Negotiation-Based TDMA Scheme for Ad Hoc Networks
from Game Theoretical Perspective
Hui Leifang, Li Jiandong*, Li Hongyan, Ma Yinghong
Broadband Wireless Communications Laboratory, Information Science Institute, State Key Laboratory of Integrated Service
Networks, Xidian University, Xi’an 710071, Shaanxi Province, P. R. China
Abstract: A negotiation-based TDMA scheme for ad
hoc networks, which was modeled as an asynchronous
myopic 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. Compared
to the traditional centralized TDMA schemes, our
scheme operates in a decentralized manner and is
scalable to topology changes. Simulation results show
that, with respect to the coloring quality, the
performance of our scheme is close to that of the
classical centralized algorithms with much lower
complexity. Moreover, there is a fairness benefit on it
compared to CSMA/CA.
Key words: TDMA scheme; asynchronous myopic
repeated game; coloring quality; classical centralized
algorithms
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 multiplexing
methods. It is especially effective for networks with
high load or deadline-sensitive traffic. Slot assignment
is 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 a
graph 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 ad
hoc 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 to
increase 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 a
sink 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 which
only 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 of
RAND[2]. It gives the same maximum slot number as
RAND, but the exchange procedure is complicated.
Owing to these drawbacks, a negotiation-based
TDMA 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. With
respect to the number of time slots required (It is the
measure of coloring/scheduling quality[3].), our
scheme has similar performance to the classical
PMNF[2] with lower computational burden and is
better 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 AND EXPRESSION
We consider an ad hoc network with several
homogeneous users randomly deployed in a square
area. Similar to other literatures, users around one user
are 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 effect
is not considered and the interference occurs when
one user is transmitting and receiving simultaneously
or receiving 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 have
packets to transmit all the time.
As mentioned earlier, a decentralized scheme is
preferred for ad hoc networks. Under this mode, even
if 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,
all users should be synchronized. In a word, one user
needs to act as a coordinator to implement these
functions, 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 .
NCi and NCi _ 2 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
aij  1 indicates time slot j (1  j  TS ) is available
for user i. Similarly, O represents the time slot
occupancy matrix. Its element Oij  1 indicates user i
occupies time slot j (1  j  TS ) to transmit with
rate Ri .
III. PROBLEM FOR MULATION
Our TDMA procedure is divided into two phases: the
negotiation phase and the collision-free transmission
phase. The latter one is also known as TDMA phase1.
During the negotiation phase, each user
continuously judges 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 negotiation
phase, 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 certain
probability (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. This
process repeats until the collision-free phase has been
achieved.
This procedure is well matched with the repeated
game[8-9]. Firstly, the problem is a repeated game
with asynchronous timing. Besides, each user can only
get its neighbor information by listening to the
channel. 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
u (d )  R 
i i
i


(1 okd ) 
Rk ( di )
i
k NCi NCi _ 2
k NCi NCi _ 2
(1)
It expresses the local throughput of user i with
current choice d i . If it selects time slot j, then di  j .
In the first term, okdi denotes whether user k occupies
time slot j, thus the product of 1  okdi reflects the
collision. It means that as long as the same decision
has been made within i’s two-hop area, user i might
get 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.
Their one-hop neighbors who receive the information
will update the record.
Accordingly, we define a potential function as
N
P(di , d  i )   ( Rk 
k 1

mNCk NCk _ 2
(1  omdk ))
(2)
It is the sum of each user's throughput, in which
d  i is the decision set for users except i.
Theorem 1. The stage game {N ,{ j},{ui }} , with
ui 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 the
particular 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 a
commonly 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 multiple
NEs for this model.
3.2.2 Optimality
Optimality here means that the number of time slots is
minimized. Ref.[2] has pointed out that the optimal
solution 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 a
normal form game as it is defined as having only a
single iteration. Convergence is much frequently
discussed 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 games
converge 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 steady
state.
IV. THE NEGOTIATION-BASED TDMA
MAC SCHEME
4.1 Frame format
A frame consists of three parts as shown in Figure 1.
Frame tail
Frame head
SI
CP
TS
TS1
TS2
……
TSm
Time Slots
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 CP value. During the negotiation
phase, CP=0; otherwise, CP=1. TS (Time Slot)
represents the number 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 system state.
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 purpose
as RTS in CSMA/CA; however, the required length
is much shorter than that of RTS. It is used to
reserve the channel;
-- CTSP (CTS Packet): It is used for similar
purposes as CTS in CSMA/CA; however, the
required length is much shorter than that of CTS. It
is used to reply to RTSP and also to silence its own
one-hop neighbors (i.e. the two-hop neighbors of
the user who sent RTSP);
-- NOTIFP (Notification Packet): It is used to
broadcast 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 the
coordinator. It can only be sent in frame tail.
4.3 Backoff issue
The backoff scheme we discussed here is similar to
that of IEEE 802.11.
In the negotiation phase, for users who try to access
channel in the current time slot, backoff is executed
first. 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 can
effectively alleviate concurrent attempts among
adjacent users.
In CSMA/CA, RTS and CTS exchanges between
source 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 occur
at common neighbors. Figure 2 is an example.
User B, C and D receive RTSP from User A. If they
reply CTSP at the same time, collisions occur at both
User A and E. For User E, the collided CTSP makes it
unaware of the reservation from User A. In order to
avoid this case, the random backoff is also adopted
when users reply CTSP.
User E
User D
User B
User A
User C
Fig.2 Topology example
The handshake and backoff procedure is depicted in
Figure 3.
RTSP
NOTIFP
A
–
CTSP
B
NAV (RTSP)
CTSP
C
–
NAV (RTSP)
CTSP
D
NAV (RTSP)
E
NAV (CTSP)
TSi starts
TSi ends
= Backoff
–
= Remaining backoff
Fig.3 Handshake and backoff procedure
At the beginning of time slot i, suppose User A and
C try to participate in the negotiation. They backoff
first (User A has a shorter counter than C’s). Then
User A sends RTSP after his backoff, while User C
cancels backoff and gives up the attempt. For the
one-hop neighbor of User A, User B, C, and D reply
with CTSP to RTSP after a random backoff and also
set NAV to keep silent. User E hears CTSP from User
B and C and 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 TSi , indicating the current
choice; Ai , the set of current available time slots and
pi , 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,
TSi  1, Ai  { j}, pi  p0 .
- Repeat of the frame:
Frame Head: SI, CP=0 and TS=N is sent by the
coordinator;
Time Slots: in time slot j, each user decides whether
to access channel according to pi . The user, who tries
to negotiate, backoffs a random time.
– If nothing has received during the backoff
period, the user, say i, sends RTSP and then
waits;
– If RTSP is received by the backoffing user, say
k, it cancels the current backoff, starts a new
backoff, sends CTSP after backoff and then
keeps silent in this slot;
– If RTSP is 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 CTSP but it is not the user
sent RTSP in 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 Ai as its current choice and broadcasts
NOTIFP;
k and m update Ak and Am respectively once
NOTIFP is received.
Users who participate in the current negotiation
decrease their pi , while other users increase
pi . If pi exceeds the given range, set
pi  p0 .
Frame Tail: each user returns FBP containing
TSi 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 pi
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 by
CP=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 hoc
network 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 the
computational burden compared to the centralized
solution. Finally, fairness is taken into consideration.
During the negotiation phase, pi 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 already
transmitted 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 to
changes? Now we give the basic mechanisms.
5.1 Coordinator alternation
The coordinator broadcasts the basic information in
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 regular transmission
range is set to 40 for each user, making every link
bidirectional. The initial access index is set to 5,
80
(TS1) 10
(TS1)
8
70
(TS1)
60
Y Coordinates
5.2 Topology change
If one user is going to leave, it informs its neighbors
in advance. Even though there is no advance
notification, 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
been negotiated, 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 by
backoffing 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 the
same 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 have
been 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 is
quite 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 method will get an
appropriate TS value but requires longer time to
achieve steady states compared to the former one.
thus p0  1/ 5 . 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/CTSP or those who do not access at all in the
current 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.
7
50
(TS1)
1
(TS1)
3
40
(TS1)
4
30
(TS1) 9
20
(TS1)
5
10
(TS1) 2
0
0
10
6 (TS1)
20
30
40
50
60
70
80
X Coordinates
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, since
TS1 to TS6 have been occupied within its two-hop
areas, it chooses TS7.
80
(TS5) 10
(TS2)
8
70
(TS4)
60
Y Coordinates
the 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
expectation in SI to indicate that a new coordinator is
wanted. The user who wants to be the coordinator
contains the application in the FBP. Then current
coordinator selects one by a predefined rule and
conveys this information in SI of next frame. Thus, all
users will be aware of the new coordinator.
The following time benchmark can barely follow
the current one or be redefined by the new
coordinator.
50
7
(TS1)
30
(TS7)
3
40
1
(TS1)
4
(TS2) 9
20
(TS6)
10
5
(TS4) 2
0
0
10
20
6 (TS3)
30
40
50
60
70
80
X Coordinates
Fig.5 Negotiation result of our scheme
From Figure 5 we know that the value of TS has
been reduced from 10 to 7 after negotiation. No user
can 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.
80
(TS6) 10
(TS3)
8
70
(TS5)
Y Coordinates
60
50
7
(TS2)
1
(TS1)
3
40
(TS2)
4
30
(TS3) 9
20
(TS7)
10
5
(TS5) 2
0
0
10
20
coloring solution. 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. N increases 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 the solution closest to DLB. Due to the
randomness, RAND/DRAND requires more time slots
than PMNF does. Our scheme only needs a little more
time slots than RAND does, while the coloring
qualities are with the same order.
6 (TS4)
42
30
40
50
60
70
80
40
X Coordinates
DLB
RAND/DRAND
PMNF
Our Scheme
38
Average Coloring Quality
36
Fig.6 Another negotiation result of our scheme
80
(TS6) 10
(TS2)
8
70
(TS3)
Y Coordinates
60
50
7
(TS1)
3
1
28
26
24
22
20
14
25
4
30
35
40
45
50
Radius
(TS4) 9
20
Fig.9 Coloring quality comparison of different radii
(TS2)
5
10
(TS3) 2
0
0
10
20
6 (TS5)
30
40
50
60
70
80
X Coordinates
Fig.7 Assignment result of PMNF
20
DLB
RAND/DRAND
PMNF
Our Scheme
18
Average Coloring Quality
30
18
(TS7)
30
32
16
(TS5)
40
34
16
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 the
increase of the radius, each user has more neighbors,
thus the time slot required are increased. The curves
have the same trend as those of Figure 8, and the
result of our scheme is also close to that of the
centralized algorithms.
VII. CONCLUSIONS
14
12
10
8
6
4
10
20
30
40
50
60
Number of Users
Fig.8 Coloring quality comparison of different scales
We also compare the coloring quality of several
schemes. RAND[2] executes easier than PMNF does
and has been used in many channel assignment
schemes. DRAND[7] has the same performance as of
RAND 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
We design a TDMA MAC scheme for ad hoc
networks, which is a negotiation-based method with
the assistance of a coordinator. Game theory has been
utilized to model the negotiation procedure as a
potential game. On coloring quality, the performance
of 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 to
CSMA/CA.
It remains future work to investigate the design rule
for the time slot length and the efficient slot
assignment method for users with different QoS
requirements.
Acknowledgements
The 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 in
part by National Science Fund for Distinguished Young
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 Grant
No.IRT0852; National Natural Science Foundation of China
under Grants No.60972047, 61072068 and 111 Project under
Grant No.B08038.
Note
1.
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, Xi’an,
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 resource
management in heterogeneous networks.
Li Jiandong, received the B.E., M.S. and Ph.D. degrees in electrical
engineering from Xidian University, Xi’an, 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 a
member 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
Distinguished Young Researcher and Changjiang Scholar from
Ministry of Science and Technology, China. His major research
interests include wireless communication theory, cognitive radio and
signal
processing.
*The
corresponding
author.
Email:
[email protected]
Li Hongyan, received her M.S. degree in control engineering from
Xi’an Jiaotong University, and the Ph.D. degree in signal and
information processing from Xidian University, Xi’an, 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, Xi’an, Shaanxi, China. Her
research interests focus on wireless communications and
human-computer interaction techniques.