IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1
Cross-Layer Aided Energy-Efficient
Opportunistic Routing in Ad Hoc Networks
Jing Zuo, Chen Dong, Hung Viet Nguyen, Soon Xin Ng, Lie-Liang Yang, and Lajos Hanzo
Abstract—Most of the nodes in ad hoc networks rely on
batteries, which requires energy saving. Hence, numerous energy-
efficient routing algorithms have been proposed for solving this
problem. In this paper, we exploit the benefits of cross-layer
information exchange, such as the knowledge of the Frame Error
Rate (FER) in the physical layer, the maximum number of
retransmissions in the Medium Access Control (MAC) layer and
the number of relays in the network layer. Energy-consumption-
based Objective Functions (OF) are invoked for calculating the
end-to-end energy consumption of each potentially available
route for both Traditional Routing (TR) and for our novel
Opportunistic Routing (OR), respectively. We also improve the
TR and the OR with the aid of efficient Power Allocation
(PA) for further reducing the energy consumption. For the TR,
we take into account the dependencies amongst the links of a
multi-hop route, which facilitates a more accurate performance
evaluation than upon assuming the links that are independent.
Moreover, two energy-efficient routing algorithms are designed
based on Dijkstra’s algorithm. The algorithms based on the
energy OF provide the theoretical bounds, which are shown to
be close to the bound found from exhaustive search, despite the
significantly reduced complexity of the former. Finally, the end-
to-end throughput and the end-to-end delay of this system are
analyzed theoretically and a new technique of characterizing the
delay distribution of OR is proposed. The simulation results show
that our energy-efficient OR outperforms the TR and that their
theoretical analysis accurately matches the simulation results.
Index Terms—Opportunistic routing, energy-efficient routing,
cross-layer, objective function, near-capacity coding, end-to-end
throughput, end-to-end delay.
I. INTRODUCTION
E
NERGY saving in wireless ad hoc networks is a salient
problem, which mitigates the problem of limited battery
supply at each node. Numerous energy-efficient algorithms
have been proposed for reducing the energy consumption [1–
10]. The authors of [5, 6] have aimed for energy saving without
considering the specifics of the network layer, the Medium
Access Control (MAC) layer or the physical layer. By contrast,
the authors of [1, 4, 7–13] invoked cross-layer optimization,
since the energy reduction is related to several layers. The
authors of [2, 5] conceived energy-efficient routing concepts.
Traditional Routing (TR) relies on a route discovery pro-
cess invoked for gleaning sufficient routing information for
Manuscript received October 9, 2012; revised March 21, August 9, and
October 17, 2013. The editor coordinating the review of this paper and
approving it for publication was V. Wong.
The authors are with the School of ECS, University of Southampton, SO17
1BJ, UK (e-mail: {jz08r, cd2g09, hvn08r, sxn, lly, lh}@ecs.soton.ac.uk).
The research leading to these results has received funding from the Euro-
pean Union’s Seventh Framework Programme (FP7/2012-2014) under grant
agreement no 288502. The financial support of the China-UK Scholarship
Council, and of the RC-UK under the auspices of the IU-ATC initiative, is
also gratefully ackno wledged.
Digital Object Identifier 10.1109/TCOMM.2013.121413.120767
the source to make meritorious routing decisions, regardless,
whether the routing protocol is proactive or reactive [14].
However, due to the rapid fluctuation of the channel con-
ditions, the routing information estimated on the basis of
the average Channel Quality Information (CQI) may become
stale, resulting in suboptimum routing. Therefore, Opportunis-
tic Routing (OR) [2–4, 8, 11, 13, 15–20] has been proposed
for avoiding this problem. In OR no pre-selected route is
employed, instead a so-called forwarder relay set is used for
forwarding the packets along a beneficial route. The near-
instantaneously varying characteristics of wireless channels
is beneficially explo ited considered by OR. Liu et al. [16]
illustrated the basic idea behind OR and categorized the
potential design criteria, including the Estimated Transmission
count (ETX), the geographic distance aided and the energy
consumption based philosophies. Biswas and Morris [11] pro-
posed an Extremely Opportunistic Routing (ExOR) scheme,
which employed the ETX metric at the destination for deciding
the priority order of selecting a relay from the potential for-
warder set. The proposed routing regime integrated the routing
protocol and the MAC protocol for the sake of increasing the
attainable throughput of multi-hop wireless networks. Their
solution [11] also exploited the less reliable long-distance
links, which would have been ignored by traditional routing
protocols. Moreover, Dubois-Ferri`ere et al. [17] conceived the
Least-Cost Anypath Routing (LCAR) regime, which finds the
optimal choice of candidate relays relying on the expected
ETX cost of forwarding a packet to the destination. This
LCAR algorithm considers the coordination of the link layer
protocols. Laufer et al. [20] proposed a ‘polynomial-time
multirate anypath’ routing algorithm and provided the proof of
its optimality. The proposed routing algorithm employed the
Expected Anypath Transmission Time (EATT) as the routing
metric, which is a generalization of the unidirectional ETX
metric that takes into account that nodes transmit at multiple
bit rates. The authors of [2, 15, 19] employed a geographic
distance based metric for choosing the potential forwarder
relay set. More specifically, Zorzi and Rao [15] proposed an
OR scheme based on random forwarding, where the specific
node, which is closest to the Destination (D) is chosen as
the Relay (R) for the n ext hop. This paper theoretically
analyzed the achievable multi-hop perfo rmance. Furthermore,
Zorzi and Rao [2] analyzed the achievable energy as well
as latency performance and provided a detailed descr iption
of a MAC scheme based on both opportunistic concepts and
on collision avoidance. Zeng et al. [19] proposed a m ultirate
OR by incorporating rate adaptation into their candidate-
selection algorithm, which was shown to achieve a higher
throughput and lower delay than the corresponding traditional
0090-6778/13$31.00
c
2013 IEEE
2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTE D FOR PUBLICATION
single-rate routing and its opportunistic single-rate routing
counterpart. The authors of [3, 4, 8] employed the energy
consumption metric for choosing the potential forwarder relay
set. More concretely, Mao et al. [3] presented an energy-
efficient OR strategy relying on sophisticated power allocation,
which prioritizes the forwarder relays by directly minimizing
the total energy consumption of all nodes. Dehghan et al. [4]
developed a minimum-energy cooperative routing based on
many-to-many cooperation and determines the optimal route
with the aid of the Bellman-Ford algorithm [21]. Wei et
al. [8] proposed an energy-conserving Assistant Opportunistic
Routing (AsOR) protocol, which classified a sequence of
nodes into three different node sets, namely, the frame node,
the assistant node and the unselected node. The frame nodes
were indispensable for decode-and-forward operation, while
the assistant nodes provided protection against unsuccessful
opportunistic transmissions. Although the authors of [3, 4, 8]
employed the energy consumption as their routing metric, they
have not provided any theoretical bounds in their performance
analysis. Moreover, these authors assumed that the number
of affordable MAC retransmissions was infinite. Against this
background, our novel contributions are:
• Two accurate energy-consumption-based OFs are con-
structed, which are used for the TR and the OR respec-
tively. We exploit the knowledge of both the Frame Error
Ratio (FER) within the physical layer, and of the number
of MAC retransmissions as well as of the number of relays
in the network layer.
• A routing algorithm is designed for the TR, which
employs our energy-consumption-based OF. Similarly, a
routing algorithm is designed for OR, which employs our
energy-consumption-based OF for ordering the relays
(from small to large) in the forwarder set. Theoretical
bounds are derived for the Normalized Energy Consump-
tion (NEC) of both the algorithms, which are shown to
be close to the ultimate bound obtained with the aid of
an exhaustive search.
• The achievable end-to-end throughput, the end-to-end
delay and the delay distribution of the system are also
evaluated theoretically.
The rest of the paper is organized as follows. Section II
theoretically analyzes the performance of the system for the
single-hop route, for the TR and for the OR. Section III
describes our energy-efficient routing algorithms conceived for
TR and OR, respectively. The delay distribution of OR is also
analyzed. Finally, Section IV analyzes the overall performance
of the system, while Section V provides our conclusions.
II. T
HEORETICAL ANALY S I S
In this paper, the transmit energy consumed by the data
packets during their transmission is considered under the
idealized simplifying assumption that the energy dissipated by
other packets, such as the routing and MAC control packets, is
negligible. Before defining the proposed energy-consumption-
based OF, the symbols used are defined.
• H: the number of hops in an established route;
• P
t
i
: the transmit power in the i-th node of the established
route;
• FER
i
: the FER of the i- th link in an established route;
• p
i
: the successful probability of the i-th link, where p
i
=
1 − FER
i
;
• N
r
: the maximum number of MAC retransmissions,
including the first transmission attempt;
• E
total
: the total energy consumption;
• E
total
:
A. FER and power allocation in a single-hop route
In our previous work [22], an accurate energy-consumption-
based OF was employed for estimating the normalized end-
to-end energy consumption for a given route under the as-
sumption that the FER, the maximum number of MAC re-
transmissions and the number of hops were known. Although
the energy-consumption-based OF representing the real-world
scenarios is beneficial, the re sultant best route still wastes
energy, since the distances between the different pairs of relays
are different. For the sake o f ensuring that the total power-
consumption is minimized, each node’s transmit power should
be different. Therefore, we invoke power control for further
reducing the energy consumption by finding the optimum
transmit power for each node.
Naturally, the channel conditions, the thermal noise level
and the distance between the transmitter and the receiver
jointly determine the FER of a link. We conducted simulations
for characterizing the FER performance versus the Signal-
to-Noise-Ratio (SNR) and followed the approach of [12]
for fitting a polynomial to the FER versus SNR curve. The
Forward Error Correction (FEC) scheme employed in our
paper is an Irregular Convolutional Coded, Unity-Rate Coded
and Quadrature Phase-Shift Keying (IrCC-URC-QPSK) [23,
24] scheme. The corresponding FER versus SNR curve was
generated with the aid of bit-by-bit simulations. The overall
FEC code rate was R
c
=0.5, the effective throughput was 1
bps (bits/symbol), the frame length was 8688 bits, the number
of transmitted frames was 10 000. The IrCC had 17 component
codes, associated with the weights [0.049, 0, 0, 0, 0, 0.24, 0.16,
0.12, 0.035, 0.102, 0, 0.071, 0.093, 0, 0.091, 0, 0 .039]
1
.We
generated the FER curve for the AWGN channel model with
the aid of simulation. According to the approach of [12], this
will allow us to determine the average FER for arbitrary fading
channels upon weighting the AWGN-FER by the PDF of the
fading channel and averaging it over the legitimate dynamic
range. More specifically, the channel model considered is
the uncorrelated, non-dispersive Rayleigh fading channel. The
average FER expression FER
Rayleigh
is determined for the
Rayleigh fading channel considered by integrating the specific
FER
AW GN
value of the AWGN channel experienced at a
given SNR af ter weighting it by the p robability of that specific
SNR, which is given by:
FER
Rayleigh
=
∞
0
e
−γ
FER
AW GN
(γ)dγ, (1)
1
The 17 c oding coefficients α
i
, i =[1, 2, ..., 17], are the 17 coding frac-
tions of the 17 corresponding component codes (subcode), the i
th
of which
having a code rate β
i
encodes the fraction α
i
of the input bit stream, where
we have [β
1
=0.1,β
2
=0.15,β
3
=0.2,β
4
=0.25,β
5
=0.3,β
6
=
0.35,β
7
=0.4,β
8
=0.45,β
9
=0.5,β
10
=0.55,β
11
=0.6,β
12
=
0.65,β
13
=0.7,β
14
=0.75,β
15
=0.8,β
16
=0.85,β
17
=0.9]. Hence,
the constraint of R
c
=
17
i=1
α
i
β
i
=0.5 is always satisfied.
ZUO et al.: CROSS-LAYER AIDED ENERGY-EFFICIENT OPPORTUNISTIC ROUTING IN AD HOC NET WORKS 3
where γ is the channel SNR, e
−γ
represents the Rayleigh
channel while the FER
AW GN
(γ) versus the SNR curve (ob-
tained by off-line simulation) is approximated by the following
four-segment FER vs SNR model representing the AWGN
channel:
FER
AW GN
(γ) ≈
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1,if0 ≤ γ<η
1
,
10a
1
log(γ)+a
2
,ifη
1
≤ γ<η
2
,
10a
3
log(γ)+a
4
,ifη
2
≤ γ<η
3
,
a
5
e
−10a
6
log(γ)
,ifγ ≥ η
3
,
(2)
with η
1
, η
2
and η
3
being the break-points of the four-segment
FER versus SNR approximation FER
AW GN
(γ). Eq. (1) and
Eq. (2) are suitable for approximating different FER curves
by appropriately setting the corresponding parameter values
invoked. Specifically, for the IrCC-URC-QPSK scheme [23]
employed, we have a
1
= −0.5889, a
2
=1.3341, a
3
=
−3.705, a
4
=3.5169, a
5
=4.4669 × 10
6
and a
6
=18.9118.
Additionally, the values of the break-points η
1
, η
2
and η
3
are
determined for the SNR points of 0.6 dB, 0.7 dB and 0.9 dB,
whose relationships are given by:
η
1
=
10
0.6
10
d
α
N
0
(4π)
2
P
t
1
λ
2
(3)
η
2
=
10
0.7
10
d
α
N
0
(4π)
2
P
t
1
λ
2
(4)
η
3
=
10
0.9
10
d
α
N
0
(4π)
2
P
t
1
λ
2
, (5)
where λ is the wavelength of light, d is the distance between
the transmitter and the receiver, N
0
is the thermal noise power
and α is the path-loss exponent. In this paper, we set α =2.
Substituting our FER versus SNR model of Eq. (2) asso-
ciated with the above-mentioned parameters into Eq. (1), we
have the following results:
• When 0 ≤ γ<η
1
,wehave
FER
I
=
η
1
0
e
−γ
dγ =1− e
−η
1
=1 − e
−
10
0.6
10
d
α
N
0
(4π)
2
P
t
1
λ
2
; (6)
• When η
1
≤ γ<η
2
,wehave
FER
II
=
η
2
η
1
(10a
1
log(γ)+a
2
)e
−γ
dγ
=a
2
(e
−η
1
− e
−η
2
)
+ a
1
η
2
η
1
10
ln 10
ln
P
t
1
λ
2
(4π)
2
d
α
γ
e
−γ
dγ; (7)
After carrying out the integration with the aid of the Euler
function of Ei(x)=
∞
−x
e
−t
t
dt [25] (8.211.1), we arrive
at:
FER
II
=a
2
(e
−η
1
− e
−η
2
)+0.6a
1
e
−η
1
− 0.7a
1
e
−η
2
+ a
1
10
ln 10
[Ei(−η
2
) − Ei(−η
1
)] ; (8)
• When η
2
≤ γ<η
3
, we have an expression similar to
Eq. (8):
FER
III
=
η
3
η
2
(10a
3
log(γ)+a
4
)e
−γ
dγ
=a
4
(e
−η
2
− e
−η
3
)+0.7a
3
e
−η
2
− 0.9a
3
e
−η
3
+ a
3
10
ln 10
[Ei(−η
3
) − Ei(−η
2
)] ; (9)
• Finally, for γ ≥ η
3
,wehave
FER
IV
=
∞
η
3
a
5
e
(−10a
6
log(γ))
e
−γ
dγ
= a
5
∞
η
3
e
−γ
e
ln
P
t
1
λ
2
(4π)
2
d
α
γ
10b
ln 10
dγ
=10
−0.9a
6
ln 10
a
5
η
3
G
2,0
1,2
η
3
10a
6
ln 10
10a
6
ln 10
− 1, 0
,
(10)
where the Meijer-G function is defined in [25] (9.301)
and we have G
2,0
1,2
x
ν
ν − 1, 0
= E
ν
(x)=
∞
1
e
−xt
t
ν
dt [26] (06.34.02.001.01).
However, the FER formula derived above does not consider
the effects of retransmissions in the MAC layer, neither does it
take into account the number of hops in the network layer. In a
realistic scenario, however, we have poor channel conditions,
a high level of interference, the effects of node mobility,
potential network congestions and so on, where the packets
are often dropped before reaching the destination. However,
the dropped packets consume a high amount of energy during
their passage through the network. Therefore, we analyze the
Normalized Energy Consumption (NEC) during a packet’s
passage from the source to the destination. We have to
consider two scenarios, namely the energy consumption E
s
when a packet is delivered successfully to the destination
and that given by E
f
when it is dropped before reaching the
destination.
The performance of a single-hop route is analyzed first,
where p
1
is the successful probability of the first hop. Then,
the probability p
s
that a packet is successfully delivered from
the Source (S) to the destination (D) within the maximum N
r
number of retransmissions is [22]
p
s
=
N
r
i
1
=1
(1 − p
1
)
i
1
−1
p
1
. (11)
By contrast, the probability p
f
that a packet is dropped before
reaching its destination is [22]
p
f
=(1− p
1
)
N
r
. (12)
Hence, the energy E
s
required for the successful transmission
of a packet and that dissipated during the transmission of a
failed packet, namely E
f
, are respectively given by
E
s
=
N
r
i
1
=1
(1 − p
1
)
i
1
−1
p
1
i
1
P
t
1
T, (13)
E
f
=(1− p
1
)
N
r
N
r
P
t
1
T, (14)
4 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTE D FOR PUBLICATION
where T is the duration of a time slot.
Consequently, the average total energy E
total
required for
transmitting a packet is E
total
= E
s
+ E
f
. Furthermore, the
total ene rgy E
total
norma lized by the successful probability
p
s
, which is the energy dissipated by the whole system during
the successful delivery of a packet to D, can be expressed as
E
total
=
E
total
p
s
=
E
s
+ E
f
p
s
. (15)
When substituting Eqs. (11), (13) and (14) into Eq. (15),
we arrive at:
E
total
=
P
t
1
p
1
T, (16)
which shows that
E
total
is independent of the number of
retransmissions in a single-hop route. Moreover, Eqs. (6), (7),
(9) and (10) illustrate the relationship between P
t
1
and p
1
.
Then,
E
total
only depends on the distance between S and D.
Therefore, optimizing the transmit power of the S may be
formulated as a convex optimization problem.
Let us set the derivative of Eq. (16) with respect to P
t
1
to
zero, yielding
1
p
1
+
P
t
1
p
2
1
d(1 − p
1
)
dP
t
1
=0
p
1
−P
t
1
=
d(1 − p
1
)
dP
t
1
, (17)
Upon manipulating Eq. (17) further by setting the derivatives
of the four parts of the four-segment approximated FER versus
SNR curve detailed in Eqs. (6), (7), (9) and (10), we arrive
at
p
1
−P
t
1
=
dF ER
1,I
dP
t
1
+
dF ER
1,II
dP
t
1
+
dF ER
1,III
dP
t
1
+
dF ER
1,IV
dP
t
1
. (18)
To elaborate a little further, based on Eq. (6), we have
dF ER
1,I
dP
t
1
=
d(1 − e
−η
1
)
dP
t
1
= −
10
0.6
10
(4π)
2
d
α
N
0
e
−
10
0.6
10
(4π)
2
d
α
N
0
P
t
1
λ
2
P
2
t
1
λ
2
. (19)
While based o n Eq. (7), we arrive at Eq. (27), where
dEi(x)
dx
=
e
x
x
[25] (8.211.1). It can be readily shown that
dF ER
1,III
dP
t
1
obeys an expression similar to Eq. (27), which is
formulated in Eq. (28).
Finally, based on Eq. (10), the 4th part at the righthand
side of Eq. (18) can be expressed as Eq. (29), where the
differentiation of Meijer ’s G function was taken from [26]
(07.34.20.0005.01). Eqs. (27), (28) and (29) are shown on
the top of next page.
Finally, when substituting Eqs. (19), (27), (28) and (29)
into Eq. (18), the optimized transmit power P
t
1
can be found.
R
2
DR
1
···
SR
H−1
Fig. 1. Test-topology having one source, one destination and (H − 1) relay
nodes.
B. Theoretical analysis of TR in an idealized network
The en e rgy consumption of an idealized multi-hop route is
analyzed in this subsection. The network topology is shown
in Fig. 1.
In Fig. 1, we have a single S, a single D and (H − 1)
Rs. The (H − 1) Rs are located between S and D.Our
previous contribution [22] an alyzed both the probability and
the total energy consumption of a packet, when it is delivered
successfully to D or when it is dropped before reaching D of
Fig. 1. However, in [22] we assumed that the transmit power
of all nodes is the same, which wasted some energy in the
realistic scenario, when the distances between each pair of
nodes was different. If the optimal distance-dependent transmit
power is found, then the NEC may be further reduced.
The probabilities p
s
and p
f
are the same as those in [22],
namely
p
s
=
N
r
i
1
=1
···
N
r
i
H
=1
(1 − p
1
)
i
1
−1
p
1
(1 − p
2
)
i
2
−1
p
2
···(1 − p
H
)
i
H
−1
p
H
, (20)
p
f
=p
f
(1) +
H
h=2
p
f
(h), (21)
where p
f
(1) is given by Eq. (12) and p
f
(h) is the probability
that a packet is dropped at the h-th hop, which is formulated
as:
p
f
(h)=
N
r
i
1
=1
···
N
r
i
h−1
=1
(1 − p
1
)
i
1
−1
p
1
···
(1 − p
h−1
)
i
h−1
−1
p
h−1
(1 − p
h
)
N
r
,h=1. (22)
Let E
i
be the energy required by node i to send a packet,
where E
i
= P
t
i
T . Then, we can show that E
s
and E
f
are
formulated as:
E
s
=
N
r
i
1
=1
···
N
r
i
H
=1
(1 − p
1
)
i
1
−1
p
1
(1 − p
2
)
i
2
−1
p
2
···(1 − p
H
)
i
H
−1
p
H
(i
1
E
1
+ i
2
E
2
+ ···+ i
H
E
H
),
(23)
E
f
=E
f
(1) +
H
h=2
E
f
(h)
=N
r
E
1
(1 − p
1
)
N
r
+
H
h=2
N
r
i
1
=1
···
N
r
i
h−1
=1
(1 − p
1
)
i
1
−1
p
1
···(1 − p
h−1
)
i
h−1
−1
p
h−1
(1 − p
h
)
N
r
(i
1
E
1
+ ···+ i
h−1
E
h−1
+ N
r
E
h
)
. (24)
Let D
s
denote the average time required for delivering
a packet successfully to D, where the source of delay is
assumed to be the Automatic Repeat reQuest (ARQ)-aided
ZUO et al.: CROSS-LAYER AIDED ENERGY-EFFICIENT OPPORTUNISTIC ROUTING IN AD HOC NET WORKS 5
retransmissions. E xplicitly, each new hop and new transmis-
sion attempt imposes a delay of unity, i.e. one time-slot
(TS). Furthermore, let D
f
be the average delay imposed on a
packet’s passage through the route, when it is dropped before
reaching its destination. Hence, the total delay is represented
as D
total
= D
s
+ D
f
. These average delays have been
quantified in [22], which are
D
s
=
N
r
i
1
=1
···
N
r
i
H
=1
(1 − p
1
)
i
1
−1
p
1
(1 − p
2
)
i
2
−1
p
2
···(1 − p
H
)
i
H
−1
p
H
(i
1
+ i
2
+ ···+ i
H
)
T, (25)
D
f
=p
f
(1)N
r
T +
H
h=2
D
f
(h), (26)
where D
f
(h) is the average delay experienced by a packet,
which is dropped during the h-th hop, expressed as
D
f
(h)=
N
r
i
1
=1
···
N
r
i
h−1
=1
(1 − p
1
)
i
1
−1
p
1
···
(1 − p
h−1
)
i
h−1
−1
p
h−1
(1 − p
h
)
N
r
(i
1
+ ···+ i
h−1
+ N
r
)
T,h =1. (30)
For the sake of simplifying Eq. (23), (24), (25) and
(26), we define A(p
i
)=(
1−(1−p
i
)
N
r
p
i
− N
r
(1 − p
i
)
N
r
)E
i
and
B(p
i
)=1− (1 − p
i
)
N
r
. Then, we have p
s
=
H
1
B(p
i
).
Furthermore, it may be readily shown that E
s
and E
f
can be
formulated alternatively as:
E
s
=
H
i=1
B(p
i
)
H
i=1
A(p
i
)
B(p
i
)
, (31)
E
f
=
H
h=2
h−1
i=1
B(p
i
)
h−1
i=1
A(p
i
)
B(p
i
)
[1 − B(p
h
)]
+ N
r
E
h
(1 − B(p
h
))
h−1
i=1
B(p
i
)
+ N
r
E
1
[1 − B(p
1
)] .
(32)
Similarly, upon defining C(p
i
)=(
1−(1−p
i
)
N
r
p
i
− N
r
(1 −
p
i
)
N
r
)T , we can express D
s
and D
f
as
D
s
=
H
i=1
B(p
i
)
H
i=1
C(p
i
)
B(p
i
)
, (33)
D
f
=
H
h=2
h−1
i=1
B(p
i
)
h−1
i=1
C(p
i
)
B(p
i
)
[1 − B(p
h
)]
+ N
r
T [1 − B(p
h
)]
h−1
i=1
B(p
i
)
+ N
r
T [1 − B(p
1
)] .
(34)
Based on the above derivation, the NEC expressed as
E
total
=
E
total
p
s
=
E
s
+E
f
p
s
can now be evaluated.
S
R
1
R
2
D
······
R
M
R
M −1
Fig. 2. A two-hop network assisted by a number of relays.
The end-to-end delay D
e2e
is given by
D
e2e
= D
s
, (35)
which represents the delay experienced by a p acket that is
successfully delivered to the destination. Moreover, the end-
to-end throughput R
e2e
is given by
R
e2e
=
p
s
D
total
=
p
s
D
s
+ D
f
. (36)
C. Theoretical analysis of OR in a random network
The TR transmits the packet along the specific pre-selected
route having the lowest estimated NEC. This pre-selected
route is determined after the estimation and comparison of
the NEC of each potential candidate route. The information
invoked for routing decisions is gleaned during the process of
route discovery, but this information may become stale owing
to node-mobility. Instead, OR considers the potential chances
of success for each candidate relay, bearing in mind their time-
variant channel conditions. Regardless of which particular
relay receives the packet from the source successfully, if this
relay has the highest priority in the forwarder relay list, it will
forward the packet to the next relay. Naturally, the challenge
in the design of the OR procedure is the beneficial selection of
the forwarder R-set, the specific priority order of the potential
forwarders and the avoidance of duplicate transmissions [16].
We assume that all the nodes in a node’s neighbor list belong
to this node’s forwarder R-list. The metric used for determin-
ing the priority order is the normalized energy required by
this particular relay for reaching D. Acknowledgement (ACK)
packets are employed for avoiding the duplicate transmissions.
The particular relay in the forwarder R-set, which has the
highest priority owing to requiring the lowest energy will send
the ACK first. The other relays, which overhear the ACK will
withdraw from the competitio n [27, 28].
A two-hop network is shown in Fig. 2, which has a
single source S, a single destination D and M relays
R
1
,R
2
, ··· ,R
M−1
,R
M
. S and D are capable of commu-
nicating with all the relays, as well as with each other.
By contrast, the M relays are unable to communicate with
each other. We stipulate the idealized simplifying assumption
furthermore that each node knows the position of all other
nodes. For each relay R
m
,m =1...M , the total average
energy consumption E
R
m
D
required for transmission from
R
m
to D is given by E
R
m
D
= E
s
R
m
D
+ E
f
R
m
D
,where
