On Throughput Efficiency of Geographic Opportunistic
Routing in Multihop Wireless Networks
∗
Kai Zeng Wenjing Lou Jie Yang D. Richard Brown III
Department of ECE
Worcester Polytechnic Institute
Worcester, MA 01609
{kzeng, wjlou, abbyyang, drb}@wpi.edu
ABSTRACT
Geographic opportunistic routing (GOR) is a new routing
concept in multihop wireless networks. In stead of picking
one node to forward a packet to, GOR forwards a packet to
a set of candidate nodes and one node is selected dynam-
ically as the actual forwarder based on the instantaneous
wireless channel condition and node position and availabil-
ity at the time of transmission. GOR takes advantages of
the spatial diversity and broadcast nature of wireless com-
munications and is an efficient mechanism to combat the
unreliable links. The existing GOR schemes typically in-
volve as many as available next-hop neighbors into the local
opportunistic forwarding, and give the nodes closer to the
destination higher relay priorities. In this paper, we focus
on realizing GOR’s potential in maximizing throughput. We
start with an insightful analysis of various factors and their
impact on the throughput of GOR, and propose a local met-
ric named expected one-hop throughput (EOT) to balance
the tradeoff between the benefit (i.e., packet advancement
and transmission reliability) and the cost (i.e., medium time
delay). We identify an upper bound of EOT and proof its
concavity. Based on the EOT, we also propose a local can-
didate selection and prioritization algorithm. Simulation re-
sults validate our analysis and show that the metric EOT
leads to both higher one-hop and path throughput than the
corresponding pure GOR and geographic routing.
Categories and Subject Descriptors
C.2.2 [Computer-Communication Networks]: Network
Protocols-Routing Protocols; C.2.1 [Computer-Communi-
cation Networks]: Network Architecture and Design -
Wireless Communication
∗
This work was supported in part by the US National
Science Foundation under grants CNS-0626601 and CCF-
0447743.
Permission to make digital or hard copies of all or part of this work for
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Qshine’07 August 14-17, 2007, Vancouver, Canada
Copyright 2007 ACM 978-1-59593-756-8 ...$5.00.
General Terms
Algorithms, Performance
Keywords
wireless, geographic, performance
1. INTRODUCTION
Routing in multihop wireless networks is a challenging is-
sue. The main difficulty lies in that wireless links can be very
unstable and unreliable [4, 15]. Traditional routing protocols
for wireless networks have followed the routing concept in
wired networks by abstracting wireless links as wired links,
and focused on finding a fixed shortest path for forwarding
packets between a pair of nodes. However, it is not an ideal
approach for wireles s networks with broadcast links of time
varying qualities. Recently, a new routing paradigm, known
as opportunistic routing [18, 11, 3] (or contention-based for-
warding [6]), was proposed to cope with the unreliability of
link quality.
The basic idea behind opportunistic routing is to integrate
the network and MAC layers such that at the network layer
a set of forwarding candidates are selected and at the
MAC layer one node is chosen as the actual relay. Owing
to the broadcast nature and spatial diversity of the wireless
medium, the probability of at least one forwarding candidate
correctly receiving the packet will increase when multiple
candidates are involved, thus improve the packet delivery
efficiency such as throughput [3, 6] or energy efficiency [11,
18, 13].
Two important issues of opportunistic routing are for-
warding candidates selection and relay priority assignment.
Several variants of opportunistic routing [18, 11, 6] lever-
age the location information of nodes to select forwarding
candidates and prioritize them. For example, in [18], all
the available next-hop neighbors that are nearer than the
sender to the destination are selected as the candidates, and
the nodes closer to the destination are given higher relay
priorities. In this paper, we mainly focus on this kind of
geographic opportunistic routing (GOR).
Intuitively, giving nodes closer to the destination higher
relay priorities will maximize the expected packet advance-
ment [13]. However it is not always the case to maximize the
throughput, especially when the packet reception ratios from
the sender to the neighbors that make large advancements
are low. Since before relaying the packet, lower-priority can-
didates always need to wait for a certain period of time to
confirm that higher-priority candidates have not relayed the
packet, it will introduce larger latency when higher-priority
candidates are very unlikely to receive the packet correctly.
On the other hand, it is also not a good strategy to in-
volve as many as possible next-hop nodes as candidates.
Although involving more forwarding candidates tends to in-
crease the packet advancement and delivery reliability, the
medium time needed for ensuring only one actual forwarder
to relay the packet is also expected to increase when more
forwarding candidates are involved. So there exists a trade-
off between the medium time [2], which is directly relative to
the throughput, and other performance goals, such as packet
advancement and delivery reliability. This trade-off is not
well studied in the existing works [18, 11, 3].
In this paper, we endeavor to study the impact of can-
didate selection, prioritization and coordination on the dis-
tance-reliability-time trade-off in GOR. We introduce a lo-
cal metric, expected one-hop throughput (EOT), to balance
these factors. We also derive an upper bound of the EOT,
and unveil its concavity, which indicates that the gained
throughput becomes marginal when the number of forward-
ing candidates keeps increasing. Based on EOT, we further
propose a heuristic algorithm to select the forwarding can-
didates and assign relay priority to them. The simulation
results validate our analysis and show that the metric EOT
leads to both higher one-hop and path throughput than the
corresponding pure GOR and geographic routing (GR).
2. PROBLEM FORMULATION
2.1 System Model
Fig. 1 shows an example of GOR. Assume node S, i.e.,
the sender, is forwarding a packet to a destination D, and
s
i
is one of S’s neighbors which are closer to D than S. Let
C be the set of s
i
which is the available next-hop node
set of S, and let N = |C|, which is the number of nodes
in C. Like geographic routing [10, 17, 9, 14], we assume S
is aware of the location information of itself, s
i
’s and D.
Define a
i
in Eq. (1) as the packet advancement toward
the destination when a packet sent by S is relayed by s
i
.
a
i
= d(S, D) − d(s
i
, D) (1)
where d(S, D) and d(s
i
, D) are the Euclidian distances be-
tween S and D and between s
i
and D, respectively.
Without loss of generality, we assume all the nodes in C
are indexed from s
1
to s
N
in descending order according to
the advancement a
i
, i.e., a
m
≥ a
n
, ∀ s
m
, s
n
where m < n.
Each link from S to s
i
is associated to a pair, (a
i
, p
i
), where
p
i
is the data packet reception ratio (PRR) from node S to
s
i
. A node is a neighbor of S when the PRR from S to
it is larger than some non-negligible probability
1
. The PRR
information on each link can be obtained by using probe
messages [4, 8] and is assumed to be independent. Let F
denote the forwarding candidate set of node S, which
includes all the nodes selected to get involved in the local
collaborative forwarding, and r = |F|. Here F is a subset of
C, while in the existing pure opportunistic routing protocols
[18, 3], F = C.
The GOR procedure is as follows: node S selects F based
on its knowledge of C (a
i
’s and p
i
’s); then broadcasts the
data packet to the forwarding candidates in F after detect-
ing the channel is idle. Candidates in F follow a specific
1
In this paper, we set the threshold as 0.1.
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Figure 1: Node S is forwarding a packet to a remote
destination D.
priority to relay the packet, that is, a forwarding candidate
will only relay the packet if it received the packet correctly
and all the no des with higher priorities failed to do so. The
actual forwarder will become a new sender and suppress all
the other potential forwarders in F. When no forwarding
candidate has successfully received the packet, the sender
will retransmit the packet if retransmission is enabled. The
sender will drop the packet when the retransmissions reach
the limit. This procedure reiterates until the packet arrives
at the destination.
2.2 Impact of Candidate Selection, Prioritiza-
tion and Coordination on Throughput
To ensure the relay priority among the forwarding candi-
dates, a MAC protocol similar to those proposed in [18, 6] is
necessary. For example, a feasible MAC protocol could pro-
ceed as following: when the sender decides the F and detects
the channel is idle for a while, it broadcasts the data packet,
in which the intended MAC address of the forwarding can-
didates and their relay priorities are included. To ensure the
candidates to follow the priorities to relay the packet, the
candidate with i
th
priority will wait (i − 1)T
ACK
(T
ACK
is
time needed for transmitting an ACK packet) time before
it sends out the ACK when it received the packet correctly
or keep silent otherwise. Here the ACK message plays two
roles, one is for acknowledgement to the sender, the other is
for suppressing lower-priority candidates. That is, whenever
a lower-priority candidate hears an ACK sent from a higher-
priority candidate, it will suppress itself from relaying the
packet. In our analysis, we assume the relay priority can
be strictly enforced, i.e., for this feasible MAC protocol, the
ACK can be correctly received by the sender and the can-
didates with probability 1. This assumption is reasonable
because typically the ACK packet is small and broadcast at
the basic rate, it is unlikely to be lost and can be transmitted
correctly for a longer range than the data packet.
We define the one-hop medium time consumed by the i
th
candidate as the time slot from the time when the sender is
going to broadcast the packet to the time when the i
th
can-
didate claims it receives the packet. Although the medium
time for locally forwarding a packet varies for different MAC
protocols, for any protocol, it can be divided into two parts.
One part is the sender delay and the other part is candidate
coordination delay, which are defined as follows:
•
T
s
: the sender delay defined in Eq. (2) which can be
further divided into three parts: channel acquisition
time (T
c
), data transmission time (T
d
) and propaga-
tion delay (T
prop
).
T
s
= T
c
+ T
d
+ T
prop
(2)
Parameter Value
Basic Bit Rate (BBR) 1Mbps
Bit Rate (BR) 11Mbps
PHY Header Size (PHS) 192bits
MAC Header Size (MHS) 272bits
T
h
PHS/BBR + MHS/BR
T
ACK
112/BR + PHS/BBR
T
SIF S
10µs
T
DIF S
50µs
Table 1: IEEE 802.11 DSSS PHY Parameter Set
For a contention-based MAC protocol (like 802.11),
T
c
is time needed for the sender to acquire the chan-
nel before it transmits the data packet, which may in-
clude the back-off time, Distributed Interframe Space
(DIFS) and time for transmitting Ready-To-Send (RTS)
packet. T
d
is equal to protocol heads transmission time
plus data payload transmission time, which is
T
d
= T
h
+ T
pl
(3)
T
prop
is the time for the signal propagating from the
sender to the candidates, which can be ignored when
electromagnetic wave is transmitted in the air.
• T
f
(i): the i
th
forwarding candidate coordination de-
lay which is the time needed for the i
th
candidate to
acknowledge the sender and suppress other potential
forwarders. Note that T
f
(i) is an increasing function
of i, since the lower-priority forwarding candidate al-
ways needs to wait and confirm that no higher-priority
candidates have relayed the packet before it takes its
turn to relay the packet.
Thus, the total medium time needed for a packet delivered
from the sender to the i
th
forwarding candidate is
t
i
= T
s
+ T
f
(i) (4)
In the following subsections, we will give examples to illus-
trate how the candidate prioritization, selection and coordi-
nation will affect the expected packet advancement, reliabil-
ity and m edium time cost, which in turn affect the one-hop
throughput.
2.2.1 Impact of Candidate Relay Priority on Through-
put
One factor that will affect the throughput is the candi-
date relay priority. We use the local forwarding example
in Fig.1, and assume a
1
to a
5
is normalized to be 1, 0.8,
0.6, 0.3, and 0.1 respectively and p
1
to p
5
is 0.1, 0.4, 0.55,
0.8, and 0.9 respectively. We use the IEEE 802.11 DSSS
PHY parameter set (in Table 1) to calculate the medium
time cost. Assuming data payload size L
pl
= 512 bytes and
ignoring the propagation delay, T
s
, T
f
(i) and t
i
are:
T
s
= T
DIF S
+ T
h
+ L
pl
/BR = 638 µs
T
f
(i) = (T
ACK
+ T
SIF S
)i = 212i µs
t
i
= 638 + 212i µs
Let’s first assume all the available next-hop neighbors
are involved in the local forwarding and candidates with
larger advancements have higher relay priorities. Assume
the sender sends sufficient large number of packets, N , then
statistically there are p
1
N numb er of packets relayed by can-
didate s
1
with packet advancement of a
1
and the correspond-
ing medium time is t
1
p
1
N. Similarly there are p
2
(1 − p
1
)N
number of packets relayed by s
2
with packet advancement
of a
2
and the corresponding medium time cost of t
2
p
2
(1 −
p
1
)N. If we define the throughput or transport capacity
[7] as the bit-meters successfully transmitted per second.
Then totally, there are L
pl
·
5
i=1
a
i
(p
i
N)
i−1
w=0
(1− p
w
) bit-
meters are successfully transmitted, and the corresponding
medium time cost is
5
i=1
t
i
(p
i
N)
i−1
w=0
(1 − p
w
) + t
5
(N ·
4
w=1
(1 − p
w
)). So from a long term point of view, the
one-hop throughput is 2.16Mbmps. However, if we assume
the forwarding priority as s
2
> s
3
> s
4
> s
5
> s
1
, we
get the one-hop throughput as 2.34M bmps, which is larger
than the previous case. This result contradicts the com-
mon sense that candidates closer to the destination should
relay packets first. The reason behind this result is that
since the largest-advancement candidate has poor link qual-
ity from the sender, in most of the times, it will not receive
the packet correctly, but lower-priority candidates always
have to wait for a period of time to confirm this situation
before they have chances to relay the packet, thus increase
the total medium time cost, which in result degrades the
throughput.
2.2.2 Impact of Candidate Selection on Throughput
Another factor that affects the throughput is the candi-
date selection. Intuitively, different candidate sets with the
same number of forwarding candidates will achieve different
throughput. For example, candidate set $s
1
, s
4
, s
5
% achieves
throughput of 1.28Mbmps, while candidate set $s
2
, s
3
, s
4
%
achieves much higher throughput of 2.35Mbmps. So we
should carefully select forwarding candidates that indeed
help improve the throughput. Furthermore, different num-
ber of forwarding candidates will also result in different
throughput. Actually, candidate set $s
2
, s
3
, s
4
% achieves the
largest throughput among all the candidate combination
and prioritization in this example. When all the available
next-hop nodes are involved as forwarding candidates, the
throughput does not increase while slightly drops. There-
fore, it is unwise to include as many as next-hop neighbors
as candidates. Rather, it may be sufficient to just involve
a few “good” candidates to achieve the maximum one-hop
throughput.
2.2.3 Impact of Candidate Coordination on Through-
put
The third key factor that will affect the throughput is
the candidate coordination delay. Here we use two extreme
cases to illustrate the potential impact of this factor on the
throughput. First, we assume this delay is negligible, that
is, the lower-priority candidates can relay the packet im-
mediately when higher-priority candidates failed to do so.
In this case, we should involve all the available next-hop
neighbors into opportunistic forwarding, because any extra
included candidates would help to improve the relay reliabil-
ity but without introducing any extra delay. We should also
give candidates closer to the destination higher relay priori-
ties, since larger-advancement candidates should always try
first in order to maximize the expected packet advancement,
even if they were unlikely to receive the packet correctly.
If they failed to relay the packet, the lower-priority can-
didates would instantaneously relay the correctly received
packet without needing to wait. On the other hand, if the
candidate coordination delay is very large comparing to the
sender delay, then it is preferable to retransmit the packet
in stead of waiting for other forwarding candidates to relay
the packet. In this case, one candidate may be optimal. So
this factor does affect the throughput, and we will discuss it
in more detail in our analysis and simulation.
2.3 Expected One-hop Throughput (EOT)
According to the analysis above, for a given forwarding
candidate set F, we now propose a new local metric, expected
one-hop throughput (EOT) (in Eq. (5)), to characterize the
local behavior of GOR in terms of bit-meter advancement
per second.
R(F
j
) = L
p
·
r
i=1
a
j
i
p
j
i
·
i−1
w=0
p
j
w
t
r
P
F
+
r
i=1
t
i
p
j
i
·
i−1
w=0
p
j
w
(5)
where F
j
= $s
j
1
, ..., s
j
r
%, which is an ordered set of the nodes
in F with priority s
j
1
> ... > s
j
r
; p
j
0
:= 0;
p
j
w
= 1 − p
j
w
;
and
P
F
=
r
i=1
(1 − p
i
)
(6)
which is the probability that none of the forwarding candi-
dates in F has successfully received the packet in one phys-
ical transmission from the sender.
The physical meaning of the EOT defined in Eq. (5) is the
expected bit advancement per second for a local GOR pro-
cedure. EOT integrates the packet advancement, relay reli-
ability, and MAC medium time cost. The intuitions to max-
imize EOT are as following: 1) as the whole path achievable
throughput is less than per-hop throughput on each link,
to maximize the local EOT is likely to increase the path
throughput; 2) the path delay is the summation of per-hop
delay, which is actually relative to the delay introduced by
transmitting the packet and coordinating the candidates. As
the per-hop delay factors (T
s
and T
f
(i)) are integrated in the
denominators of EOT, to maximize EOT is also implicitly
to decrease per-hop delay, which may further decrease the
path delay. 3) as EOT also takes into account the packet
advancement to the destination, maximizing it potentially
decreases hop counts needed to relay the packet to the des-
tination, which may lead to fewer transmissions, alleviated
interference to other flows, and decreased delay.
In the following sections, we will examine the behavior
of GOR by identifying an upper bound of the EOT and the
concavity of the maximum EOT. After that, we will prop ose
a heuristic algorithm to select the forwarding candidates and
assign the relay priority to approach an optimal EOT.
3. UPPER BOUND OF EOT AND ITS CON-
CAVITY
This section studies the performance of GOR in terms of
the EOT and we derive an upper bound of EOT.
3.1 Upper Bound of EOT
Lemma 1 introduces an upper bound of EOT as follows:
Lemma 1. Given a forwarding candidate set F, the EOT
defined in Eq. (5) is upper bounded by R
∗
defined as follows:
R
∗
= L
pl
·
r
i=1
a
i
p
i
·
i−1
w=0
p
w
T
s
(7)
Note that F is indexed according to the advancement s.t.
a
m
≥ a
n
, ∀ m < n.
Proof.
The minimum value of the denominator of Eq. (5)
is obtained when t
i
= T
s
, i.e. T
f
(i) = 0. Denote the numer-
ator of Eq. (5) as
g(F
j
) =
r
i=1
a
j
i
p
j
i
·
i−1
w=0
p
j
w
(8)
Now it is sufficient to prove that for any ordered candi-
date set F
j
, we have g(F
j
) ≤ g(F). This is equivalent to
prove that the m aximum g(F
j
) is obtained by prioritizing
the forwarding candidates according to the advancement a
j
,
s.t. a
m
≥ a
n
∀ m < n. We prove this by induction on r,
i.e., the size of F.
First, for r = 1, obviously g(F
j
) ≤ g(F).
Next, we assume g(F
j
) ≤ g(F) holds for r = M (M≥1),
we want to prove it holds for r = M+1.
For r = M+1. F can be divided into two complementary
sub-sets, A = F \ {s
m
} with M nodes and B = {s
m
} with 1
node. Then
g(F
j
) = g(A
j
) +
P
A
· g($s
m
%)
≤ R := g(A) + P
A
· g($s
m
%)
The first equality holds for the definitions of g(F
j
) and
the second inequality holds for the inductive hypothesis. So
it suffices to prove ∀ m (1 ≤ m ≤ M ), we have R ≤ g(F).
This can be proved as follows:
g(F) − R =
1
p
m
M+1
k=m+1
(a
m
− a
k
)p
m
p
k
k−1
w=0
p
w
≥ 0
where p
0
:= 1.
The inequality holds as a
m
≥ a
k
∀ m < k. So Eq. (7) is
an upper bound of Eq. (5) for any given F .
Lemma 1 basically shows that under some idealized MAC
scheduling, where the coordination delay among the for-
warding candidates is negligible, the maximum EOT can
be achieved by giving candidates closer to the destination
higher relay priorities.
3.2 Concavity of the Upper Bound of EOT
Lemma 1 gives the upper bound of EOT and the corre-
sponding relay priority rule when F is given. The followed
question is how the upper bound changes for different set of
F. We answer this question and unveil the concavity of the
upper bound of EOT in Theorem 2.
Theorem 2.
Given the available next-hop node set C with
N (N ≥ 1 ) nodes, define R
∗
(r ) as the upper bound of the
EOT by selecting any r candidates, then R
∗
(r) is an in-
creasing and concave function of r.
Proof.
2
Denote F
∗
r
as the feasible candidate set that
achieves R
∗
(r) . According to Eq. (7) and (8),
R
∗
(r) = L
pl
·
g(F
∗
r
)
T
s
(9)
Then it suffices to prove g(F
∗
r
) is an increasing and con-
cave function. It’s not difficult to see that
g(F
∗
r+1
) ≥ g($F
∗
r
, s
m
%) > g(F
∗
r
) (10)
where s
m
∈ C and s
m
/∈ F
∗
r
.
2
Due to space limit, we only provide a sketch of the proof.
To prove the concavity of g(F
∗
r
), we first proved that ∀
F
∗
r−1
, ∃ F
∗
r
, s.t.
F
∗
r−1
⊂ F
∗
r
∀ 1 ≤ r ≤ N (11)
Then according to the containing
3
property, we assume F
∗
r+1
\
F
∗
r
= {s
k
}, and F
∗
r
\ F
∗
r−1
= {s
j
}. There are two cases for
the advancement relationship between node s
k
and s
j
.
1) a
k
> a
j
. Then F
∗
r+1
, F
∗
r
and F
∗
r−1
can be represented
as
F
∗
r+1
= $A
1
, s
k
, A
2
, s
j
, A
3
%, F
∗
r
= $A
1
, A
2
, s
j
, A
3
%,
F
∗
r−1
= $A
1
, A
2
, A
3
%
where A
i
(1 ≤ i ≤ 3) is ordered node set and can be ∅. With
B := g(F
∗
r
) − g($A
1
, s
k
, A
2
, A
3
%) ≥ 0
(12)
we have
[g(F
∗
r
) − g(F
∗
r−1
)] − [ g(F
∗
r+1
) − g(F
∗
r
)]
= B + (1 − p
A
1
)(1 − p
A
2
)p
k
p
j
(a
j
− g(A
3
)) > 0
(13)
where p
A
i
is the probability of at least one node in A
i
re-
ceives the packet correctly.
Inequality (13) holds because B ≥ 0 (inequality (12)) and
a
j
− g(A
3
) > 0.
2) a
k
< a
j
. Similarly,
F
∗
r+1
= $A
1
, s
j
, A
2
, s
k
, A
3
%, F
∗
r
= $A
1
, s
j
, A
2
, A
3
%,
F
∗
r−1
= $A
1
, A
2
, A
3
%
With
B := g(F
∗
r
) − g($A
1
, A
2
, s
k
, A
3
%) ≥ 0
(14)
we have
[g(F
∗
r
) − g(F
∗
r−1
)] − [ g(F
∗
r+1
) − g(F
∗
r
)]
= B + (1 − p
A
1
)(1 − p
A
2
)p
k
p
j
(a
k
− g(A
3
)) > 0
(15)
From the analysis above, we know R
∗
(r ) is an increasing
and concave function of r.
Theorem 2 indicates that even if the coordination delay
among the forwarding candidates were negligible, the gained
throughput by increasing the number of the forwarding can-
didates would become marginal. So it may only need to
involve a small number of forwarding candidates to achieve
the best EOT.
4. HEURISTIC CANDIDATE SELECTION
ALGORITHM
A straightforward way to get the optimal F and the cor-
responding F
j
to maximize the EOT is to try all the ordered
subset of C, which runs in Ω(N !) time, where N is the num-
ber of available next-hop nodes. It is, however, not feasible
when N is large. In this section, we propose a heuristic
candidate selection and prioritization algorithm to get a so-
lution approaching the optimal EOT.
By observing Eq. (5), we can find that the candidate
achieving the maximum EOT by selecting 1 node from C
is contained in at least one feasible candidate set achieving
the maximum EOT by selecting r (1 ≤ r ≤ |C|) nodes from
3
In this paper, an ordered node set A containing another
ordered node set B means A is obtained by inserting a new
node into B but keeping the priority relationship of nodes in
B unchanged. It’s not necessary for B being a subsequence
of A.
GetMEOT(C)
1 F
m
← ∅; R
m
← 0; A ← C − F
m
;
2 while (A ,= ∅) do
3 F ← F
m
;
4 for each node s
n
∈ A
5 for i from 0 to |F
m
|
6 F
t
← Insert s
n
between F (i) and F (i + 1);
7 Get R on F
t
according to Eq. (5);
8 if (R > R
m
)
9 R
m
← R; F
m
← F
t
10 end for
11 end for
12 A ← C − F
m
;
13 end while
14 return(F
m
, R
m
);
Table 2: Pseudocode of finding an ordered candi-
date set F
m
, and the corresponding R
m
for a given
available next-hop set C
C. Because if it were not the case, we could always sub-
stitute the lowest-priority node in the optimal set (with r
nodes) to get another new candidate set which achieves an
EOT no smaller than that of the optimal set, which is a
contradiction. Then, we propose the algorithm GetMEOT
in Table 2 which finds an F based on this observation. This
algorithm greedily adds a new node into the current opti-
mal/suboptimal F containing r nodes without changing the
priorities among the r nodes to get an optimal/suboptimal
F with r +1 nodes. Finally, the candidate set with the max-
imum EOT is returned. This algorithm runs in O(|C|
3
). An
interesting result is that this algorithm almost surely finds
the actual global optimal F in our simulation.
5. PERFORMANCE EVALUATION
We validate the concavity of the upper bound of EOT
and evaluate the one-hop performance as well as the path
performance of GOR that applies the GetMEOT algorithm
by simulation. We compare the GOR with the geographic
routing which selects one neighbor with maximum a
j
p
j
[10,
9], and the pure opportunistic routing which involves all the
available next-hop nodes with nodes closer to the destination
having higher relay priorities.
5.1 Simulation Setup
We assume T
s
= T
backof f
+ T
DIF S
+ T
h
+ L
pl
/BR and
T
f
(i) = (T
ACK
+ T
SIF S
)i, which are calculated accord-
ing to Table 1, by assuming L
pl
= 512bytes. The simu-
lated network has stationary nodes uniformly distributed in
a 1200 × 1200 m
2
square region with nodes having identical
transmission power of 15dbm. The source and the desti-
nation nodes are fixed at two corners across the diagonal
of the square area. We also assume an ideal collision-free
MAC such that packet loss is only due to the randomness
of link quality, and at any time there is only one transmis-
sion scheduled. The results are averaged from 200 runs, and
in each run, there are 2000 packets delivered to the desti-
nation. To investigate the impact of node density on the
performance of these routing schemes, we vary the number
of nodes as 35, 50, 80, 100, which corresponds to different
node densities as 11, 16, 22, 34 neighbors per node.
We use the Nakagami distribution [12] to describe the
