Cooperation in Wireless Ad Hoc Networks
Vikram Srinivasan
†
, Pavan Nuggehalli
†
, Carla F. Chiasserini
‡
, Ramesh R. Rao
†
†
Department of Electrical and Computer Engineering, University of California at San Diego
email:
{vikram,pavan,rao}@cwc.ucsd.edu
‡
Dipartimento di Elettronica, Politecnico di Torino, Italy
email: chiasserini@polito.it
Abstract— In wireless ad hoc networks, nodes communicate
with far off destinations using intermediate nodes as relays.
Since wireless nodes are energy constrained, it may not be in
the best interest of a node to always accept relay requests. On
the other hand, if all nodes decide not to expend energy in
relaying, then network throughput will drop dramatically. Both
these extreme scenarios (complete cooperation and complete non-
cooperation) are inimical to the interests of a user. In this paper
we address the issue of user cooperation in ad hoc networks.
We assume that nodes are rational, i.e., their actions are strictly
determined by self interest, and that each node is associated with
a minimum lifetime constraint. Given these lifetime constraints
and the assumption of rational behavior, we are able to determine
the optimal throughput that each node should receive. We define
this to be the rational Pareto optimal operating point. We then
propose a distributed and scalable acceptance algorithm called
Generous TIT-FOR-TAT (GTFT). The acceptance algorithm is
used by the nodes to decide whether to accept or reject a relay
request. We show that GTFT results in a Nash equilibrium and
prove that the system converges to the rational and optimal
operating point.
Methods keywords – Economics (game theory), System Design.
I. INTRODUCTION
Wireless ad hoc networks have matured as a viable means
to provide ubiquitous untethered communication. In order to
enhance network connectivity, a source communicates with
far off destinations by using intermediate nodes as relays [1],
[2], [3], [4]. However, the limitation of finite energy supply
raises concerns about the traditional belief that nodes in ad hoc
networks will always relay packets for each other. Consider a
user in a campus environment equipped with a laptop. As part
of his daily activity, the user may participate in different ad
hoc networks in classrooms, the library and coffee shops. He
might expect that his battery-powered laptop will last without
recharging until the end of the day. When he participates in
these different ad hoc networks, he will be expected to relay
traffic for other users. If he accepts all relay requests, he
might run out of energy prematurely. Therefore, to extend his
lifetime, he might decide to reject all relay requests. If every
user argues in this fashion, then the throughput that each user
receives will drop dramatically. We can see that there is a
trade-off between an individual user’s lifetime and throughput.
Cooperation among nodes in an ad hoc network has been
previously addressed in [5], [6], [7], [8], [9]. In [5], nodes,
which agree to relay traffic but do not, are termed as misbe-
having. Clever means to identify misbehaving users and avoid
routing through these nodes are proposed. Their approach
consists of two applications: Watchdog and Pathrater.The
former runs on every node keeping track of how the other
nodes behave; the latter uses this information to calculate the
route with the highest reliability. In [6], [7], [8], a secure
mechanism to stimulate nodes to cooperate and to prevent
them from overloading the network is presented. The key
idea is that nodes providing a service should be remunerated,
while nodes receiving a service should be charged. Based
on this concept, an acceptance algorithm is proposed. The
acceptance algorithm is used to decide whether to accept or
reject a packet relay request. The acceptance algorithm at each
node attempts to balance the number of packets it has relayed
with the number of its packets that have been relayed by
others. The drawback of this scheme is that it involves per
packet processing which results in large overheads. In [9],
two acceptance algorithms are proposed, which are used by
the network nodes to decide whether to relay traffic on a per
session basis. The goal of these algorithms is to balance the
energy consumed by a node in relaying traffic for others with
energy consumed by other nodes in relaying traffic and to
find an optimal trade-off between energy consumption and
session blocking probability. By taking decisions on a per
session basis, the per packet processing overhead of previous
schemes is eliminated. We emphasize, however, that all the
above algorithms are based on heuristics and lack a formal
framework to analyze the optimal trade-off between lifetime
and throughput.
In this paper, we consider a finite population of N nodes
(e.g., students on a campus). Each node, depending on its type
(e.g., laptop, PDA, cell phone), is associated with an average
power constraint. This constraint can be derived by dividing its
initial energy allocation by its lifetime expectation. We assume
that time is slotted and that each session lasts for one slot.
We deal with connection-oriented traffic. At the beginning of
each slot, a source, destination and several relays are randomly
chosen out of the N nodes to form an ad hoc network (e.g.,
students in a coffee shop). The source requests the relay nodes
in the route to forward its traffic to the destination. If any of
the relay nodes rejects the request, the traffic connection is
blocked.
For each node, we define the Normalized Acceptance Rate
(NAR) as the ratio of the number of successful relay requests
generated by the node, to the number of relay requests made
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by the node. This quantity is an indication of the throughput
experienced by the node. Then, we study the optimal trade-
off between the lifetime and NARs of the nodes. In particular,
given the energy constraints and the lifetime expectation of the
nodes, we identify the feasible set of NARs. This provides us
with a set of Pareto optimal values, i.e., values of NAR such
that a node cannot improve its NAR without decreasing some
other node’s NAR. By assuming the nodes to be rational, i.e.,
that their actions are strictly determined by self interest, we
are able to identify a unique set of rational and Pareto optimal
NARs for each user.
Since users are self-interested and rational, there is no
guarantee that they will follow a particular strategy unless they
are convinced that they cannot do better by following some
other strategy. In game theoretic terms [10], we need to iden-
tify a set of strategies which constitute a Nash equilibrium
1
.
Ideally, we would like the Nash Equilibrium to result in the
rational and Pareto optimal operating point. We achieve this
by proposing a distributed and scalable acceptance algorithm,
called Generous TIT-FOR-TAT (GTFT). We prove that GTFT
is a Nash Equilibrium which converges to the rational and
Pareto optimal NARs.
To the best of our knowledge, this is the first paper applying
game theory to the problem of cooperation among nodes in
an ad hoc network for relaying traffic.
The remainder of the paper is organized as follows. We
describe the system scenario and introduce some notations
and definitions in Section II. In Section III we use rationality
arguments to derive the rational Pareto optimal values of NAR.
In Section IV we present the GTFT algorithm that leads
the nodes to operate at the rational optimal operating point.
Section V shows that the GTFT algorithm constitutes a Nash
Equilibrium and that the NARs of the nodes converge to the
rational and Pareto Optimal operating point. Numerical results
are shown and discussed in Section VI. Section VII discusses
some implementation issues of the GTFT algorithm. Finally,
Section VIII concludes the paper and points to some aspects
that will be the subject of future research.
II. S
YSTEM MODEL
We consider a finite population of N nodes distributed
among K classes. Let n
i
be the number of nodes in class
i (i =1,...K). All nodes in class i are associated with
an energy constraint, denoted by E
i
, and an expectation of
lifetime, denoted by L
i
. Based on these requirements, we
contend that nodes in class i have an average power constraint
of ρ
i
= E
i
/L
i
. We assume that ρ
1
>ρ
2
> ... > ρ
K
.The
system operates in discrete time. In each slot, any one of the
N nodes can be chosen as a source with equal probability. M
is the maximum number of relays that the source can use to
reach its destination. The probability that the source requires
l ≤ M relays is given by q(l). For the sake of simplicity,
in our study we assume q(0) = 0, i.e., there is at least one
1
A Nash equilibrium is a strategy profile having the property that no player
can benefit from unilaterally deviating from his strategy.
relay in each session. This assumption can be easily relaxed by
subtracting the energy spent in direct transmissions from the
total energy budget of each node. The l relays are chosen with
equal probability from the remaining N −1 nodes. We assume
that each session lasts for one slot. In this time interval, the
source along with the l relays forms an ad hoc network that
remains unchanged for the duration of the slot.
The source requests the relay nodes to forward its traffic to
the destination. A relay node has the option to either accept or
refuse the request. We assume that a relay node communicates
its decision to the source by transmitting either a positive or
a negative acknowledgment. If a negative acknowledgment is
sent, the traffic session is blocked. A session is said to belong
to type j, if at least one of the nodes involved belongs to
class j and the class of any other node is less than or equal
to j
2
. As an example, consider a session with two relays. Let
the source belong to class 1, the first relay to class 2 and the
second to class 1. Then, the session is of type 2. It will become
clear later in the paper that the interaction between nodes in
a session is dominated by the node with the smallest power
constraint.
A node spends energy in transmitting, receiving and pro-
cessing traffic. We assume that energy spent in transmit mode
is the dominant source of energy consumption; thus, in this
paper we consider only energy spent in transmitting traffic
3
This allows us to ignore the destination node in our model.
The energy consumed by the nodes in transmitting a session
will depend on several factors like the channel conditions, the
file size, and the modulation scheme. Here, we assume that
the energy required to relay a session is constant and equal to
1. While this is not a very reasonable assumption, it allows
us to capture the salient aspects of the problem. We believe
that the ideas presented in this paper can be extended to more
realistic settings.
Finally, for a generic node h, we denote by B
j
h
(k) the
number of relay requests made by node h for a session of
type j till time k, and by A
j
h
(k) the number of relay requests
generated by node h for a session of type j which have been
accepted till time k. Equivalently, we denote by D
j
h
(k) the
number of relay requests made to node h for a session of type
j till time k, and by C
j
h
(k) the number of relay requests made
to node h for a session of type j which have been accepted
by node h till time k.
For 1 ≤ j ≤ K and 1 ≤ h ≤ N, we define: φ
j
h
(k)=
A
j
h
(k)/B
j
h
(k), and ψ
j
h
(k)=C
j
h
(k)/D
j
h
(k). Observe that
φ
j
h
is the ratio of the number of relay requests for type j
sessions made by h which have been accepted, to the number
of requests for type j sessions made by h; thus, φ
j
h
is an
indication of the throughput experienced by h, with respect
to type j sessions. The Normalized Acceptance Rate (NAR)
2
The nodes involved in the session include the source and the relays; the
destination node is not considered.
3
We ignore the energy spent by a source in requesting nodes to relay traffic
and the energy spent by a relay in communicating its decision
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is defined as NAR = lim
k→∞
φ
j
h
(k)
4
. Note that the NAR
is defined for each node and session type, however, we have
suppressed the indices for the sake of simplicity. From the
above definitions it is clear that the throughput of a node is
determined by its values of NAR. In the following we will
equivalently refer to NARs and throughput.
III. R
ATI O NAL A N D PARETO OPTIMAL OPERATING POINT
The set of NAR values which users receive is a function of
the acceptance algorithm executed at the relays. As mentioned
above, we assume that the nodes are rational, i.e., their actions
are strictly determined by self interest. Given this assumption,
we can identify a set of NAR values such that: (i) they meet
the energy constraints of the nodes; (ii) they are Pareto optimal
values, i.e., values of NAR such that a node cannot improve
its NAR without decreasing some other node’s NAR; (iii) all
rational users will find the allocation fair to themselves and
hence will accept it.
In order to derive the feasible region of operation, we
assume that the nodes adopt a stationary policy, i.e., a node
in class i in a session of type j accepts a relay request with
probability τ
ij
. Given this stationary policy, we first write the
constraints on the energy consumption rate of the nodes, from
which we can derive the feasible set of τ
ij
s. Consider a node
p participating in a type j session (1 ≤ j ≤ K). The average
energy per slot spent by the node as a source, e
(s)
pj
, can be
written as
e
(s)
pj
=
1
N
× NAR
=
1
N
M
l=1
h
1
,...h
j
q(l)Γ(l; h
1
,...h
j
)τ
h
1
1j
...τ
h
j
jj
(1)
where:
• 1/N is the probability that node p is the source;
• Γ(l; h
1
,...h
K
) is a multivariate probability function con-
ditioned on the fact that the session belongs to type j
with l relays. h
i
refers to the number of relays of class i
participating in the session;
• τ
h
1
1j
...τ
h
j
jj
represent the probability that all the relay
nodes accept the request.
Similarly, the average energy per slot spent by the node as a
relay, e
(r)
pj
, is given by
e
(r)
pj
=
1
N
M
l=1
lq(l)
h
1
,...h
j
Γ(l − 1; h
1
,...h
j
)
τ
h
1
1j
...τ
h
j
jj
τ
class(p)j
(2)
with l/N being the probability that node p is chosen as one
of the l relays. The feasible region for the τ
ij
s is then defined
4
We don’t define this as an acceptance probability, since we don’t restrict
attention to the class of stationary acceptance algorithms
by the following set of inequalities,
K
j=1
(e
(s)
pj
+ e
(r)
pj
) ≤ ρ
class(p)
1 ≤ p ≤ N
τ
class(p)j
∈ [0, 1] 1 ≤ j ≤ K;1≤ p ≤ N, (3)
where class(p) is the class to which node p belongs. For a
feasible set of τ
ij
s, the corresponding feasible set of NARs
can be directly computed from (1). The Pareto optimal values
of the τ
ij
s can be derived by imposing the equality relation in
(3).
Normalized Acceptance Rate for Node B
Normalized Acceptance Rate for Node A
(ρ,ρ)
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111111111111111111111111
111111111111111111111111
111111111111111111111111
111111111111111111111111
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111111111111111111111111
111111111111111111111111
111111111111111111111111
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111111111111111111111111
111111111111111111111111
111111111111111111111111
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111111111111111111111111
111111111111111111111111
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111111111111111111111111
111111111111111111111111
(0,2ρ)
(2ρ,0)
Fig. 1. Feasible Region for N =2, K =1, ρ =0.5.
As an example, consider a system with two nodes, say A and
B, belonging to the same class and with a power constraint ρ.
Assume that both nodes want to transmit to an Internet access
point, and q(1) = 1,M =1. In this case, the feasible region
for the NARs is shown in Fig. 1. The Pareto optimal values of
the NARs are given by the line segment joining (0, 2ρ) with
(2ρ, 0). In fact, while operating at any of these points, both
nodes are consuming energy at the maximum allowable rate.
Therefore a node cannot increase its NAR without decreasing
the other node’s NAR.
We now show how rationality can be used to derive the
unique operating point from the set of feasible points. Ratio-
nality implies that each user wants to maximize his benefit by
expending least amount of effort (i.e., energy). In the example
in Fig. 1, it is straightforward to see that the only Pareto
optimal operating point acceptable to both rational users is
(ρ, ρ). In the case of multiple classes, nodes belonging to
different classes will have different NARs. The notion of
rationality can be extended to this case as follows. First,
consider a system with N nodes, all belonging to the same
class. By rationality, each node must possess the same value
of NAR; thus, it is a simple matter to derive the maximal
value of τ which satisfies the energy constraint as in (3). Then,
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
consider a system with n
1
nodes in class 1 and n
2
nodes in
class 2. Suppose n
1
=1; by rationality, the lone node in class
1 will not expend more energy than the remaining nodes in
class 2. This is because the node in class 1 will not receive
higher throughput if it is more generous to users in class 2
than users in class 2 are to it. Indeed, self interest dictates
that the lone node behaves as though he belongs to class 2.
Suppose now, that there are two nodes in class 1. When the
nodes in class 1 are involved in type 2 sessions, they have no
incentive to behave any differently than as if they were class 2
nodes. While, when they are involved in type 1 sessions, they
can utilize their excess energy to their mutual benefit. Thus
the rationality argument leads us to the following lemma.
Lemma 1: For a set of self-interested nodes, the rational
values of τ
ij
have the following property.
τ
ij
= τ
jj
1 ≤ i ≤ j ≤ K. (4)
Henceforth, we shall denote τ
jj
by τ
j
.
Given Lemma 1, the rational Pareto optimal values of the
τ
j
s and hence the NARs can be determined by recursively
solving the energy constraints in (3) and by using (1) and (2).
Also, for a node h belonging to class i involved in type j
sessions, we define
L
ij
=
Prob(h is served in a type j session)
Prob(h accepts to relay a type j session)
. (5)
L
ij
is the ratio of the rational Pareto optimal NAR for type j
session to τ
j
. Below some examples are provided.
A. Example 1
Consider K classes and N nodes with n
i
nodes in class
i, and q(1) = 1,M =1, i.e., the route between any source-
destination pair consists of exactly one relay node. In this case,
the session type is the maximum of the source class and the
relay class. Consider a node in class i. The average energy per
slot spent by the node as a source is as follows
e
(s)
i
=
1
N(N − 1)
i−1
k=1
n
k
τ
i
+(n
i
− 1)τ
i
+
K
l=i+1
n
l
τ
l
.
When the relay belongs to a class lower than i, the session is
of type i and if the relay belongs to a class higher than i,the
session type is the same as the class of the relay. The same
expression holds for the average energy per slot, e
(r)
i
, spent
by the node as a relay. The rational Pareto optimal τ
i
can be
derived from the set of equations below
e
(s)
i
+ e
(r)
i
= ρ
i
1 ≤ i ≤ K
τ
i
∈ [0, 1] 1 ≤ i ≤ K.
In particular, for K =1, the rational and Pareto optimal τ is
equal to Nρ/2, and the rational Pareto optimal NAR is equal
to τ .
B. Example 2
Consider a system with two classes. For simplicity assume
that no more than 2 relays are ever involved (M =2).
Consider a node in class 2. The energy spent by this node
as a source, e
(s)
2
, and as a relay, e
(r)
2
,aregivenby
e
(s)
2
=
1
N
M
l=1
q(l)τ
l
2
e
(r)
2
=
1
N
M
l=1
lq(l)τ
l
2
.
The optimal τ
2
can be found by solving the quadratic equation
e
(s)
2
+ e
(r)
2
= ρ
2
.
Now consider a node in class 1. The energy spent by this
node as a source, e
(s)
1
, and as a relay, e
(r)
1
,aregivenby
e
(s)
1
=
1
N
q(1)
n
2
N − 1
τ
2
+
1 −
n
2
N − 1
τ
1
+ q(2)
(n
1
− 1)(n
1
− 2)
(N − 1)(N − 2)
τ
2
1
+
1 −
(n
1
− 1)(n
1
− 2)
(N − 1)(N − 2)
τ
2
2
e
(r)
1
=
1
N
q(1)
n
2
N − 1
τ
2
+
1 −
n
2
N − 1
τ
1
+2q(2)
(n
1
− 1)(n
1
− 2)
(N − 1)(N − 2)
τ
2
1
+
1 −
(n
1
− 1)(n
1
− 2)
(N − 1)(N − 2)
τ
2
2
. (6)
Since we know τ
2
, we can obtain τ
1
by solving the quadratic
equation e
(s)
1
+ e
(r)
1
= ρ
1
.
Note that the method presented in these examples can be
easily extended to multiple classes and relays.
IV. T
HE GTFT ALGORITHM
In this section, we present a distributed and scalable ac-
ceptance algorithm which propels the nodes to operate at
the rational Pareto optimal NARs. We call this algorithm the
Generous TIT-FOR-TAT (GTFT) algorithm.
In a network of self-interested nodes, each node will decide
on those actions which will provide it maximum benefit.
Any strategy that leads such users to the rational optimal
NARs should possess certain features. Firstly, it cannot be
a randomized stationary policy. If a node in class i gets a
request for a type j session, then a possible course of action
would be to accept that request with probability τ
j
.Ifall
nodes were to use this policy, then the rational optimal τs
described in Section III can be used to achieve the optimal
operating point. However, a rational selfish node will exploit
the naivete of other nodes by always denying their relay
requests thereby increasing its lifetime, while keeping its NAR
constant. In other words, in our system, any stationary strategy
is dominated by the always deny behavior. Hence, stationary
strategies are not sustainable, and behavioral strategies are
required in order to stimulate cooperation. By behavioral
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
TABLE I
P
UNISHMENT MATRIX FOR THE PRISONERS DILEMMA.THE FIRST ENTRY
REFERS TO PRISONER
P1’S PRISON TERM AND THE SECOND ONE TO
PRISONER
P2’S PRISON TERM.
P1
P2
Confess Not Confess
Confess (5,5) (0,10)
Not Confess (10,0) (1,1)
strategies, we mean that a node bases its decision on the past
behavior of the nodes in the system. The second feature, which
we would like an acceptance algorithm to have, is protection
from exploitation. Finally, the algorithm must be scalable.
Our problem falls in the framework of Non-Cooperative
Game Theory [10]. There, the canonical example is that of the
Prisoners Dilemma. In this example, two people are accused
of a crime. The prosecution promises that, if exactly one
confesses, the confessor goes free, while the other goes to
prison for ten years. If both confess, then they both go to prison
for five years. If neither confesses, they both go to prison for
just a year. Table I presents the punishment matrix showing the
years of prison that the players get depending on the decision
they make. Clearly, the mutually beneficial strategy would be
for both not to confess. However, from the perspective of the
first prisoner, P1, his punishment is minimized if he confesses,
irrespective of what the other prisoner, P2, does. Since the
other prisoner argues similarly, the unique Nash Equilibrium
is the confess strategy for both prisoners. Nevertheless, if this
game were played repeatedly (Iterated Prisoners Dilemma), it
has been shown that cooperative behavior can emerge as a
Nash equilibrium. By employing behavioral strategies, a user
can base his decision on the outcomes of previous games.
This allows the emergence of cooperative equilibrium. A well
known strategy to achieve this desirable state of affairs is
the Generous TIT-FOR-TAT (GTFT) strategy [11]. In the
Generous TIT-FOR-TAT strategy, each player mimics the
action of the other player in the previous game. Each player,
however, is slightly generous and on occasion cooperates by
not confessing even if the other player had confessed in the
previous game. We have adapted the GTFT algorithm to our
problem.
In our algorithm, each node maintains a record of its
past experience by using the two variables ψ
(j)
h
and φ
(j)
h
,
h =1,...N, j =1,...K, defined in Section II. Each node
therefore maintains only information per session type and does
not maintain individual records of its experience with every
node in the network.
The decisions are always taken by the relay nodes based
only on their ψ
(j)
h
and φ
(j)
h
values. First, consider the case
with N nodes, K classes, q(1) = 1 and M =1, i.e., each
session uses only one relay. Assume that a generic node h
receives a relay request for a type j session. Let be a small
positive number. The acceptance algorithm, which we call the
GTFT algorithm is as follows.
• If ψ
(j)
h
(k) >τ
j
or φ
(j)
h
(k) <ψ
(j)
h
(k) − Reject
• Else Accept .
Thus, a request for a type j session is refused if either (i)
ψ
(j)
h
(k) >τ
j
, i.e., node h has relayed more traffic for type
j sessions than what it should, or (ii) φ
(j)
h
(k) <ψ
(j)
h
(k) − ,
i.e., the amount of traffic relayed by node h in sessions of
type j is greater than the amount of traffic relayed for node
h by others in type j sessions. Since is positive, nodes are
a little generous by agreeing to relay traffic for others even
if they have not received a reciprocal amount of help. The
GTFT algorithm has the following desirable properties. (i) It
is not a stationary strategy. (ii) Each node takes its action
based solely on locally gathered information; this prevents a
node from being exploited. (iii) Only 4K variables need to
be stored at each node, independently of N, and this makes
GTFT scalable.
Let us now consider the multiple relay case. While for the
single relay case, GTFT attempts to equalize the amount of
cooperation a node provides with the amount of cooperation
it receives, when multiple relays are used, the amount of help
rendered is always more than the amount of help received. This
is because a node is a relay more often than it is a source. We
therefore modify the GTFT algorithm as follows, and call this
version of the algorithm m-GTFT. Assume that a relay request
for a type j session arrives at node h belonging to class i.The
acceptance algorithm becomes,
• If ψ
(j)
h
(k)>τ
j
or φ
(j)
h
(k)<L
ij
ψ
(j)
h
(k) − Reject
• Else Accept
where L
ij
is defined as in (5).
V. N
ASH EQUILIBRIUM OF THE GTFT ALGORITHM
We now prove that the GTFT algorithm constitutes a Nash
Equilibrium and show that similar arguments can be extended
to prove the convergence of the m-GTFT algorithm.
We first consider the case where all nodes belong to the
same class and routes include one relay only (i.e., q(1) = 1,
M =1). For the sake of simplicity, we drop the session type
index in the following theorem.
Theorem 1: Consider a system of N nodes, with all nodes
belonging to the same class and having energy constraint ρ.
Assume q(1) = 1 and M =1. Then,
1) If all nodes except node h are employing GTFT, then
lim sup
k→∞
φ
h
(k) ≤ N
ρ
2
2) If all nodes employ GTFT, then all φ
h
(k) (h =1,...N)
converge to τ =
Nρ
2
.
Proof: See the appendix.
The first part of Theorem 1 shows that if node h tries to deviate
from the GTFT strategy, then it cannot achieve throughput
greater than the rational Pareto optimal value. The second part
of the theorem shows that GTFT results in the rational Pareto
optimal point.
We can now extend the proof to the case with multiple
classes and a single relay, i.e., K>1, q(1) = 1 and M =1.
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
