TS6 have been occupied within its two-hop

areas, it chooses TS7.

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

(TS3)(TS4)

(TS2)

(TS6)

(TS1)

(TS7)

(TS2)

(TS4)

(TS5) 10

9

8

7

6

5

4

3

2

Y

Coordinates

X Coordinates

1(TS1)

Fig.5 Negotiation result of our scheme

From Figure 5 we know that the value of TS hasbeen reduced from 10 to 7 after negotiation. No usercan choose another time slot with smaller index. The

steady state has been achieved.

Since the randomness is introduced in our scheme,the steady state is not unique. There exist several

steady states with the same number of time slots. Fig.

6 gives another result. As mentioned earlier, time slot

assignment issue is usually solved by graph coloring.Some of these schemes produce good results [2,14-

16], e.g., PMNF. Figure 7 shows its solution. From

Figure 7 we know that PMNF also needs 7 time slots.


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0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

(TS4)(TS5)

(TS3)

(TS7)

(TS2)

(TS1)

(TS3)

(TS5)

(TS6) 10

9

8

7

6

5

4

3

2

Y

Coord

inates

X Coordinates

1(TS2)

Fig.6 Another negotiation result of our scheme

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

(TS5)(TS3)

(TS4)

(TS2)

(TS7)

(TS1)

(TS2)

(TS3)

(TS6) 10

9

8

7

6

5

4

3

2

YC

oordinates

X Coordinates

1(TS5)

Fig.7 Assignment result of PMNF

10 20 30 40 50 60

4

6

8

10

12

14

16

18

20

AverageColoringQuality

Number of Users

DLB

RAND/DRAND

PMNF

Our Scheme

Fig.8 Coloring quality comparison of different scales

We also compare the coloring quality of severalschemes. RAND[2] executes easier than PMNF does

and has been used in many channel assignment

schemes. DRAND[7] has the same performance as ofRAND on coloring quality. DLB[2-3] is the degree-

based lower bound, which is defined as the maximal

user degree plus one. This lower bound is very tight

but can be used to approximate the optimal coloringsolution. Figure 8 shows the comparison for different

network scales. N users are deployed in a 100-by-100

square area. The regular transmission range is fixed to

25. The initial access index is set to the half of N. Nincreases from 10 to 60 by 10. We run 1000

simulations and take the average for each network

scale. It can be seen from Figure 8 that PMNF has thesolution closest to DLB. Due to the randomness,

RAND/DRAND requires more time slots than PMNFdoes. Our scheme only needs a little more time slots

than RAND does, while the coloring qualities are withthe same order.

25 30 35 40 45 50

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

AverageColoringQuality

Radius

DLB

RAND/DRAND

PMNF

Our Scheme

Fig.9 Coloring quality comparison of different radii

We also generate 1000 random topologies of 50

users. The initial access probability is set to 1/25. The

radius ranges from 25 to 50 by 5. Figure 9 compares

the coloring quality for different solutions. With theincrease of the radius, each user has more neighbors,

thus the time slot required are increased. The curveshave the same trend as those of Figure 8, and theresult of our scheme is also close to that of the

centralized algorithms.

VII. CONCLUSIONS

We design a TDMA MAC scheme for ad hoc

networks, which is a negotiation-based method withthe assistance of a coordinator. Game theory has been

utilized to model the negotiation procedure as a

potential game. On coloring quality, the performanceof our scheme is similar to that of the classical

centralized TDMA solutions with distributed manner.

In addition, it is scalable to the topology change.

Moreover, there is a fairness benefit on it compared toCSMA/CA.

It remains future work to investigate the design rule

for the time slot length and the efficient slotassignment method for users with different QoS

requirements.

AcknowledgementsThe authors would like to thank the reviewers for their detailed

reviews and constructive comments, which have helped

improve the quality of this paper. This work was supported inpart by National Science Fund for Distinguished Young


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Scholars under Grant No.60725105; National Key Basic

Research Program of China (973 Program) under Grant No.2009CB320404; Program for Changjiang Scholars and

Innovative Research Team in University under GrantNo.IRT0852; National Natural Science Foundation of China

under Grants No.60972047, 61072068 and 111 Project underGrant No.B08038.

Note1. In this paper, we use collision-free phase and TDMA phase

interchangeably.

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Biographies

Hui Leifang, is currently a Ph.D. candidate at Xidian University, Xian,

China. She has been an IEEE student member since 2010. She was a

visiting student to the Department of Electrical and Computer

Engineering at University of Florida from 2008 to 2009. Her research

interests include spectrum allocation, resource sharing and resourcemanagement in heterogeneous networks.

Li Jiandong, received the B.E., M.S. and Ph.D. degrees in electricalengineering from Xidian University, Xian, China, in 1982, 1985 and

1991 respectively. He has been a faculty member of

Telecommunications Engineering at Xidian University since 1985,where he is currently a professor and director of State Key Laboratory

of Integrated Service Networks. Prof. Li is a senior member of IEEE.

He was a visiting professor to the Department of Electrical and

Computer Engineering at Cornell University from 2002-2003. He was amember of Personal Communication Networks (PCN) specialist group

for China 863 Communication High Technology Program during 1993-

1994 and again 1999-2000. He also served as the General Vice Chair

for COMSOCs Chinacom 2009. He was awarded as DistinguishedYoung Researcher and Changjiang Scholar from Ministry of Science

and Technology, China. His major research interests include wirelesscommunication theory, cognitive radio and signal processing. *The

corresponding author. Email: [email protected]

Li Hongyan, received her M.S. degree in control engineering from

Xian Jiaotong University, and the Ph.D. degree in signal and

information processing from Xidian University, Xian, Shaanxi, China,in 1991 and 2000 respectively. She is currently a professor in the State

Key Laboratory of Integrated Service Networks, Xidian University. Her

research interests include wireless networking, cognitive networks,integration of heterogeneous network, and mobile ad hoc networks.

Ma Yinghong, received the B.S. degree in electronic information

science and technology and the M.S. degree in communication &information system from North China Electric Power University,

Baoding, Hebei, China, in 2003 and 2006, respectively. She is currently

a Ph.D. candidate at Xidian University, Xian, Shaanxi, China. Herresearch interests focus on wireless communications and human-

computer interaction techniques.
mailto:[email protected]:[email protected]:[email protected]

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