# Statistical Delay QoS Provisioning for Energy-Efficient Spectrum-Sharing Based Wireless Ad Hoc Sensor Networks

Authors: Yichen Wang; Wenwen Xu
Journal: Journal of Sensors
 (2016)

---

## Abstract

In this paper, we develop the statistical delay quality-of-service (QoS) provisioning framework for the energy-efficient spectrum-sharing based wireless ad hoc sensor network (WAHSN), which is characterized by thedelay-bound violation probability. Based on the established delay QoS provisioning framework, we formulate the nonconvex optimization problem which aims at maximizing the average energy efficiency of the sensor node in the WAHSN while meeting PU’s statistical delay QoS requirement as well as satisfying sensor node’s average transmission rate, average transmitting power, and peak transmitting power constraints. By employing the theories offractional programming,convex hull, andprobabilistic transmission, we convert the original fractional-structured nonconvex problem to theadditively structured parametric convex problem and obtain the optimal power allocation strategy under the given parameter via Lagrangian method. Finally, we derive the optimal average energy efficiency and corresponding optimal power allocation scheme by employing theDinkelbach method. Simulation results show that our derived optimal power allocation strategy can be dynamically adjusted based on PU’s delay QoS requirement as well as the channel conditions. The impact of PU’s delay QoS requirement on sensor node’s energy efficiency is also illustrated.


---

## Body

## 1. Introduction

With the rapid development of wireless communications technologies, wireless ad hoc sensor networks (WAHSNs), where sensor nodes have the capability of self-healing and self-organizing, have been regarded as one of the most important wireless network architectures. Specifically, WAHSNs aim at collecting information from the surrounding environment and providing the fundamental for decisions. As WAHSNs lack of central control entities as well as the open nature of wireless channels, many challenging issues in WAHSNs arise, such as routing and networking [1–6], spectrum access and admission control [7–12], and networking and communications security [13–19]. Moreover, since WAHSNs usually include large number of sensor nodes, the wireless spectrum demands for WAHSNs have been also constantly increasing. However, recent research outcomes demonstrate that the wireless spectrum has become one of the scarce resources for wireless communications due to the currently used static spectrum allocation policy where a certain portion of spectrum can only be utilized, the particular type of wireless systems [20–22].To efficiently alleviate the spectrum shortage problem, spectrum-sharing and cognitive radio technologies have been regarded as one of the most important features for the future wireless systems and thus attracted lots of research attention from both academia and industry in the past decade [23–37]. In spectrum-sharing based wireless networks, how to provide efficient quality-of-service (QoS) provisioning for primary user (PU) has been widely accepted as a critical important issue. In existing works, PU’s QoS protection is usually implemented by imposing the average/peak interference power constraint on secondary users (SU) [38–42] or guaranteeing PU’s minimum average/instantaneous transmission rate [43–46]. However, the abovementioned methods cannot provide accurate and fine-grained delay QoS protection for PU due to the following reasons. First, it is difficult to build the accurate relationship between PU’s maximum allowed interference power and its delay QoS requirement. Second, guaranteeing PU’s minimum average/instantaneous transmission rate can only reflect two extreme cases of PU’s delay requirements. Specifically, on one hand, the average transmission rate constraint only requires that a certain amount of data should be transmitted within the required time period and thus only corresponds to the loose delay requirement. On the other hand, PU’s minimum instantaneous rate requires that the transmission rate cannot be below the given threshold at any time, which implies that the instantaneous rate constraint corresponds to a very stringent delay requirement. Moreover, the minimum instantaneous transmission rate usually cannot be satisfied due to the stochastic feature of wireless channels. Consequently, there is an urgent need to establish an efficient framework to accurately describe a wide range of PU’s delay QoS requirements. Moreover, it is also crucial to develop the resource allocation scheme which can not only optimize SU’s energy efficiency, but also meet PU’s various delay requirements.Based on the above analysis, we in this paper investigate PU’s delay QoS provisioning technique over energy-efficient spectrum-sharing based WAHSNs. Specifically, by employing the theory of statistical delay QoS provisioning, we first build an efficient PU’s delay QoS protection framework which is described byPU’s queueing-delay-bound violation probability. Different from currently widely used PU’s QoS protection approaches, our adopted framework can quantitatively and accurately describe PU’s fine-grained delay requirement by a single parameter calledPU’s delay QoS exponent. Based on the established PU’s statistical delay QoS protection framework, we formulate the optimization problem aiming at maximizing the average energy efficiency of the sensor node in the WAHSN while meeting PU’s statistical delay QoS requirement as well as satisfying the sensor node’s average transmission rate and average and peak transmitting power constraints. To solve our formulated fractional-structured nonconvex problem, we first adopt thefractional programming technique to convert the original fractional-structured problem to theparametric nonconvex program. Then, by employing the theories ofconvex hull andprobabilistic transmission, we convert the parametric nonconvex problem to the equivalent convex problem and obtain the optimal power allocation strategy under the given parameter via Lagrangian method. Finally, the optimal energy efficiency of the sensor node is derived by using theDinkelbach method. Simulation results illustrate that our obtained optimal power allocation strategy can adapt to both PU’s delay QoS requirement and channel conditions. Moreover, PU’s delay QoS requirement will significantly affect sensor node’s energy efficiency.The rest of this paper is organized as follows. In Section2, we present the system model. In Section 3, we first establish PU’s statistical delay QoS protection frame and then obtain the optimal power allocation strategy which can maximize the average energy efficiency of each sensor node subject to PU’s delay QoS requirement as well as SU’s average transmission rate and average and peak transmitting power constraints. Simulation results are provided in Section 4. This paper concludes with Section 5.


## 2. System Model

We consider that one wireless ad hoc sensor network (WAHSN) coexists with one primary network by sharing a certain portion of spectrum licensed to the primary network, as shown in Figure1. Thus, the WAHSN considered in this paper can be viewed as the secondary network. Specifically, the primary network includes one primary sender (PS) and one primary receiver (PR). The WAHSN includes one fusion center and K sensor nodes. The sensor nodes collect required information from the environment and send the collected information to the fusion center. Thus, we in this paper denote the sensor node as the secondary sender (SS) and denote the fusion center as the secondary receiver (SR).Figure 1
System model for the spectrum-sharing based wireless ad hoc sensor networks.The channel gains between PS and PR, thekth SS and kth SR, PS and the kth SR, and kth SS and PR are denoted by h
p
p, h
s
s
k, h
p
s
k, and h
s
p
k, respectively, where k
∈
{
1,2
,
…
,
K
}. All channel gains are assumed to be stationary, ergodic, independent, and block fading processes and follow Rayleigh fading model. Thus, all channel gains remain unchanged within each frame with duration T
f but independently vary from one frame to another. Moreover, we also assume that PS transmits with constant power, but SS transmits with variable power. We also assume that the WAHSN is synchronized with the primary network, which means that all sensor nodes in the WAHSN know the beginning and ending instants of each frame. Note that we in this paper assume that the channel gains h
p
p and h
s
p
k (k
∈
{
1,2
,
…
,
K
}) are available for WAHSN. One possible approach for obtaining this knowledge is to perform cooperation with primary networks. Furthermore, due to the large number of sensor nodes and limited wireless spectrum resource, therandom access mechanism is employed for the WAHSN. In particular, at the beginning of each frame, each sensor node in the WAHSN tries to access the spectrum licensed to primary network with probability P
a
c
c. If only one sensor node tries to access the spectrum, the sensor node can successfully transmit the information to the fusion center; otherwise, collision will occur at the fusion center or all sensor nodes remain silent. Consequently, the probability that the given sensor node can successfully send information to the fusion center, denoted by P
s
u
c
c, is determined by(1)
P
s
u
c
c
=
P
a
c
c
1
-
P
a
c
c
K
-
1
.In this paper, we aim at providing the statistical queueing-delay QoS protection for PS, which will be detailed in the following section.


## 3. Energy-Efficient Power Allocation Strategy with PU’s Statistical Delay QoS Protection

### 3.1. Statistical Delay QoS Protection for PU

In wireless communications systems, delay includes several components, such as propagation delay, signal processing delay, and encoding/decoding delay. However, due to the stochastic nature of wireless channels which causes the highly time-varying feature of queue’s service rate, queueing-delay has been widely recognized as an important contributor for the delay uncertainty. Moreover, the dynamics of wireless channels also cause that deterministic/hard delay provisioning is often unrealistic for practical wireless systems. Consequently, statistical approach can be better suited for the queueing-delay protection in wireless networks.To perform statistical delay QoS protection for the primary network, we consider that there is a queue at PS as shown in Figure2. Specifically, the upper layer of PS delivers data to the link layer and then the received data, which will be divided into link layer frames, is stored in the queue. PS will split the link layer frames into bit-streams and deliver them to the physical layer for transmission.Figure 2
Queueing model for primary sender.Based on the statistical QoS provisioning theory, PS’s statistical QoS requirement can be described by thequeue-length bound violation probability, which can be written as [47, 48](2)
P
r
Q
p
≥
Q
t
h
≤
P
t
h
,where Q
p denotes PU’s queue-length, Q
t
h represents the predefined queue-length threshold of PU, and P
t
h denotes the required violation probability, respectively. The above inequality (2) requires that the probability of PS’s queue-length Q
p exceeding the given threshold Q
t
h cannot be larger than the targeted probability requirement P
t
h (in this paper, we assume that the queue size of PS is infinite implying that no queue overflow will happen and thus we usequeue-length bound violation probability as the QoS provisioning metric. If the queue size of PS is finite, we can employqueue-overflow violation probability
P
r
Q
u
e
u
e
O
v
e
r
f
l
o
w
≤
P
t
h for statistical QoS protection, where we require that PS’s queue-overflow probability cannot exceed the predefined threshold).If we consider delay as the performance metric, we can also convert the abovementionedqueue-length bound violation probability to the correspondingqueueing-delay-bound violation probability, which is given by [47, 48](3)
P
r
D
p
≥
D
t
h
≤
P
t
h
,where D
p denotes PU’s queueing-delay and D
t
h represents PU’s queueing-delay threshold. Similar to (2), inequality (3) requires that the probability of PS’s queue-delay D
p exceeding the given threshold D
t
h need below the targeted probability requirement P
t
h. Moreover, larger values of D
t
h and P
t
h imply looser delay requirement and smaller values of D
t
h and P
t
h mean more stringent delay constraint. Furthermore, by employing the large derivation principle, PU’squeueing-delay-bound violation probability can be approximately determined by [48](4)
P
r
D
p
≥
D
t
h
≈
e
-
θ
p
E
B
P
U
θ
p
D
t
h
,where θ
p is called the PU’s QoS exponent and E
B
P
U
(
θ
p
) is the effective bandwidth of PU’s data arrival process that determines the minimum constant service rate required to support the given data arrival process under the specified delay QoS requirement [47]. Since it is assumed that PS has constant data arrival rate R
A (nats/s/Hz) as shown in Figure 2, we have E
B
P
U
(
θ
p
)
=
T
f
B
R
A (nats/frame). Based on (3) and (4), we can obtain that if (3) is satisfied, we must have(5)
θ
p
≥
-
1
E
B
P
U
θ
p
D
t
h
l
o
g
P
t
h
,which implies that PU’s delay QoS requirement can be quantitatively described by the QoS exponent θ
p. In particular, larger values of D
t
h and P
t
h result in smaller value of θ
p and smaller values of D
t
h and P
t
h lead to larger value of θ
p. Consequently, we can derive that small value of θ
p implies loose delay QoS requirement and large value of θ
p represents stringent delay constraint. Moreover, we can also obtain that when θ
p
→
0, PS can tolerate an arbitrary long delay; when θ
p
→
∞, PS cannot allow any delay [48].By employing the theory of effective capacity, which defines the maximum sustainable constant data arrival rate for the queueing system that the data service process can support under given QoS requirement [48], we can convertPU’s queueing-delay-bound violation probability to the equivalenteffective capacity requirement. Specifically, as PS’s service rate R
p, as shown in Figure 2, is time-uncorrelated across different frames, the effective capacity of PU’s service rate process, denoted by E
C
P
U
θ
p, can be written as(6)
E
C
P
U
θ
p
=
-
1
θ
p
l
o
g
E
e
-
θ
p
T
f
B
R
p
.Then, we can obtain that PU’s queueing-delay-bound violation probability constraint is satisfied only if the effective capacity requirement(7)
E
C
P
U
θ
p
≥
T
f
B
R
Acan be met, which implies that PS’s maximum sustainable arrival rate cannot be smaller than the constant arrival rate.The theories of effective bandwidth and effective capacity provide us with a convenient yet efficient approach to perform accurate and fine-grained delay QoS protection for the queueing system. Specifically, it allows us to design the service (arrival) process of the queueing system to meet the required statistical delay QoS requirement characterized by the queue-length bound/delay-bound violation probability by using the properties and statistics of the arrival (service) process. In this paper, we mainly focus on the service rate process and effective capacity part. Consequently, when PU’s internal conditions change, such that congestion occurs, PU’s data arrival rate will correspondingly vary. In this case, WAHSN can update the statistics of PU’s data arrival rate process. Then, WAHSN can reperform power allocation (which will be detailed in the following section) such that PU’s data arrival rate processR
p is also time-uncorrelated as R
p is obtained based on the updated statistics of data arrival rate process and is the function of channel gains which are assumed to be time-uncorrelated. Moreover, if the stochastic data arrival process for PS is considered, we can apply the theory of effective bandwidth to determine the minimum constant service rate E
B
P
U
θ
p needed for the given data arrival process. Then, the statistical QoS requirement for PS becomes E
C
P
U
θ
p
≥
E
B
P
U
θ
p, which implies that the effective capacity of the service rate process cannot be smaller than the effective bandwidth of the data arrival process.


### 3.2. Optimization Problem Formulation

Recall that the successful access probability of each sensor node isP
s
u
c
c as given by (1). Consequently, the service rates of PS-PR and SS-SR links for any given frame, denoted by R
p
P
s
θ
p
,
h (nats/s/Hz) and R
s
P
s
θ
p
,
h (nats/s/Hz), respectively, are determined by (we omit the index k
∈
{
1
,
2
,
…
,
K
} for brevity and the SS-SR link denotes the link from the sensor node to fusion center)(8)
R
p
P
s
θ
p
,
h
=
l
o
g
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
l
o
g
1
+
h
p
p
P
p
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
c
,
(9)
R
s
P
s
θ
p
,
h
=
l
o
g
1
+
h
s
s
P
s
θ
p
,
h
h
p
s
P
p
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
0
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
c
,respectively, where h
≜
h
s
s
,
h
p
s
,
h
s
p
,
h
p
p is the network gain vector (NGV), P
p is PS’s constant transmitting power, P
s
θ
p
,
h denotes SS’s transmitting power as the function of PU’s QoS exponent θ
p and NGV h, and σ
2 represents the variance of additive white Gaussian noise (AWGN). Then, PS’s effective capacity of the service rate process given by (8) is determined by(10)
E
C
P
U
θ
p
=
-
1
θ
p
l
o
g
P
s
u
c
c
·
E
e
-
θ
p
T
f
B
l
o
g
1
+
h
p
p
P
p
/
h
s
p
P
s
θ
p
,
h
+
σ
2
+
1
-
P
s
u
c
c
·
E
e
-
θ
p
T
f
B
l
o
g
1
+
h
p
p
P
p
/
σ
2
.In this paper, we aim at maximizing the sensor node’s normalized average energy efficiency while meeting PS’s statistical delay QoS requirement as well as satisfying sensor node’s average data transmission rate constraint and the average and peak transmitting power constraints, which can be mathematically formulated as(11)
P
1
Γ
E
E
o
p
t
=
m
a
x
P
s
θ
p
,
h
≥
0
E
h
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
E
h
ξ
·
P
s
u
c
c
·
P
s
θ
p
,
h
+
P
c
i
r
,
(12)
s
.
t
.
E
h
f
P
s
θ
p
,
h
+
e
-
β
R
A
≥
0
,
(13)
E
h
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
≥
R
t
h
,
(14)
E
h
P
s
u
c
c
·
P
s
θ
p
,
h
≤
P
a
v
,
(15)
P
s
θ
p
,
h
≤
P
p
k
,
∀
h
, where(16)
f
P
s
θ
p
,
h
=
-
P
s
u
c
c
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
+
σ
2
-
β
-
1
-
P
s
u
c
c
1
+
h
p
p
P
p
σ
2
-
β
,
β
=
T
f
B
θ
p denotes the normalized PS’s QoS exponent, R
t
h denotes the minimum required transmission rate of the SS-SR link, ξ is the amplifier coefficient, P
c
i
r represents the power consumption for hardware components, and P
a
v and P
p
k denote the maximum allowed average and peak transmitting power for SS, respectively. Moreover, (12) represents PU’s effective capacity requirement which is obtained based on (7) and (10).


### 3.3. Optimal Power Allocation Scheme

As the numerator and denominator of the objective function given by (11) are concave and affine, respectively, we can adoptfractional programming to solve problem (
P
1
) [49]. Specifically, by introducing the nonnegative parameter η, we can construct the new optimization problem, which is given by(17)
P
2
J
η
=
m
a
x
P
s
θ
p
,
h
,
η
≥
0
E
h
J
η
,
P
s
θ
p
,
h
,
η
,
(18)
s
.
t
.
E
h
f
P
s
θ
p
,
h
,
η
+
e
-
β
R
A
≥
0
,
(19)
E
h
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
,
η
≥
R
t
h
,
(20)
E
h
P
s
u
c
c
·
P
s
θ
p
,
h
,
η
≤
P
a
v
,
(21)
P
s
θ
p
,
h
,
η
≤
P
p
k
,
∀
h
,where(22)
J
η
,
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
,
η
-
η
ξ
·
P
s
u
c
c
·
P
s
θ
p
,
h
,
η
+
P
c
i
r
. Note that as we introduced the parameter η to convert the fractionally structured objective function to the above linearly additive function, the sensor node’s power allocation strategy P
s should also be the function of parameter η, where we use P
s
θ
p
,
h
,
η to replace P
s
θ
p
,
h in the above problem (
P
2
).We can easily prove that the objective function of problem(
P
2
) and SU’s average transmission rate constraint given by (13) are both concave. Moreover, SU’s average and peak transmitting power constraint given by (14) and (15), respectively, are both affine. Thus, the convexity of (
P
2
) is determined by that of PU’s statistical delay QoS requirement given by (12). We can obtain from (16) that(23)
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
=
-
P
s
u
c
c
·
β
h
s
p
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
2
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
-
β
-
1
<
0
,
d
2
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
2
=
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
h
s
p
β
-
1
h
p
p
P
p
-
2
h
s
p
P
s
θ
p
,
h
,
η
-
2
σ
2
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
h
p
p
P
p
+
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
, which demonstrate that f
P
s
θ
p
,
h
,
η is a decreasing function of P
s
θ
p
,
h
,
η but is not concave over P
s
θ
p
,
h
,
η as d
2
f
P
s
θ
p
,
h
,
η
/
d
P
s
θ
p
,
h
,
η
2
≤
0 does not hold. However, we can determine the unique inflexion point P
i
n
f for f
P
s
θ
p
,
h
,
η, which is given by(24)
P
i
n
f
=
β
-
1
h
p
p
P
p
-
2
σ
2
2
h
s
p
. Then, we can obtain that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
(
-
∞
,
P
i
n
f
] and is convex for P
s
θ
p
,
h
,
η
∈
[
P
i
n
f
,
∞
). Consequently, although problem (
P
2
) is not convex, we can still solve this nonconvex problem by employing theconvex hull andprobabilistic transmission techniques. Before discussing how to derive the optimal power allocation strategy, we first briefly describe the theories ofconvex hull andprobabilistic transmission as follows. (i)
Convex Hull. For a nonconvex region, all convex combination of the points in the region is defined as its convex hull. Moreover, it has been shown that the boundary function of the convex hull for any given two-dimensional plane is the straight line segment [50].
(ii)
Probabilistic Transmission. Denote y
=
v
(
x
) as the straight line with the end points (
x
1
,
v
(
x
1
)
) and (
x
2
,
v
(
x
2
)
). Then, any point on the line can be achieved by ξ
(
x
1
,
v
(
x
1
)
)
+
(
1
-
ξ
)
(
x
2
,
v
(
x
2
)
), where ξ
∈
[
0,1
] denotes the probability that we use point (
x
1
,
v
(
x
1
)
) and (
1
-
ξ
) represents the probability for using point (
x
2
,
v
(
x
2
)
).Based on the abovementioned techniques, we can convert the nonconvex problem(
P
2
) to the equivalent strictly concave problem, which can be analyzed from the following three cases.Case 1.
IfP
i
n
f
<
0 or P
i
n
f
∈
0
,
P
p
k but the following inequality(25)
f
P
p
k
-
f
0
P
p
k
>
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
0is satisfied, we can obtain the following conclusions. (1)
IfP
i
n
f
<
0, it is easy to derive that f
P
s
θ
p
,
h
,
η is convex over P
s
θ
p
,
h
,
η
∈
0
,
P
p
k. Thus, based on the definition of convex function, we can obtain that f
P
s
θ
p
,
h
,
η in the range of 0
,
P
p
k is below the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
).
(2)
IfP
i
n
f
∈
0
,
P
p
k, it is obvious that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
0
,
P
i
n
f but is convex for P
s
θ
p
,
h
,
η
∈
P
i
n
f
,
P
p
k. However, (25) implies that f
P
s
θ
p
,
h
,
η in the range of 0
,
P
p
k is also below the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
).
Consequently, the boundary function for the convex hull of functionf
P
s
θ
p
,
h
,
η in the range of 0
,
P
p
k, denoted by f
~
P
s
θ
p
,
h, is the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
), which can be mathematically written as(26)
f
~
P
s
θ
p
,
h
,
η
=
f
0
+
f
P
p
k
-
f
0
P
p
k
P
s
θ
p
,
h
,
η
. Then, based on the probabilistic transmission technique, we can achieve all the points along the boundary function f
~
P
s
θ
p
,
h
,
η, where sensor node’s transmitting power P
s
θ
p
,
h
,
η can only equal 0 and P
p
k. Therefore, sensor node’s transmission rate R
s
(
P
s
θ
p
,
h
,
η
) also needs to be modified. We denote sensor node’s reformulated transmission rate by R
~
s
(
P
s
θ
p
,
h
,
η
). Then, by employing the probabilistic transmission technique, when P
s
θ
p
,
h
,
η
∈
0
,
P
p
k, R
~
s
(
P
s
θ
p
,
h
,
η
) is given by(27)
R
~
s
P
s
θ
p
,
h
,
η
=
P
s
θ
p
,
h
,
η
P
p
k
l
o
g
1
+
h
s
s
P
p
k
h
p
s
P
p
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
0
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
cwhich implies that the reformulated transmission rate R
~
s
(
h
,
P
s
θ
p
,
h
,
η
) is also the straight line segment with two end points 0
,
R
s
(
0
) and P
p
k
,
R
s
(
P
p
k
). That is to say, when one sensor node accesses the spectrum via random access, the probability that the sensor node uses power P
p
k for transmission is P
s
θ
p
,
h
,
η
/
P
p
k and the probability that the sensor node gives up the transmission opportunity is 1
-
P
s
θ
p
,
h
,
η
/
P
p
k.Case 2.
IfP
i
n
f
>
P
p
k, we can derive that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
0
,
P
p
k. Therefore, in this case, the boundary function f
~
P
s
θ
p
,
h
,
η for the convex hull is equal to f
P
s
θ
p
,
h
,
η and thus we have(28)
f
~
P
s
θ
p
,
h
,
η
=
-
P
s
u
c
c
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
-
β
-
1
-
P
s
u
c
c
1
+
h
p
p
P
p
σ
2
-
β
.Correspondingly, in this case, sensor node’s reformulated transmission rate R
~
s
P
s
θ
p
,
h
,
η is also equal to the original rate R
s
P
s
θ
p
,
h
,
η; that is,(29)
R
~
s
P
s
θ
p
,
h
,
η
=
l
o
g
1
+
h
s
s
P
s
θ
p
,
h
,
η
h
p
s
P
p
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
0
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
c
.Note that as f
P
s
θ
p
,
h
,
η is concave, the boundary function for the convex hull is exactly the same as the original function f
P
s
θ
p
,
h
,
η and thus it is not necessary to adopt probabilistic transmission technique in this case.Case 3.
IfP
i
n
f
∈
0
,
P
p
k and the following inequality(30)
f
P
p
k
-
f
0
P
p
k
<
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
0is satisfied, we have that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
0
,
P
i
n
f and is convex for P
s
θ
p
,
h
∈
P
i
n
f
,
P
p
k. However, different from Case 1 in which function f
P
s
θ
p
,
h
,
η is completely below the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
) in the range of 0
,
P
p
k, there must exist one unique cross-point denoted by P
c
r
o in this case such that (i)
functionf
P
s
θ
p
,
h
,
η isabove the straight line with end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
) for P
s
θ
p
,
h
,
η
∈
0
,
P
c
r
o;
(ii)
functionf
P
s
θ
p
,
h
,
η isbelow the straight line with end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
) for P
s
θ
p
,
h
,
η
∈
P
c
r
o
,
P
p
k.
Thus, based on the definition of convex hull, to determine the boundary function for the convex hull of functionf
P
s
θ
p
,
h
,
η, we need to find the unique tangent point for f
P
s
θ
p
,
h
,
η and the straight line with two end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
). We denote the tangent point by P
t
a
n and then P
t
a
n can be determined by(31)
f
P
p
k
-
f
0
P
p
k
=
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
t
a
n
.Based on the obtained tangent point P
t
a
n, the boundary function f
~
P
s
θ
p
,
h
,
η is equal to the original function f
P
s
θ
p
,
h
,
η when P
s
θ
p
,
h
,
η
∈
0
,
P
t
a
n and becomes a straight line when P
s
θ
p
,
h
∈
P
t
a
n
,
P
p
k. Consequently, the boundary function f
~
P
s
θ
p
,
h
,
η can be mathematically written as(32)
f
~
P
s
θ
p
,
h
,
η
=
f
P
s
θ
p
,
h
,
η
,
P
s
θ
p
,
h
∈
0
,
P
t
a
n
,
f
P
t
a
n
+
f
P
p
k
-
f
P
t
a
n
P
p
k
-
P
t
a
n
P
s
θ
p
,
h
-
P
t
a
n
,
P
s
θ
p
,
h
,
η
∈
P
t
a
n
,
P
p
k
.Equation (32) implies that the probabilistic transmission technique only needs to be used to realize the straight line with the end points P
t
a
n
,
f
P
t
a
n and P
p
k
,
f
P
p
k. Then, by adopting the probabilistic transmission technique, we can also modify sensor node’s transmission rate as(33)
R
~
s
P
s
θ
p
,
h
,
η
=
R
s
P
s
θ
p
,
h
,
η
,
P
s
θ
p
,
h
,
η
∈
0
,
P
t
a
n
,
R
s
P
t
a
n
+
R
s
P
p
k
-
R
s
P
t
a
n
P
p
k
-
P
t
a
n
P
s
θ
p
,
h
,
η
-
P
t
a
n
,
P
s
θ
p
,
h
,
η
∈
P
t
a
n
,
P
p
k
.
Based on the above analysis, we can convert the nonconvex problem(
P
2
) to the strictly concave problem, which is determined by(34)
P
3
J
~
η
=
m
a
x
P
s
θ
p
,
h
,
η
≥
0
E
h
J
~
η
,
P
s
θ
p
,
h
,
η
,
(35)
s
.
t
.
E
h
f
~
P
s
θ
p
,
h
,
η
+
e
-
β
R
A
≥
0
,
(36)
E
h
P
s
u
c
c
·
R
~
s
P
s
θ
p
,
h
,
η
≥
R
t
h
,
(37)
E
q
s
.
20
a
n
d
21
,where(38)
J
~
η
,
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
·
R
~
s
P
s
θ
p
,
h
,
η
-
η
ξ
·
P
s
u
c
c
·
P
s
θ
p
,
h
,
η
+
P
c
i
r
.Recall that R
~
s
(
P
s
θ
p
,
h
,
η
) denotes sensor node’s reformulated transmission rate under the probabilistic transmission given by (27), (29), and (33), respectively. f
~
(
P
s
θ
p
,
h
,
η
) represents the boundary function for the convex hull given by (26), (28), and (32), respectively. As the above problem (
P
3
) is strictly concave, we can obtain the optimal solution via Lagrangian method. Specifically, we can construct the Lagrangian function, denoted by L
(
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
), as(39)
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
=
E
h
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
-
η
P
c
i
r
+
λ
e
β
R
A
-
μ
R
t
h
+
δ
P
a
v
,where(40)
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
1
+
μ
R
~
s
P
s
θ
p
,
h
,
η
+
λ
f
~
P
s
θ
p
,
h
,
η
-
η
ξ
P
s
u
c
c
+
δ
P
s
θ
p
,
h
,
η
,and λ, μ, and δ are Lagrangian multipliers associated with constraints (35), (36), and (14), respectively. We denote the optimal solution of problem (
P
3
) by P
s
∗
θ
p
,
h
,
η. Then, based on the Karush-Kuhn-Tucker (KKT) conditions, we can obtain(41)
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
P
s
∗
θ
p
,
h
,
η
∈
0
,
P
p
k
,
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
s
∗
θ
p
,
h
,
η
<
0
,
i
f
P
s
∗
θ
p
,
h
,
η
=
0
,
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
s
∗
θ
p
,
h
,
η
>
0
,
i
f
P
s
∗
θ
p
,
h
,
η
=
P
p
k
,where(42)
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
1
+
μ
d
R
~
s
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
+
λ
d
f
~
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
-
η
ξ
P
s
u
c
c
-
δ
.Denote the optimal Lagrangian multipliers byλ
∗, μ
∗, and δ
∗, respectively. Then, the optimal solution P
s
∗
θ
p
,
h
,
η for problem (
P
3
) is determined as follows:Case 1.
(43)
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
<
0
,
P
p
k
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
>
0
.Case 2.
(44)
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
0
<
0
,
P
p
k
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
P
p
k
>
0
,
P
~
1
,
o
t
h
e
r
w
i
s
e
,where P
~
1
∈
(
0
,
P
p
k
) and it is the solution to the following equality:(45)
P
s
u
c
c
1
+
μ
d
R
~
s
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
+
λ
d
f
~
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
-
η
ξ
P
s
u
c
c
-
δ
=
0
.Case 3.
(46)
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
0
<
0
,
P
p
k
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
P
t
a
n
>
0
,
P
~
2
,
o
t
h
e
r
w
i
s
e
,where P
~
2
∈
(
0
,
P
t
a
n
) is the solution to d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
/
d
P
s
θ
p
,
h
,
η
=
0.
Although (43)–(46) determine the optimal solution of problem (
P
3
), we can observe that our obtained optimal solution belongs to the range that the boundary function f
~
(
P
s
θ
p
,
h
,
η
) overlaps with the original function f
(
P
s
θ
p
,
h
,
η
). Consequently, (43)–(46) also determine the optimal solution of nonconvex problem (
P
2
).
As we convert the original EE-maximization problem(
P
1
) with the fractionally structured objective function to the parametric problem (
P
2
) with the linearly additive objective function by employing the fractional programming approach, we also need to determine the optimal solution to the original problem (
P
1
) based on (43)–(46). According to the fractional programming theory, if we can find the parameter η
∗ such that (47)
J
η
∗
=
E
h
J
η
∗
,
P
s
θ
p
,
h
,
η
∗
P
s
θ
p
,
h
,
η
∗
=
P
s
∗
θ
p
,
h
,
η
∗
=
0
,then we must have Γ
E
E
o
p
t
=
η
∗, where Γ
E
E
o
p
t denotes sensor node’s maximum energy efficiency and can be determined by Algorithm 1. Note that our obtained optimal power allocation scheme can be dynamically adjusted based on PU’s QoS exponent θ
p. Consequently, how to derive PU’s QoS exponent is a critically important issue. In realistic wireless systems, each type of services/applications for a particular wireless system usually has similar delay QoS requirement. Then, WAHSN can determine the corresponding values of θ
p based on the service types of primary networks. Moreover, we in this paper mainly focus on the single link scenario. However, our work can also extend to the multilink scenario. Specifically, on one hand, if each PU’s licensed channel is only allowed to be shared by one SS-SR link, then our proposed power allocation scheme can be directly applied. On the other hand, if each PU’s licensed channel is allowed to be shared by all N SS-SR links, we can also adopt the theories of convex hull and probabilistic transmission to convert the original nonconvex problem to the equivalent convex problem, where the inflexion point in the single link scenario is replaced by the inflexion hyperplane in the multilink scenario.Algorithm 1:Algorithm to determine sensor node’s maximum average energy efficiency.
(1)Initialization: n
=
0 and η
0 where η
0 satisfies
J
η
0
=
E
h
J
η
0
,
P
s
θ
p
,
h
,
η
0
>
0
(2)for  n
=
0,1
,
2
,
…
do
(3)  Letη
=
η
n
(4)  Solve problem(
P
2
) with η
n, where the optimal solution denoted P
s
∗
θ
p
,
h
,
η
n is determined by (43)–(46).
(5)if  |
J
η
n
|
=
|
E
h
{
J
(
η
n
,
P
s
∗
θ
p
,
h
,
η
0
)
}
|
>
ε
then
(6)   Update parameterη
n by
η
n
+
1
=
η
n
-
J
η
n
J
′
η
n
=
E
h
{
P
s
u
c
c
⋅
R
s
P
s
∗
θ
p
,
h
,
η
n
}
E
h
{
ξ
⋅
P
s
u
c
c
⋅
P
s
∗
θ
p
,
h
,
η
n
}
+
P
c
i
r
(7)   Letn
=
n
+
1
(8)else
(9)   The optimal power allocation strategyP
s
o
p
t
θ
p
,
h and the sensor node’s maximum average
energy-efficiencyΓ
E
E
o
p
t are determined by
P
s
o
p
t
θ
p
,
h
=
P
s
∗
θ
p
,
h
,
η
n
Γ
E
E
o
p
t
=
η
n
=
E
h
{
P
s
u
c
c
⋅
R
s
P
s
∗
θ
p
,
h
,
η
n
}
E
h
{
ξ
⋅
P
s
u
c
c
⋅
P
s
∗
θ
p
,
h
,
η
n
}
+
P
c
i
r
(10)end if
(11)end for
(12)return  P
s
o
p
t
θ
p
,
h and Γ
E
E
o
p
t


## 3.1. Statistical Delay QoS Protection for PU

In wireless communications systems, delay includes several components, such as propagation delay, signal processing delay, and encoding/decoding delay. However, due to the stochastic nature of wireless channels which causes the highly time-varying feature of queue’s service rate, queueing-delay has been widely recognized as an important contributor for the delay uncertainty. Moreover, the dynamics of wireless channels also cause that deterministic/hard delay provisioning is often unrealistic for practical wireless systems. Consequently, statistical approach can be better suited for the queueing-delay protection in wireless networks.To perform statistical delay QoS protection for the primary network, we consider that there is a queue at PS as shown in Figure2. Specifically, the upper layer of PS delivers data to the link layer and then the received data, which will be divided into link layer frames, is stored in the queue. PS will split the link layer frames into bit-streams and deliver them to the physical layer for transmission.Figure 2
Queueing model for primary sender.Based on the statistical QoS provisioning theory, PS’s statistical QoS requirement can be described by thequeue-length bound violation probability, which can be written as [47, 48](2)
P
r
Q
p
≥
Q
t
h
≤
P
t
h
,where Q
p denotes PU’s queue-length, Q
t
h represents the predefined queue-length threshold of PU, and P
t
h denotes the required violation probability, respectively. The above inequality (2) requires that the probability of PS’s queue-length Q
p exceeding the given threshold Q
t
h cannot be larger than the targeted probability requirement P
t
h (in this paper, we assume that the queue size of PS is infinite implying that no queue overflow will happen and thus we usequeue-length bound violation probability as the QoS provisioning metric. If the queue size of PS is finite, we can employqueue-overflow violation probability
P
r
Q
u
e
u
e
O
v
e
r
f
l
o
w
≤
P
t
h for statistical QoS protection, where we require that PS’s queue-overflow probability cannot exceed the predefined threshold).If we consider delay as the performance metric, we can also convert the abovementionedqueue-length bound violation probability to the correspondingqueueing-delay-bound violation probability, which is given by [47, 48](3)
P
r
D
p
≥
D
t
h
≤
P
t
h
,where D
p denotes PU’s queueing-delay and D
t
h represents PU’s queueing-delay threshold. Similar to (2), inequality (3) requires that the probability of PS’s queue-delay D
p exceeding the given threshold D
t
h need below the targeted probability requirement P
t
h. Moreover, larger values of D
t
h and P
t
h imply looser delay requirement and smaller values of D
t
h and P
t
h mean more stringent delay constraint. Furthermore, by employing the large derivation principle, PU’squeueing-delay-bound violation probability can be approximately determined by [48](4)
P
r
D
p
≥
D
t
h
≈
e
-
θ
p
E
B
P
U
θ
p
D
t
h
,where θ
p is called the PU’s QoS exponent and E
B
P
U
(
θ
p
) is the effective bandwidth of PU’s data arrival process that determines the minimum constant service rate required to support the given data arrival process under the specified delay QoS requirement [47]. Since it is assumed that PS has constant data arrival rate R
A (nats/s/Hz) as shown in Figure 2, we have E
B
P
U
(
θ
p
)
=
T
f
B
R
A (nats/frame). Based on (3) and (4), we can obtain that if (3) is satisfied, we must have(5)
θ
p
≥
-
1
E
B
P
U
θ
p
D
t
h
l
o
g
P
t
h
,which implies that PU’s delay QoS requirement can be quantitatively described by the QoS exponent θ
p. In particular, larger values of D
t
h and P
t
h result in smaller value of θ
p and smaller values of D
t
h and P
t
h lead to larger value of θ
p. Consequently, we can derive that small value of θ
p implies loose delay QoS requirement and large value of θ
p represents stringent delay constraint. Moreover, we can also obtain that when θ
p
→
0, PS can tolerate an arbitrary long delay; when θ
p
→
∞, PS cannot allow any delay [48].By employing the theory of effective capacity, which defines the maximum sustainable constant data arrival rate for the queueing system that the data service process can support under given QoS requirement [48], we can convertPU’s queueing-delay-bound violation probability to the equivalenteffective capacity requirement. Specifically, as PS’s service rate R
p, as shown in Figure 2, is time-uncorrelated across different frames, the effective capacity of PU’s service rate process, denoted by E
C
P
U
θ
p, can be written as(6)
E
C
P
U
θ
p
=
-
1
θ
p
l
o
g
E
e
-
θ
p
T
f
B
R
p
.Then, we can obtain that PU’s queueing-delay-bound violation probability constraint is satisfied only if the effective capacity requirement(7)
E
C
P
U
θ
p
≥
T
f
B
R
Acan be met, which implies that PS’s maximum sustainable arrival rate cannot be smaller than the constant arrival rate.The theories of effective bandwidth and effective capacity provide us with a convenient yet efficient approach to perform accurate and fine-grained delay QoS protection for the queueing system. Specifically, it allows us to design the service (arrival) process of the queueing system to meet the required statistical delay QoS requirement characterized by the queue-length bound/delay-bound violation probability by using the properties and statistics of the arrival (service) process. In this paper, we mainly focus on the service rate process and effective capacity part. Consequently, when PU’s internal conditions change, such that congestion occurs, PU’s data arrival rate will correspondingly vary. In this case, WAHSN can update the statistics of PU’s data arrival rate process. Then, WAHSN can reperform power allocation (which will be detailed in the following section) such that PU’s data arrival rate processR
p is also time-uncorrelated as R
p is obtained based on the updated statistics of data arrival rate process and is the function of channel gains which are assumed to be time-uncorrelated. Moreover, if the stochastic data arrival process for PS is considered, we can apply the theory of effective bandwidth to determine the minimum constant service rate E
B
P
U
θ
p needed for the given data arrival process. Then, the statistical QoS requirement for PS becomes E
C
P
U
θ
p
≥
E
B
P
U
θ
p, which implies that the effective capacity of the service rate process cannot be smaller than the effective bandwidth of the data arrival process.


## 3.2. Optimization Problem Formulation

Recall that the successful access probability of each sensor node isP
s
u
c
c as given by (1). Consequently, the service rates of PS-PR and SS-SR links for any given frame, denoted by R
p
P
s
θ
p
,
h (nats/s/Hz) and R
s
P
s
θ
p
,
h (nats/s/Hz), respectively, are determined by (we omit the index k
∈
{
1
,
2
,
…
,
K
} for brevity and the SS-SR link denotes the link from the sensor node to fusion center)(8)
R
p
P
s
θ
p
,
h
=
l
o
g
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
l
o
g
1
+
h
p
p
P
p
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
c
,
(9)
R
s
P
s
θ
p
,
h
=
l
o
g
1
+
h
s
s
P
s
θ
p
,
h
h
p
s
P
p
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
0
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
c
,respectively, where h
≜
h
s
s
,
h
p
s
,
h
s
p
,
h
p
p is the network gain vector (NGV), P
p is PS’s constant transmitting power, P
s
θ
p
,
h denotes SS’s transmitting power as the function of PU’s QoS exponent θ
p and NGV h, and σ
2 represents the variance of additive white Gaussian noise (AWGN). Then, PS’s effective capacity of the service rate process given by (8) is determined by(10)
E
C
P
U
θ
p
=
-
1
θ
p
l
o
g
P
s
u
c
c
·
E
e
-
θ
p
T
f
B
l
o
g
1
+
h
p
p
P
p
/
h
s
p
P
s
θ
p
,
h
+
σ
2
+
1
-
P
s
u
c
c
·
E
e
-
θ
p
T
f
B
l
o
g
1
+
h
p
p
P
p
/
σ
2
.In this paper, we aim at maximizing the sensor node’s normalized average energy efficiency while meeting PS’s statistical delay QoS requirement as well as satisfying sensor node’s average data transmission rate constraint and the average and peak transmitting power constraints, which can be mathematically formulated as(11)
P
1
Γ
E
E
o
p
t
=
m
a
x
P
s
θ
p
,
h
≥
0
E
h
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
E
h
ξ
·
P
s
u
c
c
·
P
s
θ
p
,
h
+
P
c
i
r
,
(12)
s
.
t
.
E
h
f
P
s
θ
p
,
h
+
e
-
β
R
A
≥
0
,
(13)
E
h
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
≥
R
t
h
,
(14)
E
h
P
s
u
c
c
·
P
s
θ
p
,
h
≤
P
a
v
,
(15)
P
s
θ
p
,
h
≤
P
p
k
,
∀
h
, where(16)
f
P
s
θ
p
,
h
=
-
P
s
u
c
c
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
+
σ
2
-
β
-
1
-
P
s
u
c
c
1
+
h
p
p
P
p
σ
2
-
β
,
β
=
T
f
B
θ
p denotes the normalized PS’s QoS exponent, R
t
h denotes the minimum required transmission rate of the SS-SR link, ξ is the amplifier coefficient, P
c
i
r represents the power consumption for hardware components, and P
a
v and P
p
k denote the maximum allowed average and peak transmitting power for SS, respectively. Moreover, (12) represents PU’s effective capacity requirement which is obtained based on (7) and (10).


## 3.3. Optimal Power Allocation Scheme

As the numerator and denominator of the objective function given by (11) are concave and affine, respectively, we can adoptfractional programming to solve problem (
P
1
) [49]. Specifically, by introducing the nonnegative parameter η, we can construct the new optimization problem, which is given by(17)
P
2
J
η
=
m
a
x
P
s
θ
p
,
h
,
η
≥
0
E
h
J
η
,
P
s
θ
p
,
h
,
η
,
(18)
s
.
t
.
E
h
f
P
s
θ
p
,
h
,
η
+
e
-
β
R
A
≥
0
,
(19)
E
h
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
,
η
≥
R
t
h
,
(20)
E
h
P
s
u
c
c
·
P
s
θ
p
,
h
,
η
≤
P
a
v
,
(21)
P
s
θ
p
,
h
,
η
≤
P
p
k
,
∀
h
,where(22)
J
η
,
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
·
R
s
P
s
θ
p
,
h
,
η
-
η
ξ
·
P
s
u
c
c
·
P
s
θ
p
,
h
,
η
+
P
c
i
r
. Note that as we introduced the parameter η to convert the fractionally structured objective function to the above linearly additive function, the sensor node’s power allocation strategy P
s should also be the function of parameter η, where we use P
s
θ
p
,
h
,
η to replace P
s
θ
p
,
h in the above problem (
P
2
).We can easily prove that the objective function of problem(
P
2
) and SU’s average transmission rate constraint given by (13) are both concave. Moreover, SU’s average and peak transmitting power constraint given by (14) and (15), respectively, are both affine. Thus, the convexity of (
P
2
) is determined by that of PU’s statistical delay QoS requirement given by (12). We can obtain from (16) that(23)
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
=
-
P
s
u
c
c
·
β
h
s
p
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
2
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
-
β
-
1
<
0
,
d
2
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
2
=
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
h
s
p
β
-
1
h
p
p
P
p
-
2
h
s
p
P
s
θ
p
,
h
,
η
-
2
σ
2
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
h
p
p
P
p
+
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
, which demonstrate that f
P
s
θ
p
,
h
,
η is a decreasing function of P
s
θ
p
,
h
,
η but is not concave over P
s
θ
p
,
h
,
η as d
2
f
P
s
θ
p
,
h
,
η
/
d
P
s
θ
p
,
h
,
η
2
≤
0 does not hold. However, we can determine the unique inflexion point P
i
n
f for f
P
s
θ
p
,
h
,
η, which is given by(24)
P
i
n
f
=
β
-
1
h
p
p
P
p
-
2
σ
2
2
h
s
p
. Then, we can obtain that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
(
-
∞
,
P
i
n
f
] and is convex for P
s
θ
p
,
h
,
η
∈
[
P
i
n
f
,
∞
). Consequently, although problem (
P
2
) is not convex, we can still solve this nonconvex problem by employing theconvex hull andprobabilistic transmission techniques. Before discussing how to derive the optimal power allocation strategy, we first briefly describe the theories ofconvex hull andprobabilistic transmission as follows. (i)
Convex Hull. For a nonconvex region, all convex combination of the points in the region is defined as its convex hull. Moreover, it has been shown that the boundary function of the convex hull for any given two-dimensional plane is the straight line segment [50].
(ii)
Probabilistic Transmission. Denote y
=
v
(
x
) as the straight line with the end points (
x
1
,
v
(
x
1
)
) and (
x
2
,
v
(
x
2
)
). Then, any point on the line can be achieved by ξ
(
x
1
,
v
(
x
1
)
)
+
(
1
-
ξ
)
(
x
2
,
v
(
x
2
)
), where ξ
∈
[
0,1
] denotes the probability that we use point (
x
1
,
v
(
x
1
)
) and (
1
-
ξ
) represents the probability for using point (
x
2
,
v
(
x
2
)
).Based on the abovementioned techniques, we can convert the nonconvex problem(
P
2
) to the equivalent strictly concave problem, which can be analyzed from the following three cases.Case 1.
IfP
i
n
f
<
0 or P
i
n
f
∈
0
,
P
p
k but the following inequality(25)
f
P
p
k
-
f
0
P
p
k
>
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
0is satisfied, we can obtain the following conclusions. (1)
IfP
i
n
f
<
0, it is easy to derive that f
P
s
θ
p
,
h
,
η is convex over P
s
θ
p
,
h
,
η
∈
0
,
P
p
k. Thus, based on the definition of convex function, we can obtain that f
P
s
θ
p
,
h
,
η in the range of 0
,
P
p
k is below the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
).
(2)
IfP
i
n
f
∈
0
,
P
p
k, it is obvious that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
0
,
P
i
n
f but is convex for P
s
θ
p
,
h
,
η
∈
P
i
n
f
,
P
p
k. However, (25) implies that f
P
s
θ
p
,
h
,
η in the range of 0
,
P
p
k is also below the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
).
Consequently, the boundary function for the convex hull of functionf
P
s
θ
p
,
h
,
η in the range of 0
,
P
p
k, denoted by f
~
P
s
θ
p
,
h, is the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
), which can be mathematically written as(26)
f
~
P
s
θ
p
,
h
,
η
=
f
0
+
f
P
p
k
-
f
0
P
p
k
P
s
θ
p
,
h
,
η
. Then, based on the probabilistic transmission technique, we can achieve all the points along the boundary function f
~
P
s
θ
p
,
h
,
η, where sensor node’s transmitting power P
s
θ
p
,
h
,
η can only equal 0 and P
p
k. Therefore, sensor node’s transmission rate R
s
(
P
s
θ
p
,
h
,
η
) also needs to be modified. We denote sensor node’s reformulated transmission rate by R
~
s
(
P
s
θ
p
,
h
,
η
). Then, by employing the probabilistic transmission technique, when P
s
θ
p
,
h
,
η
∈
0
,
P
p
k, R
~
s
(
P
s
θ
p
,
h
,
η
) is given by(27)
R
~
s
P
s
θ
p
,
h
,
η
=
P
s
θ
p
,
h
,
η
P
p
k
l
o
g
1
+
h
s
s
P
p
k
h
p
s
P
p
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
0
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
cwhich implies that the reformulated transmission rate R
~
s
(
h
,
P
s
θ
p
,
h
,
η
) is also the straight line segment with two end points 0
,
R
s
(
0
) and P
p
k
,
R
s
(
P
p
k
). That is to say, when one sensor node accesses the spectrum via random access, the probability that the sensor node uses power P
p
k for transmission is P
s
θ
p
,
h
,
η
/
P
p
k and the probability that the sensor node gives up the transmission opportunity is 1
-
P
s
θ
p
,
h
,
η
/
P
p
k.Case 2.
IfP
i
n
f
>
P
p
k, we can derive that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
0
,
P
p
k. Therefore, in this case, the boundary function f
~
P
s
θ
p
,
h
,
η for the convex hull is equal to f
P
s
θ
p
,
h
,
η and thus we have(28)
f
~
P
s
θ
p
,
h
,
η
=
-
P
s
u
c
c
1
+
h
p
p
P
p
h
s
p
P
s
θ
p
,
h
,
η
+
σ
2
-
β
-
1
-
P
s
u
c
c
1
+
h
p
p
P
p
σ
2
-
β
.Correspondingly, in this case, sensor node’s reformulated transmission rate R
~
s
P
s
θ
p
,
h
,
η is also equal to the original rate R
s
P
s
θ
p
,
h
,
η; that is,(29)
R
~
s
P
s
θ
p
,
h
,
η
=
l
o
g
1
+
h
s
s
P
s
θ
p
,
h
,
η
h
p
s
P
p
+
σ
2
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
P
s
u
c
c
,
0
w
i
t
h
p
r
o
b
a
b
i
l
i
t
y
1
-
P
s
u
c
c
.Note that as f
P
s
θ
p
,
h
,
η is concave, the boundary function for the convex hull is exactly the same as the original function f
P
s
θ
p
,
h
,
η and thus it is not necessary to adopt probabilistic transmission technique in this case.Case 3.
IfP
i
n
f
∈
0
,
P
p
k and the following inequality(30)
f
P
p
k
-
f
0
P
p
k
<
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
0is satisfied, we have that f
P
s
θ
p
,
h
,
η is concave for P
s
θ
p
,
h
,
η
∈
0
,
P
i
n
f and is convex for P
s
θ
p
,
h
∈
P
i
n
f
,
P
p
k. However, different from Case 1 in which function f
P
s
θ
p
,
h
,
η is completely below the straight line with the end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
) in the range of 0
,
P
p
k, there must exist one unique cross-point denoted by P
c
r
o in this case such that (i)
functionf
P
s
θ
p
,
h
,
η isabove the straight line with end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
) for P
s
θ
p
,
h
,
η
∈
0
,
P
c
r
o;
(ii)
functionf
P
s
θ
p
,
h
,
η isbelow the straight line with end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
) for P
s
θ
p
,
h
,
η
∈
P
c
r
o
,
P
p
k.
Thus, based on the definition of convex hull, to determine the boundary function for the convex hull of functionf
P
s
θ
p
,
h
,
η, we need to find the unique tangent point for f
P
s
θ
p
,
h
,
η and the straight line with two end points 0
,
f
(
0
) and P
p
k
,
f
(
P
p
k
). We denote the tangent point by P
t
a
n and then P
t
a
n can be determined by(31)
f
P
p
k
-
f
0
P
p
k
=
d
f
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
t
a
n
.Based on the obtained tangent point P
t
a
n, the boundary function f
~
P
s
θ
p
,
h
,
η is equal to the original function f
P
s
θ
p
,
h
,
η when P
s
θ
p
,
h
,
η
∈
0
,
P
t
a
n and becomes a straight line when P
s
θ
p
,
h
∈
P
t
a
n
,
P
p
k. Consequently, the boundary function f
~
P
s
θ
p
,
h
,
η can be mathematically written as(32)
f
~
P
s
θ
p
,
h
,
η
=
f
P
s
θ
p
,
h
,
η
,
P
s
θ
p
,
h
∈
0
,
P
t
a
n
,
f
P
t
a
n
+
f
P
p
k
-
f
P
t
a
n
P
p
k
-
P
t
a
n
P
s
θ
p
,
h
-
P
t
a
n
,
P
s
θ
p
,
h
,
η
∈
P
t
a
n
,
P
p
k
.Equation (32) implies that the probabilistic transmission technique only needs to be used to realize the straight line with the end points P
t
a
n
,
f
P
t
a
n and P
p
k
,
f
P
p
k. Then, by adopting the probabilistic transmission technique, we can also modify sensor node’s transmission rate as(33)
R
~
s
P
s
θ
p
,
h
,
η
=
R
s
P
s
θ
p
,
h
,
η
,
P
s
θ
p
,
h
,
η
∈
0
,
P
t
a
n
,
R
s
P
t
a
n
+
R
s
P
p
k
-
R
s
P
t
a
n
P
p
k
-
P
t
a
n
P
s
θ
p
,
h
,
η
-
P
t
a
n
,
P
s
θ
p
,
h
,
η
∈
P
t
a
n
,
P
p
k
.
Based on the above analysis, we can convert the nonconvex problem(
P
2
) to the strictly concave problem, which is determined by(34)
P
3
J
~
η
=
m
a
x
P
s
θ
p
,
h
,
η
≥
0
E
h
J
~
η
,
P
s
θ
p
,
h
,
η
,
(35)
s
.
t
.
E
h
f
~
P
s
θ
p
,
h
,
η
+
e
-
β
R
A
≥
0
,
(36)
E
h
P
s
u
c
c
·
R
~
s
P
s
θ
p
,
h
,
η
≥
R
t
h
,
(37)
E
q
s
.
20
a
n
d
21
,where(38)
J
~
η
,
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
·
R
~
s
P
s
θ
p
,
h
,
η
-
η
ξ
·
P
s
u
c
c
·
P
s
θ
p
,
h
,
η
+
P
c
i
r
.Recall that R
~
s
(
P
s
θ
p
,
h
,
η
) denotes sensor node’s reformulated transmission rate under the probabilistic transmission given by (27), (29), and (33), respectively. f
~
(
P
s
θ
p
,
h
,
η
) represents the boundary function for the convex hull given by (26), (28), and (32), respectively. As the above problem (
P
3
) is strictly concave, we can obtain the optimal solution via Lagrangian method. Specifically, we can construct the Lagrangian function, denoted by L
(
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
), as(39)
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
=
E
h
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
-
η
P
c
i
r
+
λ
e
β
R
A
-
μ
R
t
h
+
δ
P
a
v
,where(40)
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
1
+
μ
R
~
s
P
s
θ
p
,
h
,
η
+
λ
f
~
P
s
θ
p
,
h
,
η
-
η
ξ
P
s
u
c
c
+
δ
P
s
θ
p
,
h
,
η
,and λ, μ, and δ are Lagrangian multipliers associated with constraints (35), (36), and (14), respectively. We denote the optimal solution of problem (
P
3
) by P
s
∗
θ
p
,
h
,
η. Then, based on the Karush-Kuhn-Tucker (KKT) conditions, we can obtain(41)
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
P
s
∗
θ
p
,
h
,
η
∈
0
,
P
p
k
,
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
s
∗
θ
p
,
h
,
η
<
0
,
i
f
P
s
∗
θ
p
,
h
,
η
=
0
,
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
θ
p
,
h
,
η
=
P
s
∗
θ
p
,
h
,
η
>
0
,
i
f
P
s
∗
θ
p
,
h
,
η
=
P
p
k
,where(42)
d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
=
P
s
u
c
c
1
+
μ
d
R
~
s
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
+
λ
d
f
~
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
-
η
ξ
P
s
u
c
c
-
δ
.Denote the optimal Lagrangian multipliers byλ
∗, μ
∗, and δ
∗, respectively. Then, the optimal solution P
s
∗
θ
p
,
h
,
η for problem (
P
3
) is determined as follows:Case 1.
(43)
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
<
0
,
P
p
k
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
>
0
.Case 2.
(44)
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
0
<
0
,
P
p
k
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
P
p
k
>
0
,
P
~
1
,
o
t
h
e
r
w
i
s
e
,where P
~
1
∈
(
0
,
P
p
k
) and it is the solution to the following equality:(45)
P
s
u
c
c
1
+
μ
d
R
~
s
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
+
λ
d
f
~
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
-
η
ξ
P
s
u
c
c
-
δ
=
0
.Case 3.
(46)
P
s
∗
θ
p
,
h
,
η
=
0
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
0
<
0
,
P
p
k
,
i
f
d
L
λ
∗
,
μ
∗
,
δ
∗
,
P
s
θ
p
,
h
,
η
d
P
s
θ
p
,
h
,
η
P
s
∗
θ
p
,
h
,
η
=
P
t
a
n
>
0
,
P
~
2
,
o
t
h
e
r
w
i
s
e
,where P
~
2
∈
(
0
,
P
t
a
n
) is the solution to d
L
λ
,
μ
,
δ
,
P
s
θ
p
,
h
,
η
/
d
P
s
θ
p
,
h
,
η
=
0.
Although (43)–(46) determine the optimal solution of problem (
P
3
), we can observe that our obtained optimal solution belongs to the range that the boundary function f
~
(
P
s
θ
p
,
h
,
η
) overlaps with the original function f
(
P
s
θ
p
,
h
,
η
). Consequently, (43)–(46) also determine the optimal solution of nonconvex problem (
P
2
).
As we convert the original EE-maximization problem(
P
1
) with the fractionally structured objective function to the parametric problem (
P
2
) with the linearly additive objective function by employing the fractional programming approach, we also need to determine the optimal solution to the original problem (
P
1
) based on (43)–(46). According to the fractional programming theory, if we can find the parameter η
∗ such that (47)
J
η
∗
=
E
h
J
η
∗
,
P
s
θ
p
,
h
,
η
∗
P
s
θ
p
,
h
,
η
∗
=
P
s
∗
θ
p
,
h
,
η
∗
=
0
,then we must have Γ
E
E
o
p
t
=
η
∗, where Γ
E
E
o
p
t denotes sensor node’s maximum energy efficiency and can be determined by Algorithm 1. Note that our obtained optimal power allocation scheme can be dynamically adjusted based on PU’s QoS exponent θ
p. Consequently, how to derive PU’s QoS exponent is a critically important issue. In realistic wireless systems, each type of services/applications for a particular wireless system usually has similar delay QoS requirement. Then, WAHSN can determine the corresponding values of θ
p based on the service types of primary networks. Moreover, we in this paper mainly focus on the single link scenario. However, our work can also extend to the multilink scenario. Specifically, on one hand, if each PU’s licensed channel is only allowed to be shared by one SS-SR link, then our proposed power allocation scheme can be directly applied. On the other hand, if each PU’s licensed channel is allowed to be shared by all N SS-SR links, we can also adopt the theories of convex hull and probabilistic transmission to convert the original nonconvex problem to the equivalent convex problem, where the inflexion point in the single link scenario is replaced by the inflexion hyperplane in the multilink scenario.Algorithm 1:Algorithm to determine sensor node’s maximum average energy efficiency.
(1)Initialization: n
=
0 and η
0 where η
0 satisfies
J
η
0
=
E
h
J
η
0
,
P
s
θ
p
,
h
,
η
0
>
0
(2)for  n
=
0,1
,
2
,
…
do
(3)  Letη
=
η
n
(4)  Solve problem(
P
2
) with η
n, where the optimal solution denoted P
s
∗
θ
p
,
h
,
η
n is determined by (43)–(46).
(5)if  |
J
η
n
|
=
|
E
h
{
J
(
η
n
,
P
s
∗
θ
p
,
h
,
η
0
)
}
|
>
ε
then
(6)   Update parameterη
n by
η
n
+
1
=
η
n
-
J
η
n
J
′
η
n
=
E
h
{
P
s
u
c
c
⋅
R
s
P
s
∗
θ
p
,
h
,
η
n
}
E
h
{
ξ
⋅
P
s
u
c
c
⋅
P
s
∗
θ
p
,
h
,
η
n
}
+
P
c
i
r
(7)   Letn
=
n
+
1
(8)else
(9)   The optimal power allocation strategyP
s
o
p
t
θ
p
,
h and the sensor node’s maximum average
energy-efficiencyΓ
E
E
o
p
t are determined by
P
s
o
p
t
θ
p
,
h
=
P
s
∗
θ
p
,
h
,
η
n
Γ
E
E
o
p
t
=
η
n
=
E
h
{
P
s
u
c
c
⋅
R
s
P
s
∗
θ
p
,
h
,
η
n
}
E
h
{
ξ
⋅
P
s
u
c
c
⋅
P
s
∗
θ
p
,
h
,
η
n
}
+
P
c
i
r
(10)end if
(11)end for
(12)return  P
s
o
p
t
θ
p
,
h and Γ
E
E
o
p
t


## 4. Simulation Results

In this section, we will evaluate the performance of our proposed energy-efficient power allocation scheme by simulations. Specifically, in our simulations, we set that the bandwidthB
=
1
0
5 Hz, the frame duration T
f
=
2ms, the number of sensor nodes K
=
10, PU’s constant transmitting power P
p
=
100mW, sensor node’s maximum allowed average transmitting power P
a
v
=
100mW, and sensor node’s maximum allowed peak transmitting power P
p
k
=
150mW. Moreover, we also set PU’s data arrival rate R
A
=
1.5 nats/s/Hz, the amplifier coefficient of the sensor node ξ
=
0.2, the constant circuit power of the sensor node P
c
i
r
=
50mW, and the noise power σ
2
=
10mW.Figure3 shows the average energy efficiency of the sensor node achieved by our proposed optimal power allocation scheme as the function of PU’s QoS requirement described by PU’s QoS exponent θ
p with different values of sensor node’s access probability P
a
c
c. We can observe from Figure 3 that, under the given access probability P
a
c
c, the sensor node can achieve the highest energy efficiency for the small value of PU’s QoS exponent θ
p implying loose delay QoS requirement. Moreover, the energy efficiency of the sensor node decreases as the value of PU’s QoS exponent θ
p increases, which denotes that the sensor node can only achieve lower energy efficiency when PU’s delay QoS requirement becomes stringent. Furthermore, we can also observe that the sensor node can only achieve zero energy efficiency; that is, the sensor node stops its transmission, when PU’s delay QoS requirement becomes extremely stringent resulting in large value of θ
p. Figure 3 also illustrates that under the given value of θ
p, that is, the given PU’s delay QoS requirement, the sensor node’s energy efficiency is the decreasing function of the access probability P
a
c
c. The main reason is that the collision among sensor nodes increases for larger value of P
a
c
c. Consequently, the sensor node will waste more energy on spectrum access process and thus can only achieve lower energy efficiency.Figure 3
Optimal average energy efficiency of the sensor node versus PU’s QoS exponentθ
p under different values of access probability P
a
c
c.To more explicitly demonstrate the relationship between the sensor node’s achievable energy efficiency and PU’s delay QoS requirement, Figure4 depicts the energy efficiency that the sensor node can achieve under our proposed power allocation scheme as the function of PU’s delay threshold D
t
h and its maximum allowed violation probability P
t
h under different values of the access probability P
a
c
c where we can observe similar phenomenon as in Figure 3. In particular, the sensor can achieve higher energy efficiency under the larger values of D
t
h and P
t
h but can only get lower energy efficiency under the smaller values of D
t
h and P
t
h. Such phenomenon can be explained based on the analysis in Section 3.1. Specifically, we have obtained that the larger values of D
t
h and P
t
h result in smaller value of θ
p implying looser PU’s delay requirement. On the contrary, the smaller values of D
t
h and P
t
h cause larger value of θ
p denoting more stringent PU’s delay demand. Therefore, we can obtain the curves demonstrated in Figure 4.Figure 4
Optimal average energy efficiency of the sensor node as the function of PU’s maximum allowed delay-boundD
t
h and violation probability threshold P
t
h under different values of access probability P
a
c
c.Figure5 shows the optimal energy efficiency achieved by the sensor node under the proposed power allocation scheme versus the number of sensor nodes K and the access probability P
a
c
c. We can observe from Figure 5 that, under the given number of sensor nodes, the energy efficiency achieved by the sensor node first increases as the value of P
a
c
c increases and then decreases while keeping increasing the value of P
a
c
c. Such the phenomenon can be explained as follows. Specifically, on one hand, when the value of P
a
c
c is small, the probability that all sensor nodes stay silent in each slot is large such that the spectrum resource is not efficiently utilized by the WAHSN. Consequently, increasing the value of P
a
c
c will improve the energy efficiency of the sensor node when the value of P
a
c
c is small. On the other hand, when the value of P
a
c
c is large, the probability that the collision among sensor nodes in each slot is high as each sensor node will try to access the spectrum with large probability. Therefore, the energy efficiency of the sensor node will be degraded if we keep increasing the value of P
a
c
c. Based on the above analysis, we can conclude that it is critically important to find the optimal access probability as the energy efficiency achieved by the sensor node is highly related to P
a
c
c. Consequently, to achieve the aforementioned goal, Figure 6 shows the relationship between the optimal access probability of the sensor node and the number of sensor nodes. We can observe from Figure 6 that the optimal access probability is the decreasing function of the number of sensor nodes. This is because when the number of sensor nodes is small, the competition among sensor nodes is not fierce and thus the sensor node can access the spectrum with high probability. On the contrary, when the number of sensor nodes is small, the sensor node needs to lower the access probability to avoid high collision probability.Figure 5
Optimal average energy efficiency of the sensor node as the function of the number of sensor nodes and the access probabilityP
a
c
c.Figure 6
Optimal access probability of the sensor node that achieves the optimal average energy efficiency versus the number of sensor nodes.Figure7 depicts the energy efficiency of the sensor node as the function of PU’s data arrival rate under different PU’s delay QoS requirements and values of sensor node’s access probability. We can observe from Figure 7 that, under the same PU’s delay QoS requirement and sensor node’s access probability, the energy efficiency achieved by the sensor node decreases as PU’s data arrival rate increases. This is because that under the same PU’s delay QoS requirement, that is, the identical value of PU’s QoS exponent θ
p, increasing PU’s data arrival rate requires larger PU’s service rate, which demands the sensor node to introduce smaller interference power on PR. Consequently, the sensor node can only achieve lower throughput and thus get smaller energy efficiency while increasing PU’s data arrival rate. Moreover, we can also observe similar phenomenon that the sensor node can achieve higher energy efficiency under smaller values of θ
p and P
a
c
c but can only get lower energy efficiency under larger values of θ
p and P
a
c
c.Figure 7
Optimal average energy efficiency of the sensor node versus PU’s data arrival rateR
A under different values of PU’s QoS exponent θ
p and sensor node’s access probability P
a
c
c.Figure8 shows the sensor node’s energy efficiency versus PU’s transmitting power under different values of PU’s QoS exponent θ
p and sensor node’s access probability P
a
c
c. We can observe from this figure that, under the same PU’s delay QoS requirement and sensor node’s access probability, the sensor node’s energy efficiency dynamically varies with PU’s transmitting power. Specifically, when PU’s transmitting power is small, the sensor node’s energy efficiency is zero because the sensor node needs to stop its transmission to guarantee PU meeting the targeted delay QoS requirement and thus can only get the zero energy efficiency. When PU’s transmitting power is increasing, the sensor node can obtain nonzero energy efficiency. However, if we keep increasing PU’s transmitting power, the sensor node’s energy efficiency decreases. This is mainly because larger PU’s transmitting power will impose larger interference on the sensor node. Consequently, the throughput achieved by the sensor node decreases and thus can only get lower energy efficiency.Figure 8
Optimal average energy efficiency of the sensor node versus PU’s transmitting powerP
p under different values of PU’s QoS exponent θ
p and sensor node’s access probability P
a
c
c.


## 5. Conclusions

In this paper, we developed the statistical delay quality-of-service (QoS) provisioning framework for the energy-efficient spectrum-sharing based wireless ad hoc sensor network (WAHSN). Based on the established PU’s delay QoS provisioning framework, we further formulated the nonconvex optimization problem which aims at maximizing the average energy efficiency of the sensor node while meeting PU’s statistical delay QoS requirement as well as satisfying the sensor node’s average transmission rate and average and peak transmitting power constraints. To solve our formulated nonconvex optimization problem, we first usedfractional programming to convert the original fractional-structured problem to theparametric nonconvex program. Then, by adopting the theories ofconvex hull andprobabilistic transmission, we converted the parametric nonconvex problem to the equivalent convex problem and obtain the optimal power allocation strategy under the given parameter via Lagrangian method. Finally, we derived the optimal average energy efficiency of the sensor node in the WAHSN by employing theDinkelbach method. Simulation results show that our derived optimal power allocation strategy can be dynamically adjusted based on PU’s delay QoS requirement as well as the channel conditions. Moreover, the impact of PU’s delay QoS requirement on sensor node’s energy efficiency is also illustrated.



---

*Source: 6545791-2016-11-14.xml*