joint throughput maximization and power control in poisson
TRANSCRIPT
arX
iv:1
611.
0854
6v1
[cs
.NI]
25
Nov
201
6
Joint Throughput Maximization and Power Control in
Poisson CoopMAC Networks
Masoumeh Sadeghi, Homa Nikbakht, Amir Masoud Rabiei, and VahidShah-Mansouri
School of Electrical and Computer Engineering, College of Engineering
University of Tehran, P.O. Box 14395–515, Tehran, IranE-mail: masoume.sadeghi, homanikbakht, rabiei, [email protected]
Abstract
A cooperative medium access control (CoopMAC) network with randomlydistributed helpers is considered. We introduce a new helper selection schemewhich maximizes the average throughput while maintaining a low power con-sumption profile in the network. To this end, all transmissions are assumedto be performed using power control. We assume that each node can esti-mate the channel between itself and a receiving node. Then, it evaluates theminimum transmission power needed to achieve the desired signal to noiseratio (SNR). If the required transmission power is less than the maximumtransmission power of the node, the communication is regarded as success-ful. Otherwise, the transmission is canceled. In order to increase the averagethroughput, we assume that the cooperative link with the highest transmis-sion rate is chosen from those that can successfully forward the source signalto destination. Also, when there are several links with the same rates, theone with minimum required power is given the highest priority. Assumingthat the helpers are distributed as a Poisson point process with fixed inten-sity, we derive exact expressions for the average throughput and the powerconsumption of the network. Simulation results show that our scheme isable to significantly increase the throughput in comparison to the conven-tional CoopMAC network. It is also able to reduce the power consumptioncompared to a network with no power control approach.
Keywords: Cooperative medium access control (CoopMAC), IEEE802.11b, Poisson point process, Power control, stochastic geometry.
Preprint submitted to Elsevier November 28, 2016
1. Introduction
In wireless networks with battery powered nodes, the rate versus powerconsumption has always been a trade-off. The higher the rate, the lowerthe network lifetime. The power control mechanisms are used to achieve de-sired rate levels with optimizing the lifetime of the nodes and consequentlythe network. Power control can also be seen as a tool to provide efficientspectral reuse by alleviating the level of interference in the network. Re-cently, a noticeable part of studies is dedicated to incorporate power controltools into the IEEE 802.11 standard specifically in the MAC schemes. Twodifferent cases are considered for power control in MAC protocols, namelysimilar power levels and varying power levels. When the same level of poweris assumed for all the links, symmetric links are created in the network[16].Symmetric links along with the simplicty of the algorithms are the mainadvantage of this design. However, it is shown that this case is a time con-suming approach since nodes must declare their agreement with sharing thesame power level before any transmission. We shall notice that using thesame power level while interference is involved, cannot guarantee symmetry[22, 18].
In [18], a power control scheme based on ready–to–send (RTS) and clear–to–receive (CTS) handshaking mode, is presented. In this scheme, the sourcenode specifies its current transmit power level while sending RTS packet and,in turn, recognizes the desired level of transmit power through the receivedCTS packet from the destination. Therefore, besides maintaining the de-manded signal-to-noise ratio (SNR), the data is transmitted at appropriatepower level. In power control MAC protocol that has been proposed in [5], theacknowledgment (ACK) and data packets are transmitted in a lower powerlevel than RTS and CTS packets in order to save the energy. It has beenshown that the throughput of this protocol degrades since by decreasing thepower level of ACK and data packets, their transmission range and carriersensing zone will be limited [10]. To overcome these deficiencies, a variationof this protocol has been presented in [10] where data packets are transmittedusing a maximum power level periodically to be sensible for nodes in carriersensing zone.
In [1] a new power control MAC protocol has been proposed in which eachnode saves the required transmit power level of its neighbors after the firstRTS / CTS handshake in a table. The maintained table looked up before acommunication happens in order to change the transmit power in every node
2
related to its destination dynamically. The battery power would be saved inthis case; however, in mobile scenarios it does not work efficiently He andLi [6]. Transmitting busy tone pulses to control the power has been alsoinvestigated in power controlled multiple access (PCMA) protocol Monkset al. [13]. Destination node, while receiving the data packet, sends a set ofperiodical busy tones through a different channel with the power equal tothat of the maximum additional noises that it can tolerate. Other nodes bylistening to both channels can bound their transmit power according to thecurrent signal and busy tones’ strength. It has been investigated that this isnot a collision-free scheme in time of transmitting the ACK packet Lin et al.[11].
The aforementioned power control MAC protocols are non–cooperativewith multi-rate scenarios. Recently, cooperation communications as a promis-ing method have emerged to enhance the performance of IEEE 802.11 basedMAC protocols. The benefits of cooperation in MAC layer, mainly in termsof throughput, are presented in [14, 12, 23, 15]. In [12], a cooperative MACprotocol, called CoopMAC, based on RTS-CTS scheme has been proposedto exploit the multi–rate attribute of MAC protocols. Their goal was to pro-vide cooperation communication link for low date rate links such that thethroughput increases. A similar scenario has also been investigated in [23] toenhance the throughput performance of ad hoc networks. However, the addi-tional overhead of cooperative control packets has been neglected in [12] and[23]. Handling a busy tone channel in a CoopMAC scheme to avoid collisionhas been adopted in [20]. An approach to separate the mobile helpers fromthe static ones in a CoopMAC network has been presented in [9]. Also, in [8],a game theoretic based method has been introduced to examine a CoopMACprotocol with incentive design.
The existing CoopMAC protocols mainly focus on throughput perfor-mance improvement through different helper selection scenarios, and powercontrol mechanisms with regard to minimizing the energy consumption andnetwork lifetime have been conducted in a limited number of research ef-forts in the literature. However, in energy constrained wireless networks,energy conservation is a critical issue that should be taken in to account. In[21], a distributed energy-adaptive location-based CoopMAC (DEL-CMAC)protocol with a novel approach to enhance the performance of the mobilead hoc networks (MANET) from the lifetime and energy perspectives hasbeen investigated. In this scheme, among potential helpers, the one withminimum transmitting power is defined as the best helper. DEL-CMAC is
3
known to prolong the network lifetime significantly. A total power controlCoopMAC (TPC-MAC) protocol has been proposed in [19] to minimize theoverall transmit power in wireless sensor networks. In this protocol, an ex-tended CTS packet with a non–cooperative power field is applied. Similarly,the best helper in this case is a potential node that minimizes the transmit-ting power.
In our previous work [17], in order to improve the throughput performanceof the CoopMAC protocols in the presence of shadowing, we proposed a newhelper selection approach in which the potential helpers are divided intoseveral tiers based on their distances from the source and destination nodes.From the helper’s tiers with the maximum cooperative transmission rate,the helper that is less affected by shadowing is chosen for cooperation. Wederived upper and lower bounds on the average cooperative throughput forthe proposed CoopMAC protocol and observed that the performance of ourproposed scheme was quite close to the upper bound.
In this paper, we propose a new CoopMAC scheme with a novel helper se-lection approach and an efficient power control mechanism to simultaneouslyimprove the overall throughput and the energy consumption. In our coopera-tive protocol, among potential helpers that maximize the overall throughput,the helper with the minimum transmission power is defined as the best helper.We assume that in the network, the helpers are distributed randomly basedon a homogeneous two-dimensional Poisson point process (PPP) with a fixeddensity. We derive an expression for cooperative throughput and the powerconsumption in our random CoopMAC protocol.
The remainder of this paper is organized as follows. In Section 2.1, thesystem model is defined. Our helper selection approach is proposed in Sec-tion 2.2. Throughput performance analysis and average power consumptionof our helper selection algorithm are presented in Section 3 and Section 4,respectively. Simulation results are provided in Section 5 to demonstrate thesuperiority of our scheme over a conventional CoopMAC protocol. Finally,the paper is concluded in Section 6.
2. Cooperative MAC Model
In this section, we present our CoopMAC system model as well as themethod the helpers are chose in the network by the source node.
4
S D
+11Mbps
5.5Mbps
2Mbps
1Mbps
++
+
+
+
++
+
+
+
+
++
+
+
+
+ +
+
+
H
C
LSHLHD
LSD
48.2
67.1
74.7
100
Figure 1: System model.
2.1. System Model
A CoopMAC protocol is considered, where all transmissions are per-formed according to an IEEE 802.11b Standard. In this standard, the trans-mission rate of a link depends on the distance between the source and desti-nation nodes. Hence, all point-to-point links of the network can be classifiedinto four groups of rates. Table 1 shows these rates along with the cor-responding link distances. Fig. 1 illustrates a typical Cooperative MACnetwork where a source node, S, intends to communicate with a destinationnode, D, at the distance of LSD. This communication can be performed ei-ther through a direct link between S and D or by making use of any availablehelper node, H, via a dual-hop communication. Helpers are assumed to bedistributed over the region C, according to a homogeneous two-dimensionalPoisson point process (PPP) with fixed density λ. Also, LSH and LHD rep-resent the source-to-helper and helper-to-destination distances, respectively.
Let tSH and tHD show the transmission time of one bit over S–H and H–D
5
Table 1: Classification of the point-to-point links in the IEEE 802.11b Stan-dard (for BER ≥ 10−5).
Link Group distance (m) Rate (Mbps)1 [0, 48.2) 112 [48.2, 67.1) 5.53 [67.1, 74.7) 24 [74.7, 100) 1
links, respectively. Assuming helper has single transceiver and employs adecode-and-forward scheme, the total achievable rate from cooperative linkof S–H–D, denoted by RCoop, is given by [12]
RCoop =1
tSH + tHD=
11
RSH+ 1
RHD
=RSH RHD
RSH +RHD, (1)
where RSH and RHD are the rates of S–H and H–D links, respectively. Deduc-ing from Table 1 and eq.(1), it can be seen that different rates are achievablevia cooperative links.
According to Table 1, values of RSH and RHD are relative to the distancesof helper from the source and destination. For instance, assuming that helperH has a S-H link of group 1 and H-D link of group 2, i.e, 0 ≤ LSH < 48.2and 48.2 ≤ LHD < 67.1, that results in RSH = 11Mbps and RHD = 5.5Mbps.In this case RCoop by considering (1) is 3.67Mbps. Table 2 categorizes thehelpers into 6 groups, namely H1 through H6, based on the amount of RCoop
that they can provide.Another characteristic that determines the performance of a link is its
average channel gain. We consider channels to be modeled by path-loss andfading. Therefore, when a signal is transmitted over a link, the received SNRat the receiver is given by [2, Eq. 1]
Γ =P K0
N0 dαΩ, (2)
where P is the transmission power, d is the link distance, α is the path-lossexponent of the system and Ω is the fading power of the channel.Values ofΩ for different channels are identical independent random variables with cu-mulative distribution function (CDF) of FΩ(.) that assumed to be identical
6
Table 2: Classification of helpers based on their relative position from sourceand destination
Helper’sGroup
LSH (m) RSH (Mbps) LHD (m) RHD (Mbps)RCoop (Mbps)
H1 [0, 48.2) 11 [0, 48.2) 11 5.5
H2[0, 48.2) 11 [48.2, 67.1) 5.5
3.67[48.2, 67.1) 5.5 [0, 48.2) 11
H3 [48.2, 67.1) 5.5 [48.2, 67.1) 5.5 2.75
H4[0, 48.2) 11 [67.1, 74.7) 2
1.69[67.1, 74.7) 2 [0, 48.2) 11
H5[67.1, 74.7) 5.5 [67.1, 74.7) 2
1.47[67.1, 74.7) 2 [48.2, 67.1) 5.5
H6 [67, 1, 74.7) 2 [67, 1, 74.7) 2 1
for different channels in our network. Also, K0 and N0 are constants thatconsidered to be the same for all links of network; the earlier referring to an-tennas gain and the latter to the power of the additive Gaussian noise at thereceiver of the link. In order to decrease the power consumption, all trans-missions are considered to be performed by the minimum required power. Atransmission is assumed to be successful, if its corresponding received SNR atthe end of the link is greater than a threshold, γth. Consequently, from (2), itcan be deduced that the minimum power needed for successful transmissionover S–D link is
PminSD =
N0 LαSD
K0ΩSDγth, (3)
where ΩSD is the fading power over S–D. Similarly, the minimum requiredpower for successful transmission over S–H and H–D links are respectivelygiven by
PminSH =
N0 LαSH
K0Ω1γth (4)
PminHD =
N0 LαHD
K0Ω2γth, (5)
7
where Ω1 and Ω2 are fading powers of S–H and H–D links, respectively. De-ducing from (4) and (5), the total required power for successful transmissionover S–H–D is
PminH =
N0 LαSH
K0Ω1
γth +N0 L
αHD
K0Ω2
γth. (6)
We note that in our model, a maximum transmission power of Pmax hasbeen taken into account for all nodes, including the source, the destinationand the helpers. If the required power for successful communication in a linkexceeds this value, the transmission would be discarded. As a result, theprobability of successful communication over direct link, can be computed as
PSuccDir (LSD) = PrPmin
SD ≤ Pmax
= PrN0 L
αSD
K0ΩSDγth ≤ Pmax
= Pr
ΩSD ≥ N0 γthK0 Pmax
LαSD
= 1− FΩ(ALαSD), (7a)
where
A =N0 γthK0 Pmax
. (7b)
Similarly, the probability of successful transmission over S–H–D link is givenby
PSuccH (LSH, LHD) = PrPmin
SH ≤ Pmax PrPminHD ≤ Pmax
= (1− FΩ(ALαSH))× (1− FΩ(ALα
HD)) . (8)
The performance of a link is evaluated by its throughput which is introducedas the rate of successful transmission and is given by
T = R× PSucc, (9)
where R is the corresponding rate of the link, and PSucc is its probability ofsuccessful transmission.
2.2. Helper Selection Scheme
In case that there are more than one helper that can cooperate in thetransmission, the source needs to select the best helper based on an objective.
8
In this section, we propose a helper selection scheme that can increase thethroughput and can decrease power consumption of the network. In thisarticle, helper H and its corresponding cooperative path are called Potential,provided that the transmission over S–H–D is successful, i.e., Pmin
SH ≤ Pmax
and PminHD ≤ Pmax. To this end, we consider the following essential conditions
for the helper, which is being chosen for cooperation:
• The helper must be potential (i.e., PminSH ≤ Pmax and Pmin
HD ≤ Pmax).
• The corresponding cooperative link should have an equal or greatertransmission rate than the direct link.
• In the cases that the transmission rate of the cooperative link is equalto the direct link, the source plus helper power consumption must beless than the direct link.
Suppose there are more than one helper that can satisfy the above conditions.In such cases, to maximize the throughput, the helper with the maximumRCoop would be chosen for cooperation. In addition, to satisfy the powercontrol purposes, when several helpers can provide the largest RCoop, the onethat minimizes the power consumption should be selected for cooperation.If no helper could meet the above mentioned conditions, it would be morebeneficial to perform the transmission through the direct link.
According to the above mentioned conditions, only those groups of helpersthat provide an equal or higher rate than the direct link are asked to par-ticipate in cooperation. In Table 3, we define five different communicationcases, based on the direct link transmission rate and helper’s group that canassist them. Also, the beneficial helper’s groups for each case are illustratedin Fig. 2, where Si (i = 1, .., 5) represents a sample source node of the ithcase. It should be mentioned that for Case 5, in which the distance betweenthe source and destination nodes is more than 96.4 meters, no helper fromH1 group exists in the network.
Upon receipt of an RTS from the source, the destination node is able tocalculate the transmission power and the rate of the direct link. Then, itattaches them to the CTS packet to inform the source. Each idle helper thatoverhears the RTS and extended CTS respectively, evaluates the channelcondition between itself and the source and destination, and examines thementioned conditions for the helper. Any helper, that can satisfy thosethree conditions, is a potential helper and can be used for cooperation. In
9
Table 3: The link types and their corresponding beneficial helper groups.
Link Case LSD (m) RSD (Mbps) Beneficial Helper(s)
1 [0, 48.2) 11 None
2 [48.2, 67.1) 5.5 H1
3 [67.1, 74.7) 2 H1,H2,H3
4 [74.7, 96.4] 1 H1,H2,H3,H4,H5,H6
5 (96.4, 100] 1 H2,H3,H4,H5 ,H6
D
S1
S2 S3
48.2
67.1
74.7
100
48.2
48.2
67.1
H1
H2
H3
H4
H5
H6
(a)
DS4 S5
48.2
67.1
74.7
96.4
100
48.2
67.1
74.7
48.2
67.1
74.7
(b)
Figure 2: Operating regions of different groups of potential helpers (all dis-tances are in meter).
our proposed helper selection algorithm, the helper with the highest ratehas priority for cooperation. Then, among the cooperative paths with thesame rate, the one with minimum total power consumption is chosen forcooperation. To this end, we employ a timer based approach for finding thesuitable helper. Upon hearing the CTS, each helper that could satisfy threeabove conditions, runs a timer according to some parameters. When its timeris expired, the node transmits the helper-ready-to-send (HTS) packet. If thehelper receives HTS from another node before expiry of its own timer, it
10
cancels its timer. We define timer duration for a helper as
T = ϑ (PminH + 2 (i− 1)Pmax), (10)
where ϑ is a normalizing constant, PminH is the required power of source
plus helper for successful transmission, and i represents the group of helper.According to equation (10), we can see that the timer of each helper isdefinitely greater than the helpers with higher rate, and lower than those withless rates. For example, the timer of the helpers of the first and second groupare respectively equal to ϑPmin
H and ϑPminH + 2Pmax. Since the maximum
possible value of PminH is 2Pmax, we can be sure that the timer of helpers in first
group are always smaller than that for helpers of second group, hence, desiredorder would be established in helper selection. Also, among the helperswith the same rate, the helper with minimum required power has the lowesttimer duration, and highest priority for being selected for cooperation. Thehelper whose timer expires first, sends a helper-ready-to-send (HTS) packetto the source and is chosen for cooperation. Meanwhile, other potentialhelpers must cancel their timers as soon as they receive an HTS packet.If an HTS packet is received by source, the required power is evaluatedfrom the strength of the HTS packet and data would be transmitted tothe selected helper. If the HTS packet is not received after a certain timeduration, the data would be transmitted through the direct link, supposingthat Pmin
SD < Pmax. Our proposed CoopMAC protocols have been presentedfor the source, destination and helper nodes, in Algorithms 1, 2 and 3,respectively.
3. Performance Analysis
In this section, we evaluate the average throughput, which is the rate ofsuccessful transmission, for the proposed algorithm in different cases. Assumethat PLoss
i denotes the probability that no potential helper of the ith groupexists. Also assume that Ci is the region in which the helpers of GroupHi are located. Then, the helpers in this region are distributed accordingto a homogeneous two-dimensional PPP with density λ. Suppose that thenumber and the positions of the helpers in Ci are known. Then, PLoss
i equalsthe probability that none of GroupHi helpers being potential for cooperation.
11
Algorithm 1 Best helper selection and data transmission protocol for source
1: Initialization:
2: send an RTS packet3: if CTS packet received then goto Step 74: else
5: goto Step 16: end if
7: if HTS packet received then
8: Evaluate PminSH from HTS
9: send data to the helper with transmission power of PminSH goto Step 19
10: else
11: goto Step 1312: end if
13: Extract PminSD from CTS
14: if PminSD < Pmax then
15: send data to the destination with transmission power PminSD
16: else
17: goto Step 2018: end if
19: if an ACK packet is not received then
20: perform a random backoff and goto Step 121: end if
22: Transmission Complete
Algorithm 2 Data receiving protocol for destination
1: Silent
2: if an RTS packet received then
3: evaluate PminSD and RSD from RTS
4: attach PminSD and RSD to CTS
5: send CTS packet and goto Step 76: end if
7: if data packet received then
8: send ACK packet and goto Step 19: else
10: goto Step 111: end if
12
Algorithm 3 Cooperation protocol for helpers
1: Silent
2: if an RTS packet received then
3: evaluate PminSH and RSH from RTS and goto Step 5
4: end if
5: if CTS packet received then
6: evaluate PminHD and RHD from the strength of CTS
7: extract PminSD and RSD from CTS and goto Step 9
8: end if
9: if One of the conditions defined in Steps 10 and 11 holds, then goto
Step 1310: Pmin
SH < Pmax , PminHD < Pmax , R
Coopi > RSD
11: PminSH < Pmax , P
minHD < Pmax , P
minSH + Pmin
HD < PminSD , RCoop
i = RSD
12: end if
13: set timer according to eq.(10)14: if HTS packet received then
15: reset timer and goto Step 116: else if timer reaches zero then
17: send HTS packet and goto Step 1918: end if
19: if data packet received then
20: send data to the destination with power of PminSH and goto Step 1
21: else
22: goto Step 123: end if
13
By averaging on different realizations of helper’s placements, we have
PLossi = E
∏
Hj∈Ci
(1− PSuccH (Lj,1, Lj,2))
, (11)
where Hj is the jth helper of the system, located at the distance of Lj,1 andLj,2 form the source and destination nodes, respectively and PSucc
H (·, ·) canbe obtained from (8). Using [7, Eq. 3.35], for PPP random variables, (11)can be rewritten as
PLossi = exp
(
−λ
ˆ
Ci
PSuccH (LSH, LHD) ds
)
. (12)
As proved in appendix Appendix A, (12) can be evaluated for i = 1, ..., 6 as
PLoss1 = exp (−λ S(48.2, 48.2, LSD)) , (13a)
PLoss2 = exp (−2λ S(67.1, 48.2, LSD))× (PLoss
1 )−2, (13b)
PLoss3 = exp (−λ S(67.1, 67.1, LSD))× (PLoss
1 PLoss2 )−1, (13c)
PLoss4 = exp (−2λ S(74.7, 48.2, LSD))× (PLoss
1 )−2 × (PLoss2 )−1, (13d)
PLoss5 = exp (−2λ S(67.1, 74.7, LSD))× (PLoss
1 PLoss2 PLoss
3 )−2 × (PLoss4 )−1,
(13e)
PLoss6 = exp (−λ S(74.7, 74.7, LSD))× (PLoss
1 PLoss2 PLoss
3 PLoss4 PLoss
5 )−1,(13f)
where S(·, ·, ·) is given by
S(u, v, x) ,
ˆ θ(u,v,x)
−θ(u,v,x)
ˆ u
0
PSuccH (r ,
√r2 + x2 − 2 r x cos θ) r dr dθs
+
ˆ θ(v,u,x)
−θ(v,u,x)
ˆ v
0
PSuccH (r ,
√r2 + x2 − 2 r x cos θ) r dr dθd
+ 2 x2
ˆ θ(u,v,x)
0
ˆ min(θ(v,u,x),π−θd)
0
PSuccH
(
x sin(θd)
sin(θs + θd),
x sin(θs)
sin(θs + θd)
)
× sin(θs) sin(θd)
sin3(θs + θd)dθs dθd (13g)
14
and θ(·, ·, ·) is
θ(u, v, x) , arccos(v2 − u2 − x2
2 u v
)
. (13h)
In the following, we use (13a) through (13f) and also (7a) to evaluate thethroughput of the links in different cases defined in Table 3.
3.1. Case 1(
0 ≤ LSD < 48.2)
As discussed earlier, for this type of links, communications are performedonly through the direct path with transmission rate of 11 Mbps. As a result,the throughput can be obtained from (9) as
T1 = PSuccDir (LSD)× 11(Mbps). (14)
3.2. Case 2(
48.2 ≤ LSD < 67.1)
In this case, among cooperative links of H1 and the direct link, the pathwith the minimum required power would be selected for cooperation. Recallthat H1 is the first group of helpers where 0 ≤ LSH < 48.2, 0 ≤ LHD < 48.2and RCoop = 5.5Mbps. Any selected path of this case has the rate of 5.5Mbps. Hence, the total achieved throughput would be
T2 = PSucc × 5.5
= (1−PLoss)× 5.5(Mbps), (15)
where PLoss is the probability of the event in which, no potential path existsamong the cooperative paths of the first group and direct link
PLoss = PLoss1 ×PLoss
Dir (LSD), (16)
where PLossDir (LSD) is the outage probability of the direct link that can be
obtained byPLoss
Dir (LSD) = 1− PSuccDir (LSD).
Substituting (16) in (15), T2 is given by
T2 =(
1− PLoss1 × PLoss
Dir (LSD))
× 5.5 (Mbps). (17)
3.3. Case 3(
67.1 ≤ LSD < 74.7)
As discussed before, transmission path selection for Case 3 links is per-formed based on an order. First of all, the H1 group would be searched for a
15
potential cooperative path with minimum transmission power. If no helper isfound, the search is continued among helpers of H2 and H3, respectively. Fi-nally, when all previous searches failed, the transmission would be performedthrough the direct link. So, the average throughput in this case is
T3 =
3∑
i=1
(
(
i−1∏
j=1
PLossj )PSucc
i RCoopi
)
+
( 3∏
j=1
PLossj
)
PSuccDir (LSD)× 2(Mbps),
(18)where RCoop
i is the cooperative transmission rate that can be achieved fromhelpers of Hi group which is defined in Table 2.
3.4. Case 4(
74.7 ≤ LSD < 96.4)
In this case, the search for a potential helper would be done among helpersof H1 through H5, respectively. If no potential helper found, the path withminimum required power will be selected among cooperative paths of H6 anddirect link. Hence, the average throughput in this case is
T4=
5∑
i=1
(
(
i−1∏
j=1
PLossj )PSucc
i RCoopi
)
+
( 5∏
j=1
PLossj
)
(1−PLossDir (LSD)PLoss
6 )×1(Mbps).
(19)
3.5. Case 5(
96.4 ≤ LSD ≥ 100)
This case is actually the same as the previous case, except that thereis not any helper of H1 group for this kind of the links. So, the averagethroughput is
T5=
5∑
i=2
(
(
i−1∏
j=2
PLossj )PSucc
i RCoopi
)
+
( 5∏
j=2
PLossj
)
(1−PLossDir (LSD)PLoss
6 )×1(Mbps).
(20)
4. Average Power Consumption
In this section, we investigate the average of the power consumptionfor our proposed helper selection scheme. We investigate three scenarios ofthe proposed scheme, namely direct transmission, cooperative transmission,direct-cooperative transmission.
16
4.1. Direct Transmission
Depending on the distance of LSD, one step of the scheme may includetransmission over S–D link. In this scenario, the required power for transmis-sion would be equal to Pmin
SD , defined in (3). As mentioned before, the maxi-mum transmission power of each node of system is equal to Pmax. Therefore,the transmission power of this scenario, denoted by PSD, is given by
PSD =
PminSD Pmin
SD ≤ Pmax
0 PminSD > Pmax.
(21)
We denote the complementary cumulative distribution function (CCDF) ofPSD by YSD(x) as
YSD(x) , PrPSD > x. (22)
By substituting (21) in (22), YSD(x) can be computed as
YSD(x) = Prx < PminSD ≤ Pmax (23a)
= PrPminSD ≤ Pmax − PrPmin
SD ≤ x (23b)
substituting (3) into (23b)
YSD(x) = PrN0 LαSD
K0ΩSDγth ≤ Pmax − PrN0 L
αSD
K0ΩSDγth ≤ x (24a)
= PrΩSD > ALαSD − PrΩSD > A Pmax
xLαSD (24b)
= 1− FΩ(ALαSD)−
(
1− FΩ(APmax
xLαSD)
)
(24c)
= FΩ(APmax
xLαSD)− FΩ(ALα
SD) 0 ≤ x ≤ Pmax. (24d)
CCDF function YSD(x) is used in evaluating the CDF of power consumptionin the network later.
4.2. Cooperative Transmission
Consider the scenario in which among the potential helpers of group i,the one with the minimum required power is selected for cooperation. LetPH,i denote the the total transmission power of the selected path in thisscenario and let YH,i(x) denote the related CCDF. As discussed before, thedistribution of the helpers on Ci is a two-dimensional PPP random variable
17
with density λ. Let k and k1 denote the total number of helpers in Ci andthe number of those helpers which are potential for cooperation, respectively.Since PH,i is the minimum transmission power of the potential cooperativepaths in Hi, it can be represented as
PH,i = minP jH,i | j = 1, ..., k1, (25)
where PjH,i is the total power consumption of the jth potential cooperative
path of group i. YH,i(x) is equal to the probability of the event, in which,the required power of all potential helpers in Ci are greater than x. From (6),we can see that the required power of each cooperative path depends on itshelper location and fading power of the S–H (Ω1) and H–D links (Ω2). Thefading power of all the channels are identical independent random variables.Also, from the properties of PPPs, we know that if the number of helpersin region Ci is known, their location would be independent uniform randomvariables over Ci. We deduce that the required power for all the cooperativepaths are identical random variables. Hence, YH,i(x) can be evaluated as
YH,i(x) =∞∑
J=1
J∑
I=1
PrPH,i ≥ x , k = J , k1 = I (26a)
=
∞∑
J=1
Prk = JJ
∑
I=1
PrPH,i ≥ x , k1 = I | k = J (26b)
=
∞∑
J=1
e−λ|Ci|(λ|Ci|)JJ !
J∑
I=1
(
J
I
)
µI1 µ
J−I2 (26c)
=
∞∑
J=1
e−λ|Ci|(λ|Ci|)JJ !
[(µ1 + µ2)J − µJ
2 ] (26d)
= e−λ |Ci| (1−µ1−µ2) − e−λ |Ci| (1−µ2) (26e)
= e−λ |Ci| (1−µ2)(
eλ |Ci|µ1 − 1)
, (26f)
where µ1 is the probability of the event in which, a helper with uniformrandom position over Ci, is potential for cooperation and also, requires a totaltransmission power greater than x. As shown in appendix Appendix B, µ1
18
is
µ1 =1
|Ci|
ˆ
Ci
G(LSH, LHD, x) ds, (27a)
where
G(LSH, LHD, x) =
´ x
0
BLαSH
a2fΩ(
BLαSH
a) [[FΩ(
BLαHD
x−a)− FΩ(
BLαHD
Pmax)] da 0 ≤ x < Pmax
+´
Pmax
x
BLαSH
a2fΩ(
BLαSH
a) 1− FΩ(
BLαHD
Pmax)] da
´
Pmax
x−Pmax
BLαSH
a2fΩ(
BLαSH
a) [FΩ(
BLαHD
x−a)− FΩ(
BLαHD
Pmax)] da Pmax ≤ x < 2Pmax
0 x ≥ 2Pmax.
(27b)
where B , N0 γth/K0. µ2 can be computed as
µ2 =1
|Ci|
ˆ
Ci
1− PSuccH (LSH, LHD) ds. (28)
Substituting (27a) and (28) in (26f), we have
YH,i(x)=exp
(
−λ
ˆ
Ci
PSuccH (LSH, LHD)ds
)(
exp
(
λ
ˆ
Ci
G(LSH, LHD, x)ds
)
− 1
)
.
(29)Using (12) and (29), we have
YH,i(x) = PLossi × (Gi(x)− 1), (30a)
where
Gi(x) , exp
λ
ˆ
Ci
G(LSH, LHD, x) ds
. (30b)
From Appendix A, Gi(x) is obtained for i = 1, . . . , 6 as
G1(x) = exp (λD(48.2, 48.2, LSD, x)) (31a)
G2(x) = exp (2λD(67.1, 48.2, LSD, x)) (G1(x))−2 (31b)
19
G3(x) = exp (λD(67.1, 67.1, LSD, x)) (G1(x)G2(x))−1 (31c)
G4(x) = exp (2λD(74.7, 48.2, LSD, x)) (G1(x)√
G2(x))−2 (31d)
G5(x) = exp (2λD(67.1, 74.7, LSD, x)) (G1(x)G2(x)G3(x)√
G4(x))−2 (31e)
G6(x) = exp (λD(74.7, 74.7, LSD, x)) (G1(x)G2(x)G3(x)G4(x)G5(x))−1,(31f)
where D(·, ·, ·, ·) is defined by
D(a, b, c, x) ,
ˆ θ(a,b,c)
−θ(a,b,c)
ˆ u
0
G(r ,√r2 + c2 − 2 r c cos θ, x) r dr dθs
+
ˆ θ(a,b,c)
−θ(a,b,c)
ˆ v
0
G(r ,√r2 + c2 − 2 r c cos θ, x) r dr dθd
+ 2 c2ˆ θ(a,b,c)
0
ˆ min(θ(a,b,c),π−θd)
0
G
(
csin(θd)
sin(θs + θd), c
sin(θs)
sin(θs + θd), x
)
× sin(θs) sin(θd)
sin3(θs + θd)dθs dθd, (31g)
and θ(·, ·, ·) defined in (13h).
4.3. Joint Direct and Cooperative Transmission
Under some circumstances, the last step of the proposed scheme includesselecting a potential link with minimum required power, among cooperativepaths of Hi and the direct link. Let P i
SD,H denote the power consumptionof the selected path in this scenario. Let Yi
SD,H(.) denote the CCDF ofthis variable. This scenario is the combination of the two previous scenarioswhere from the direct link and the selected path of the second scenario,the potential one with minimum required power is selected for transmission.If the direct path is not a potential link for successful transmission, thisscenario is similar to the cooperation scenario in Subsection B with powerconsumption of PH,i. Also, in scenario that no potential helper exists inHi, this scenario is equivalent to the direct path scenario in Subsection Awhere power consumption is equal to PSD. Supposing that the direct link isa potential transmission link and at least one potential helper exists in Hi,the transmission power of this scenario would be equal to minPSD , PH,i,
20
where PSD and PH,i are both greater than zero. So, it can be deduced that
YiSD,H(x) = PrPH,i > x PrPSD > x+ P i
Loss PrPSD > x+PLoss
Dir (LSD) PrPH,i > x. (32)
Using the definition of CCDF, YiSD,H(x) can be obtained as
YiSD,H(x) = YSD(x)Y
iH(x) + PLoss
i YiH(x) + PLoss
Dir (LSD)YSD(x). (33)
4.4. The CDF of power consumption
We denote the CDF function of the consumed power in ith case (definedin Table 3) by F i
P (x). Based on the path selection scheme proposed for eachscenario, and using equations (22) and definition of CDF function, F i
P (x) canbe computed for i = 1, ..., 5 as
F 1P (x) = 1− YSD(x) (34a)
F 2P (x) = 1− Y
1SD,H(x) (34b)
F 3P (x) = 1−
3∑
i=1
YiH(x)
i−1∏
j=1
PLossj − YSD(x)
3∏
j=1
PLossj (34c)
F 4P (x) = 1−
5∑
i=1
YiH(x)
i−1∏
j=1
PLossj − Y
6SD,H(x)
5∏
j=1
PLossj (34d)
F 5P (x) = 1−
5∑
i=2
YiH(x)
i−1∏
j=2
PLossj − Y
6SD,H(x)
5∏
j=2
PLossj . (34e)
Using the fact that the average of a positive random variable x is equal to´∞
01−FX(x) dx, the average power consumption of each case would be equal
to
Pi =
ˆ 2Pmax
0
1− F iP (x) dx. (35)
5. Numerical Results
In this section, we confirm our analytical derivations and investigate theperformance of our scheme via computer simulation. Table 4 shows theparameters settings used for simulations. The results are obtained via 2millions independent realizations of the system.
21
Density of helpers (λ) ×10-3
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ave
rage
Thr
ough
put
2
3
4
5
6
7
8
Simul. proposed CoopMAC with LSD
=60 m
Simul. coventional CoopMAC with LSD
=60 m
Simul. proposed CoopMAC with LSD
=70 m
Simul. conventional CoopMAC with LSD
=70 m
Analytical results
Figure 3: The average throughput as a function of λ for cases 1 and 2 (LSD =60 and 70 meters).
Density of helpers (λ) ×10-3
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ave
rage
Thr
ough
put
2
3
4
5
6
7
8
Simul. Proposed CoopMAC , LSD
=80 m
Simul. conventional CoopMAC , LSD
=80 m
Simul. Proposed CoopMAC , LSD
=98 m
Simul. conventional CoopMAC , LSD
=98 m
Analytical results
Figure 4: The average throughput as a function of λ for cases 2 and 3 (LSD =80 and 98 meters).
22
Table 4: The simulation parameters.
Parameter Pt Pth α E(Ω) K Fading model
Value 1 mW −98 dBm 3 1 −40 dB Rayleigh
Figs. 3 and 4 illustrate the average throughput performance of the net-work as a function of λ, for different values of the distance between source anddestination (LSD = 60, 70, 80 and 98 meters), representing cases 2, 3, 4 and5, respectively. In this part, our proposed protocol has been compared to theconventional CoopMAC scheme, in which, a helper is randomly chosen fromthe table of helpers, with overall transmission rate greater than the directlink. Since the throughput results of our scheme and conventional CoopMACare the same for the first case, simulation results of this case has not beenillustrated. As can be seen, compared to the conventional CoopMAC, the av-erage throughput of our scheme has been improved significantly for cases 2 to5. We can see that by increasing the λ, the performance of our scheme is alsogetting better. It is because the source node has more opportunity to find acooperative path with the highest rate to send its packet. Also, we can seethat, by increasing the density of the helpers in the network, throughput ofthe conventional method does not change considerably and has a fixed value.This is because of the fact increasing the number of helpers in the networkleads to more successful transmissions and throughput as a result. However,for the networks with enough number of successful transmissions, this in-crease is not noticeable. Also, due to the uniform distribution of helpers,increasing lambda does not affect the average rate of the randomly selectedpath.
Fig. 6 depicts the CDF of the power consumption in a network usingour proposed CoopMAC scheme, for LSD = 60, 70, 80 and 98 meters. Aswe can see, the simulation results follow the analytical results for all cases.Also, it has been shown that the statistics of power consumption varies indifferent cases. As the distance between the source and destination rises, thedensity of power consumption moves toward higher values and its averagevalue increases as a result.
Fig. 5 illustrates the average power consumption of the network as afunction of helpers density, for LSD = 60, 70, 80 and 98 meters. In a networkthat power control is not being used, the transmission power of each coop-erative link would be equal to 2Pmax. In this figure, the power efficiency of
23
Density of helpers (λ) ×10-3
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ave
rage
pow
er c
onsu
mpt
ion
(mw
)
0
0.5
1
1.5
2
2.5
LSD
=60 m
LSD
=70 m
LSD
=80 m
LSD
=98 m
without power control
Figure 5: The average power consumption as a function of λ for LSD =60, 70 80 and 98.
x ×10-3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
FP(x
)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Simul. LSD
=60 m
Simul. LSD
=70 m
Simul. LSD
=80 m
Simul. LSD
=98 m
Analytical results
Figure 6: The CDF of the power consumption of network, for λ = 2.5×10−3
and LSD = 70, 80 and 98
24
our proposed scheme has been compared with conventional metgod whereno power control is being used. As we see, the average power consumptionis significantly lower in all cases. Besides, as expected, by decreasing thedistance between the source and destination, the consumed power will bereduced. In our scheme, Increasing density of helpers , leads to a rise innumber of successful transmissions and also the chance of finding less powerconsuming links. So, we can deduce that in networks with low number ofsuccessful transmissions (referred to the cases where the number of helpersis low and the distance between source and destination is high), increasingthe value of λ leads to the rise of successful transmissions, and power con-sumption as a result. On the other hand, when the number of successfultransmissions is high enough, increasing the density of helpers leads to lowerpower consumption in each transmission and reduction of average consumedpower consequently. As we see in Fig. 5, for LSD = 98, increasing λ leads topower consumption rises at first, and when the number of successful trans-missions reaches a high level, it starts to decrease . Also, for LSD = 70 and80, where the distance of the source and destination is shorter, the curve hasa descending trend.
6. Conclusion
In this paper, we considered a CoopMAC network in a Poisson field ofdecode-and-forward helpers and general fading environment. We proposeda new helper selection scheme which aims to improve the throughput andpower efficiency of the network. We first classified helpers into six groups,based on their corresponding rates defined in IEEE 802.11b. Also, a powercontrol mechanism was proposed in which all transmissions were performedwith the minimum required power. In our scheme, the highest priority isgiven to the links with the highest cooperate rate and with minimum powerconsumption. Three different protocols namely, transmission, cooperationand reception were proposed for the source, helper and destination nodes,respectively. Simulations results showed that our scheme can outperformthe conventional CoopMAC protocol in having superior average throughput.Also, it was shown that the power consumption in our scheme reduces sig-nificantly, compared to the networks with fixed power transmission.
25
7. References
References
[1] Agarwal, S., Katz, R. H., Krishnamurthy, S. V., Dao, S. K., Oct. 2001.Distributed power control in ad-hoc wireless networks. In: IEEE Inter-national Symposium on Personal and Indoor Mobile Radio Commun.conf. (PIMRC). Vol. 2. pp. 59–66.
[2] Behnad, A., Rabiei, A. M., Beaulieu, N. C., Jan. 2013. On performanceof cooperative communication systems with spatial random relays. IEEETrans. Commun. 61 (1), 97–107.
[3] Fish, J. C. L., 1909. Coordinates of Elementary Surveying. StanfordUniversity Press.
[4] Gao, X., Behnad, A., Wang, X., Jun. 2014. Performance analysis ofdecode-and-forward dual-hop opportunistic relaying with power control.In: IEEE Int. Conf. Commun. Vol. 1. pp. 4832–4837.
[5] Gomez, J., Campbell, A. T., Naghshineh, M., Bisdikian, C., Nov. 2001.Conserving transmission power in wireless ad hoc networks. In: IEEEInt.l Conf. Network Protocols. pp. 24–34.
[6] He, X., Li, F. Y., Oct. 2009. Cooperative RTS/CTS MAC with re-lay selection in distributed wireless networks. In: IEEE Ultra ModernTelecommun. Workshops Conf. (ICUMT). St. Petersburg, pp. 1–8.
[7] Hwang, K.-S., Ko, Y.-C., Alouini, M.-S., Dec. 2008. Outage probabilityof cooperative diversity systems with opportunistic relaying based ondecode-and-forward. IEEE Trans. Wireless Commun. 7 (12), 5100–5107.
[8] Ju, P., Song, W., 2015, to be published. Repeated game analysis for co-operative MAC with incentive design for wireless networks. IEEE Trans.Veh. Technol.
[9] Ju, P., Song, W., Zhou, D., Apr. 2013. An enhanced cooperative MACprotocol based on perceptron trainings. In: IEEE Wireless Commun.Networking Conf. Shanghai, China, pp. 404–409.
26
[10] Jung, E., Vaidya, N. H., Sep. 2002. A power control mac protocol forad hoc networks. In: IEEE Mobile Computing and Networking conf.(MOBICOM). pp. 36–47.
[11] Lin, X.-H., Kwok, Y.-K., Lau, V. K. N., Oct. 2003. Power control forIEEE 802.11 ad hoc networks: issues and a new algorithm. In: IEEEParallel Processing conf. pp. 249–256.
[12] Liu, P., Tao, Z., Narayanan, S., Korakis, T., Panwar, S. S., Feb. 2007.CoopMAC:a cooperative MAC for wireless LANs. IEEE Trans. Com-mun. 25 (2), 340–354.
[13] Monks, J. P., Bharghavan, V., Hwu, W. M. W., Apr. 2001. A power con-trolled multiple access protocol for wireless packet networks. In: IEEEComputer and Commun. Societies conf. (INFOCOM). Vol. 1. pp. 219–228.
[14] Narayanan, S., Liu, P., Panwar, S. S., Mar. 2005. On the advantages ofmulti-hop extensions to the IEEE 802.11 infrastructure mode. In: IEEEWireless Commun. Networking Conf. Vol. 1. pp. 132–138.
[15] Narayanan, S., Panwar, S. S., Jan. 2006. When two-hop meets VoFi.In: IEEE Consumer Commun. Networking Conf. (CCNC). Vol. 1. pp.297–302.
[16] Narayanaswamy, S., Kawadia, V., Srinicas, R. S., Kumar, P. R., 2002.Power control in ad hoc networks: Theory, architecture, algorithm andimplementation of the COMPOW protocol. In: IEEE European WirelessConf. pp. 156–162.
[17] Nikbakht, H., Rabiei, A. M., Shah-Mansouri, V., 2016. A new ap-proach for helper selection and performance analysis in Poisson Coop-MAC networks. IEEE Trans. Wireless Commun. submitted, available athttps://arxiv.org/abs/1610.00974.
[18] Pursley, M. B., Russell, H. B., Wysocarski, J. S., May 2000. Energy-efficient transmission and routing protocols for wireless multiple-hop networks and spread-spectrum radios. In: IEEE EuroComm,IEEE/AFCEA. pp. 1–5.
27
[19] Rui, X., Cao, X., Yang, J., Aug. 2015. A total power control cooperativeMAC protocol for wireless sensor networks. In: IEEE Int. Conf. CloudComputing and Security. Nanjing, China, pp. 390–403.
[20] Shan, H., Wang, P., Zhuang, W., Wang, Z., Dec. 2008. Cross-layer coop-erative triple busy tone multiple access for wireless networks. In: IEEEGlobal Telecommun. Conf. New Orleans, LO, pp. 1–5.
[21] Wang, X., Li, J., Mar. 2015. Improving the network lifetime of MANETsthrough cooperative MAC protocol design. IEEE Trans. Parallel andDistributed Systems 26 (4), 1010–1020.
[22] Wu, C., Bertsekas, D. P., Mar. 2001. Distributed power control algo-rithms for wireless networks. IEEE Trans. Veh. Technol. 50 (2), 504–514.
[23] Zhu, H., Cao, G., Sep. 2006. rDCF: A relay-enabled medium access con-trol protocol for wireless ad hoc networks. IEEE Trans. Mobile Com-puting. 5, 1201–1214.
Appendix A. Proof Eq. (13a) through (13f), and (31a) through
(31g)
We denote Cr1,r2 as a part of system region which is determined by
Cr1,r2 =
0 ≤ LSH ≤ r10 ≤ LHD ≤ r2,
(A.1)
where LSH and LHD are distances of the helper from the source and destina-tion nodes, respectively. As illustrated in Fig. A.7, we see that Cr1,r2 can berepresented by the combination of three other areas, as follows:
Cr1,r2 = Cr1,θ1a + Cr2,θ2
b − Cθ1,θ2c , (A.2)
where Cr1,θ1a ,Cr2,θ2 and Cθ1,θ2
c are the areas respectively specified in Fig. A.7bthrough Fig. A.7d. According to the cosines laws, θ1 and θ2 are obtained asa function of r1 , r2 and LSD as
θ1 = arccos(r22 − r21 − L2
SD
2 r1 r2) (A.3)
θ2 = arccos(r21 − r22 − L2
SD
2 r1 r2). (A.4)
28
S D
+
2θ1 2θ2M
r1r2
LSH LHD
Cr1,r2
(a)
S D
+
2θ1θs M
r1r2
LSH
Cr1,θ1a
(b)
S D
+
2θ2θdM
r2r1
LHD
Cr2,θ2b
(c)
S D
+
2θ1 2θ2θdθs M
r1r2
Cθ1,θ2c
(d)
Figure A.7: A represent of Cr1,r2 by combination of Cr1,θ1a , Cr2,θ2
b and Cθ1,θ2c .
According to the rules of integral, we can deduce from (A.2) that
ˆ
Cr1,r2
g(LSH, LHD)ds =
ˆ
Cr1,θ1a
g(LSH, LHD)ds+
ˆ
Cr2,θ2b
g(LSH, LHD)ds
29
−ˆ
Cθ1,θ2c
g(LSH, LHD)ds, (A.5)
where g(.) is an arbitrary function of LSH and LHD. In order to evaluate (A.5),an appropriate coordinate system should be considered for each region. Asshown in Fig. A.7b, in a polar coordinate system with center of S, each pointof the Cr1,θ1
a can be uniquely represented by the pair of (LSH, θs). Using thelaw of cosines, LHD can be obtained as
LHD =√
L2SH + L2
SD − 2LSH LSD cos θs, (A.6)
also, ds is given byds = LSH dLSH dθs. (A.7)
We can determine Cr1,θ1a as
Cr1,θ1a =
0 ≤ LSH ≤ r1−θ1 ≤ θs ≤ θ1,
(A.8)
and by using equations (A.6) , (A.7) and (A.8), it can be deduced that
ˆ
Cr1,θ1a
g(LSH, LHD) ds =
ˆ θ1
−θ1
ˆ r1
0
g(LSH , LHD)LSH dLSH dθs. (A.9)
Similarly, in a polar coordinate system with center of D, each point of Cr2,θ2b
(shown in Fig. A.7c) can be uniquely determined by the pair of (LHD, θd).Making use of cosines laws, LSH is
LSH =√
L2HD + L2
SD − 2LHD LSD cos θd, (A.10)
also, ds is defined byds = LHD dLHD dθd, (A.11)
and Cr2,θ2b can be specified as
Cr2,θ2b =
0 ≤ LHD ≤ r2−θ2 ≤ θd ≤ θ2.
(A.12)
30
Deducing from equations (A.10) , (A.11) and (A.12) we have
ˆ
Cr2,θ2b
g(LSH, LHD) ds =
ˆ θ2
−θ2
ˆ r2
0
g(LSH, LHD)LHD dLHD dθd, (A.13)
where g(.) is an arbitrary function of LSH and LHD.In order to evaluate the integral of
´
g(LSH, LHD) ds over the surface ofCθs,θdc , we use a biangular coordinate system [3] given the poles of S and D,
where each point is uniquely represented by a pair of (θs, θd) that shown inFig. A.7d. Also, due to the symmetry of this model the integral can onlybe evaluated for the upper half of Cθs,θd
c plane over S–D line. Using thiscoordinate system, the upper half of Cθ1,θ2
c , named Cθ1,θ2c,up , is determined by
Cθ1,θ2c,up =
0 ≤ θs ≤ θ10 ≤ θd ≤ θ20 ≤ θs + θd ≤ π.
(A.14)
According to the law of sines, LSH and LHD are given by
LSH = LSDsin(θd)
sin(θs + θd)(A.15)
LHD = LSDsin(θs)
sin(θs + θd), (A.16)
and directly from [4, Eq. 21], ds can be obtained as
ds = L2SD|
sin(θs) sin(θd)
sin3(θs + θd)|. (A.17)
From equations (A.14) through (A.17) , we have
ˆ
Cθ1,θ2c
g(LSH, LHD) ds = 2L2SD
ˆ θ2
0
ˆ min(θ1,π−θd)
0
g (LSH, LHD)sin(θs) sin(θd)
sin3(θs + θd)dθs dθd.
(A.18)
31
By Substituting equations (A.9) through (A.18) in (A.5), finally
ˆ
Cr1,r2
g(LSH, LHD) ds = H(r1, r2, LSD), (A.19)
where θ1 and θ2 are obtained from (A.3) and (A.4), and H(., ., .) is
H(u, v, x) ,
ˆ −θ(u,v,x)
−θ(u,v,x)
ˆ u
0
g(r ,√r2 + x2 − 2 r x cos θ) r dr dθs (A.20a)
+
ˆ θ(v,u,x)
−θ(v,u,x)
ˆ v
0
g(r ,√r2 + x2 − 2 r x cos θ) r dr dθd
+2 x2
ˆ θ(u,v,x)
0
ˆ min(θ(v,u,x),π−θd)
0
g
(
xsin(θd)
sin(θs + θd), x
sin(θs)
sin(θs + θd)
)
×sin(θs) sin(θd)
sin3(θs + θd)dθs dθd,
where
θ(a, b, c) , arccos(b2 − a2 − c2
2 a b). (A.20b)
As illustrated in Fig. 2, we can see that each area of Ci (i = 1, ..., 6) canbe written as a combination of Cr1,r2s with different values of r1 and r2.Therefore, we have
C1 = C48.2,48.2 (A.21a)
C2 = C67.1,48.2 + C48.2,67.1 − 2C1 (A.21b)
C3 = C67.1,67.1 − C1 − C2 (A.21c)
C4 = C48.2,74.7 + C74.7,48.2 − 2C1 − C2 (A.21d)
C5 = C67.1,74.7 + C74.7,67.1 − 2C1 − 2C2 − 2C3 − C4 (A.21e)
C6 = C74.7,74.7 − C1 − C2 − C3 − C4 − C5. (A.21f)
We denote D as
Di ,
ˆ
Ci
g(LSH, LHD) ds i = 1, ..., 6 (A.22)
where LSH and LHD are distances from the source and the destination respec-
32
tively, and ds is the surface element. Using the equations (A.20a) through(A.21f), and also some mathematical manipulations, it can be summarizedas
D1 = H(48.2, 48.2, LSD) (A.23a)
D2 = H(67.1, 48.2, LSD) +H(48.2, 67.1, LSD)− 2D1 (A.23b)
D3 = H(67.1, 67.1, LSD)−D1 −D2 (A.23c)
D4 = H(74.7, 48.2, LSD) +H(48.2, 74.7, LSD)− 2D1 −D2 (A.23d)
D5 = H(67.1, 74.7, LSD) +H(74.7, 67.1, LSD)− 2D1 − 2D2 − 2D3 −D4
(A.23e)
D6 = H(74.7, 74.7, LSD)−D1 −D2 −D3 −D4 −D5 (A.23f)
From equation (12), we can deduce that PLossi would be equal to exp(−λDi)
by substituting g(LSH, LHD) = PSuccH (LSH, LHD), which leads to equations
(13a) to (13f). Also, by replacing g(LSH, LHD) = G(LSH, LHD, x), Gi(x) wouldbe equal to exp(λDi), that resulted in equations (31a) to (31g).
Appendix B. Proof of eq. (27a) and (27b)
Consider a helper, H, at the distances of LSH and LSD from the sourceand the destination respectively. We consider G(LSH, LHD, x) as the proba-bility of the event in which, S-H-D is a potential cooperative path with therequired transmission power greater than x. As discussed earlier, helper Hwould be potential for cooperation, supposing that the required powers forsuccessful transmission over S–H and H–D are both less than Pmax. As aresult, G(LSH, LHD, x) can be represented as
G(LSH, LHD, x) = PrPminSH < Pmax , P
minHD < Pmax , P
minSH + Pmin
HD > x. (B.1)
We can see that for x ≥ 2Pmax, equation (B.1) would be equal to zero.In order to evaluate G(LSH, LHD, x) for 0 ≤ x < 2Pmax, we first obtainthe probability distribution functions (PDF) of Pmin
SH and PminHD . We denote
FP,1(x) as the CDF of PminSH , which can be evaluated as follows:
FP,1(x) = PrPminHD ≤ x
= Pr
N0 LαSH
K0Ω1
γth ≤ x
33
= 1− Pr
Ω1 <N0 L
αSH γth
K0 x
= 1− FΩ
(BLαSH
x
)
, (B.2a)
where
B ,N0 γthK0
. (B.2b)
The PDF function of PminSH , can be obtained by differentiating from equation
(B.2a) as
fP,1(x) =BLα
SH
x2fΩ
(BLαSH
x
)
. (B.3)
With the same trend, CDF and PDF functions of PminHD , can be obtained
respectively as
FP,2(x) = 1− FΩ(BLα
HD
x) (B.4)
fP,2(x) =BLα
HD
x2fΩ(
BLαHD
x). (B.5)
Using equations (B.3) and (B.4), G(LSH, LHD, x) can be evaluated for x ∈[Pmax, 2Pmax] as follows:
G(LSH, LHD, x)
=
ˆ ∞
0
fP,1(a) PrPminSH < Pmax,P
minHD < Pmax,P
minSH + Pmin
HD > x|PminSD = ada
=
ˆ
Pmax
0
fP,1(a) Prx− a < PminHD < Pmax da (B.6)
Supposing that 0 ≤ x < Pmax, it can be deduced that x − a is always lessthan < Pmax. Also, due to the fact that Pmin
HD has a positive value, for a > x,the value of Prx− a < Pmin
HD < Pmax would be equal to PrPminHD < Pmax.
Hence, for 0 ≤ x < Pmax one can obtain (B.6) as
G(LSH, LHD, x) =
ˆ x
0
fP,1(a) Pr< xPminHD < Pmax da
+
ˆ
Pmax
x
fP,1(a) PrPminHD < Pmax
34
=
ˆ x
0
fP,1(a) (FP,2(Pmax)− FP,2(x− s)) da
+
ˆ
Pmax
x
fP,1(a)FP,2(Pmax)da (B.7)
Substituting (B.3), (B.4) in (B.7), one can obtain the G(LSH, LHD, x) for0 ≤ x < Pmax
G(LSH, LHD, x) =
ˆ x
0
BLαSH
a2fΩ(
BLαSH
a) [FΩ(
BLαHD
x− a)− FΩ(
BLαHD
Pmax)] da
+
ˆ
Pmax
x
BLαSH
a2fΩ(
BLαSH
a) [1− FΩ(
BLαHD
Pmax)] da (B.8)
Supposing that Pmax ≤ x < 2Pmax, it would be clear that x − a > 0 fora < Pmax. Also if x − a > Pmax, the equation (B.6) would be equal to zero.As a result, for Pmax ≤ x < 2Pmax. one cane obtain (B.6) as
G(LSH, LHD, x) =
ˆ
Pmax
x−Pmax
fP,1(a) Prx− a < PminHD < Pmax da (B.9)
=
ˆ
Pmax
x−Pmax
fP,1(a) [FP,2(Pmax)− FP,2(x− a)] da
=
ˆ
Pmax
x−Pmax
BLαSH
a2fΩ(
BLαSH
a) [FΩ(
BLαHD
x− a)− FΩ(
BLαHD
Pmax)] da.
(B.10)
To summarize, G(LSH, LHD, x) can be presented as
G(LSH, LHD, x) =
´ x
0
BLαSH
a2fΩ(
BLαSH
a)[[FΩ(
BLαHD
x−a)− FΩ(
BLαHD
Pmax)]da 0 ≤ x < Pmax
+´
Pmax
x
BLαSH
a2fΩ(
BLαSH
a)1− FΩ(
BLαHD
Pmax)]da
´
Pmax
x−Pmax
BLαSH
a2fΩ(
BLαSH
a)[FΩ(
BLαHD
x−a)− FΩ(
BLαHD
Pmax)]da Pmax ≤ x < 2Pmax
0 x ≥ 2Pmax.
(B.11)
35
In equation (26f), µ1 can be evaluated by integrating from G(LSH, LHD, x)over the surface of Ci, which leads to (27a) and (27b).
36