A spatial slottedAloha protocol in wireless networks for group communications
 Mingyu Lee^{1},
 Yunmin Kim^{1} and
 TaeJin Lee^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s136380170928x
© The Author(s) 2017
Received: 3 November 2016
Accepted: 10 August 2017
Published: 24 August 2017
Abstract
In a largescale group communication networks, we propose a slottedAlohabased access control (SAAC) scheme with the optimal transmission probability (TP) using the stochastic geometry. Since the performance of SAAC scheme depends on the TP of the members in a group, the analytical model presents a joint view for the downlink (DL) and the uplink (UL) with TP of the members in a group. We present simple and closed form expressions for the DL and UL joint probability by using Poisson point process (PPP). Furthermore, we analyze the dynamic TP and the optimal TP to maximize the DL/UL joint probability, for which the inclusion of a member and the transmission of the member is determined by the DL and UL thresholds. Since the requirement of the DL/UL joint probability varies with the type of services, the optimal TP has to be determined for an efficient group communication service. The performance of the SAAC scheme with the optimal TP is demonstrated, and it is shown to be superior over the other schemes.
Keywords
1 Introduction
In a largescale group communication network, some nodes may form a cluster and clusters communicate with one another. For example, in a tactical group communication network, a commander monitors the conditions of the soldiers. A soldier equips with a monitoring system [1], and the monitored status is transmitted to the base camp through an unmanned aerial vehicle (UAV) [2]. If all the soldiers transmit their information at the same time to the UAV, the network congestion will occur increasingly as the number of soldiers increases. Thus, a hierarchical system in which relays or vehicles collect the information of the soldiers and they transmit the collected information to the UAV could be desirable. In the system, the soldiers are grouped to communicate with a relay or a vehicle, and they act as the group members and the relay or the vehicle becomes the group leader. The group leader has to cover the group members as many as possible within its communication range. The members have to choose the best group leader among the candidate group leaders. The group leader will control the transmissions of the group members within its coverage.
With a massive number of nodes, the group leaders and the members form a largescale network and they can be modeled by the Poisson point process (PPP). PPP is a spatial point process, which is widely used to model a wireless network. However, a largescale network with group communications in which group leaders and members send and receive information has not been studied. Since the members within the coverage of a leader are able to transmit for communications, the locations of the members are no longer independent to the location of the leader. Moreover, the interference model should be different from that in [3] because the interference from the covered area of a leader is important compared to that from the noncovered area of a leader. In addition, the covered members of a group may decide whether to transmit or not by a transmission control (TC) value, which is globally determined by the leaders in a centralized manner. Thus, a new network model to capture the behavior of hierarchical group communications needs to be setup.
Since wireless resources in a group communication network are shared in a distributed manner, the performance depends on the medium access control (MAC) scheme. In this sense, transmission probability (TP), which is decided by the TC value, should be carefully chosen. Since the TP of nodes in an Alohabased scheme may be adaptively determined by channel variation, local topology, and target utility, a new MAC protocol can be designed by using the optimal TP which is derived from the PPP models in group communication networks.
In this paper, we first derive the dynamic TP which is determined by the statistical distributions of group leaders and the DL coverage probability. We focus on the special case to derive the closedform expressions. The coverage probability and the average numbers of members per leader in the DL are derived for the dynamic TP. The dynamic TP has been known to be optimal in terms of the MAC layer. However, the performance using the dynamic TP has not been evaluated by considering geometrical distributions of network elements.
Next, we propose the optimal TP which maximizes the downlink (DL) and uplink (UL) joint probability. The DL/UL joint probability is that the transmission of the leader and the transmission of a member are performed within the DL coverage threshold and the UL transmission threshold. The probability is determined by the probability density function (pdf) of the distance between the leader and a member which is within the coverage of the leader. The pdf is derived and is utilized to model the UL probability and the DL/UL joint probability. Since the DL/UL joint probability is maximized for a certain distance between the leader and a member, the closedform expressions of the optimal TP for the distance can be derived. In a practical system, the distance is usually hard to be determined. In our proposed scheme, however, a leader can determine the optimal TP by using the average number of members per leader.
Finally, we present the performance of the DL/UL joint probability and the average achievable rate for a target distance in our model. In addition, the average achievable rates for the different TPs are provided. From the analytical model, the effect of TP in a group communication network is provided. Furthermore, a new policy to determine the optimal TP is proposed by considering the geometrical effects of network elements. For the group communication network, the performance at a target distance is maximized by the proposed slottedAlohabased access control (SAAC) scheme.

We propose a new SAAC scheme with the optimal TP to maximize the DL/UL joint probability for group communication network.

We define pdf of the distance between a leader and its covered member for a given DL threshold, and the closed form expression of the optimal TP to maximize the DL/UL joint probability for a target member.

We evaluate the performance of the proposed SAAC scheme with the traditional schemes.
The rest of this paper is organized as follows. Section 2 reviews the recent PPP studies of DL, UL, and the MAC perspectives. In Section 3, we explain the proposed SAAC scheme and the other SAAC schemes in group communication networks. Section 4 analyzes the dynamic TP and the optimal TP by using the PPP model and presents an access scheme with the optimal TP. Section 5 presents numerical results to demonstrate the access control schemes with different TPs. Finally, we conclude in Section 6.
2 Related works
Modeling of the DL of a largescale network using stochastic geometry has been studied in [4, 5]. The performance of the DL system depends on the locations of base stations (BSs), and it has been derived by modeling BSs as a PPP [6–9]. The general expressions for the coverage probability and achievable rate at a typical user are presented in [4]. In some special cases, the closedform expressions, which give an intuitive view for the DL system, are provided. In [5], the association probability for a certain type of BSs which are modeled as multiple PPPs with different spatial densities and system parameters is studied. The analytical model in [5] presents the outage and spectral efficiency performance for the DL system deploying different types of BSs. Most PPP models for the DL system are analyzed from the view point of a typical user with a DL communication threshold, which shows that the tractable models are fairly accurate compared to actual systems. The performance of the resource management schemes in the DL (e.g., coverage expansion [5], fractional frequency reuse (FFR) [10], and resource partitioning [11]) is analyzed by using the PPP.
The analytical model using PPP for the UL of a largescale network has also been developed. The models in [12–14] are developed by dividing Voronoi cells from the perspective of users which are modeled as a PPP. Each user has its own BSs and UL resources for transmission. The transmission power is controlled by themselves according to its locations. For a typical transmission, the other transmissions are assumed as interference. In another approach for the UL model in [3], cells are divided with respect to BSs. The set of active users which satisfy the cutoff threshold for the UL is able to communicate with their BSs. The cutoff threshold for the UL is the average received power required at the serving BS. For a typical active user operating on an allocated channel in its serving BS, the other active users on the channel in the other BSs are assumed to be as interferences. The model provides tractability by assuming that the interfering users constitute a PPP. Since the correlations among users occur by the locations of the associated BSs and the tagged channel, the assumption is required to provide the insights for the cellular UL performance. The performance of the UL with a resource management scheme also has been analyzed with the PPP as in the DL [13].
For the clustered network scenario, [15] and [16] is researched. In these works, clusters are formed based on parent PPP points. To model the clustered networks Mattĕrn cluster process (MCP) and Thomas cluster process (TCP) is used. The interference from the inter and the intra clusters are separately analyzed and approximated using the clustered process properties. The groups or clusters are already formed as a point process without the DL/UL joint model. And the process of forming clusters is not considered when leaders and members are independently deployed.
In addition, the stochastic geometry modeling has been developed by focusing on MAC schemes, especially Alohabased schemes [17–19]. Since a node in an ad hoc network may operate as a transmitter or a receiver, an analytical model using PPP can be classified by the model of a receiver [6]. If each transmitter node has its corresponding receiver, the model is called as “Poisson bipolar model” [20–23]. In the Poisson bipolar model, the transmitters are modeled as a PPP and are divided into two nonoverlapping subsets, whether a transmitter does its role at a specific time or not by using a TP. If the transmitter and receiver are independently modeled as PPPs, the model is called as “independent receiver model.” In the model, a transmitter is controlled by TP and the transmission of the transmitter can be received by all subsets of the receivers [6]. When the transmitter is modeled as a PPP and the receiver is modeled as the subset of transmitters by using a specific condition, the model is called as “mobile ad hoc network model.” If the set of active transmitters is classified by TP, the set of inactive transmitters at a slot is the set of receivers [24]. Note that a source node has to send data to its far destination node in a mobile ad hoc network. Then, the intermediate nodes are required to relay data to its destination node. If some of the inactive transmitters are in the direction from a source node to a destination node, they can be the receivers of the transmitter, i.e., the source node. In [22], the optimal TPs are derived for the maximum throughput medium access, the maxmin fairness medium access, and the proportional fairness medium access. In [21], the performance of the Aloha protocol which is optimized for the proportional fair medium access is analyzed. The authors in [24] propose a spatial reuse Aloha protocol for a multihop network with the optimal TP which maximizes the density of progress, the mean total distance traversed in one hop by all transmissions initialized in some unit area. However, the optimal TP for the group communications with the DL coverage and the UL transmission has not been studied.
3 SlottedAlohabased access control schemes in group communication networks
In this section, we introduce the proposed SAAC scheme which utilizes the optimal TP maximizing the DL/UL joint probability for the target distance. The target distance can be varied by the policy of the proposed SAAC scheme. If the policy of the proposed SAAC scheme is to maximize the coverage, the optimal TP is determined for maximizing the performance of the outermost members. In the example, the interference for the outermost members should be decreased to satisfy the UL threshold. The distance between the leader and the outermost member is affected by the DL threshold and the interference from the other leaders. Furthermore, since the number of members in the coverage of the leader increases by extending the coverage, the interference should be controlled by TP. So the proposed SAAC scheme computes the optimal TP which is determined by considering the DL threshold, the UL threshold, the density of the leaders, the density of the members, and the target distance.
Summary of notations
Notation  Description 

l  Leader at the origin in the uplink 
l _{ i }  Leader i 
m  Typical member in the downlink 
m _{ i }  Member i 
m _{ c }  Set of covered members 
Φ _{ l }  PPP of leaders 
Φ _{ m }  PPP of members 
\(\Phi _{m_{c}}\)  Set of locations of covered members 
h _{ x,y }  Channel effect from x to y 
λ _{ l }  Intensity of leaders 
λ _{ m }  Intensity of members 
P _{ l }  Transmission power of leaders 
P _{ m }  Transmission power of members 
α  Pathloss exponent 
p _{ d,u }  Downlink, uplink joint probability 
p _{ u }  Uplink coverage probability 
p _{ d }  Downlink coverage probability 
T _{ d }  Downlink threshold 
T _{ u }  Uplink threshold 
p _{ a }  Access control probability 
r  Distance between m and its nearest leader 
r _{ d,i }  Distance between a leader i and a typical member 
r _{ u,i }  Distance between a member i and a typical leader 
r _{ d u,i }  Distance between a covered member i and a typical leader 
r _{ tar }  Target distance 
\(\bar {r}_{max}\)  Average maximum distance between l and its m _{ c } 
I _{ l }  Cumulative interference from the leaders 
I _{ m }  Cumulative interference from the members 
\(I_{m_{c}}\)  Cumulative interference from the covered members 
τ  Transmission probability (TP) 
τ ^{ d y n }  TP of the dynamic framed SAAC 
τ ^{ f i x }  TP of the fixed framed SAAC 
K  Frame size 
K ^{ d y n }  Frame size of the dynamic framed SAAC 
K ^{ f i x }  Frame size of the fixed framed SAAC 
S I R _{ d }  SIR at a typical member in the downlink 
S I R _{ u }  SIR at a typical leader in the uplink 
S I R _{ d,u }  SIR at a typical leader in the joint DL/UL transmission 
\(\bar {N}_{m}\)  Average number of the covered members per leader 
E _{ ar }  Average achievable rate of covered members 
where r _{ u,i } is the distance between a member i and a typical leader.
The SAAC scheme is assumed to work in a group communication network where the time slots are synchronized. One viable solution may be equipping global positioning system (GPS) modules on the leaders and the members since the signals from the GPS satellites provide the rather accurate timing and location information. There may be also several methods which can provide the synchronization to the members without GPS modules. Once a network is grouped into several clusters, all nodes can be bounded by the parentchildren relationship. All members can be synchronized to the leader by periodically exchanging synchronization and acknowledgment packets with their corresponding leaders using pairwise synchronization method [25, 26]. The location information may be piggybacked at the synchronization and acknowledgment packet for simultaneous time synchronization and location information distribution. Depending on the velocity of the nodes, the period between the synchronization and the location information distribution can be adjusted accordingly.
where K is the frame size which represents the number of data slots. In the SAAC scheme using K, a covered member selects and accesses a slot among the K slots. In general, 1/K is modeled as the access probability of the contenders in a slot. The framed SAAC scheme consists of the fixed framed SAAC scheme and the dynamic framed SAAC scheme. The fixed framed SAAC scheme utilizes the fixed TP which is determined by the fixed frame size [27]. Let τ ^{ f i x } and K ^{ f i x } denote the TP and the frame size of the fixed framed SAAC, respectively. The fixed framed SAAC scheme utilizes τ ^{ f i x }=1/K ^{ f i x }. The dynamic framed SAAC scheme utilizes the dynamic TP determined by the frame size which is dynamically changed according to the average number of the covered members per leader. The frame size in the dynamic framed SAAC scheme is known to be optimal when the frame size is equal to the number of contenders [18, 28]. Let τ ^{ d y n } and K ^{ d y n } denote the TP and the frame size of the dynamic framed SAAC scheme, respectively. Then the dynamic framed SAAC scheme utilizes τ ^{ d y n }=1/K ^{ d y n }.
In UL, the transmission of a member occurs when the member is in the coverage of the leader and determines whether the covered member participates in the UL transmission by τ and T _{ u }. The target member in group 2 transmits data to the leader. The leader in group 2 receives interference signal from the member in group 2 as well as the members in groups 1 and 3. To decode the UL transmission of the target member, the interference signals have to be controlled by the optimal TP τ ^{∗} which is determined by λ _{ l }, λ _{ m }, T _{ d }, T _{ u }, and r _{ tar }.
4 Optimal transmission probability for group communications
In this section, we derive the dynamic TP and the optimal TP. To derive the dynamic TP, we need the DL coverage probability and the average number of the covered members per leader. For the optimal TP, we need to derive the UL coverage probability and the DL/UL joint probability for a target distance. The target distance is the distance between a leader and a target member for which the DL/UL joint probability is maximized. Finally, we develop an analytical model for the performance of the SAAC schemes using the dynamic TP and the optimal TP. From the model, we derive an optimal TP of the SAAC scheme.
where r is the distance between a member and its nearest leader. The members are divided into the covered members and the noncovered members by T _{ d }. The following lemma provides the probability that a member is in the coverage of the nearest leader.
Lemma 1
Proof
See Appendix A.1 Proof of Lemma 1. □
Lemma 1 shows that p _{ d }(r,T _{ d }) decreases by increasing r, T _{ d }, and λ _{ l }. Since the received signal strength from the nearest leader decreases as r increases, p _{ d }(r,T _{ d }) decreases. For a certain r, the SIR value may not be higher than T _{ d } for large T _{ d }. Since the increase of λ _{ l } causes the increase of interference for a certain r, p _{ d }(r,T _{ d }) decreases. Lemma 1 implies that the number of the covered members decreases by increasing T _{ d } for both p _{ d }(r,T _{ d }) and p _{ d }(T _{ d }). The expressions in Lemma 1 yield a closedform when α=4, since \(\zeta _{l}(T_{d})=\sqrt {T_{d}}\left (\frac {\pi }{2}\arctan \left (\frac {1}{\sqrt {T_{d}}}\right)\right)\).
From Lemma 1, we derive the dynamic TP, which determines the transmission probability of the covered members.
Theorem 1
Proof
If we assume that S is the area of an entire network, \(\bar {N}_{m}\) is obtained by dividing the average number of the covered members in S by the average number of the leaders in S. Since the intensity of the covered members is λ _{ m } p _{ d }(T _{ d }), the average number of the covered members per leader \(\bar {N}_{m}=\frac {\lambda _{m}p_{d}(T_{d})S}{\lambda _{l}S} =\frac {\lambda _{m}p_{d}(T_{d})}{\lambda _{l}}\) [5]. □
Theorem 1 shows that τ ^{ d y n } is adaptive to the number of the covered members per leader. We expect that τ ^{ d y n } decreases as \(\bar {N}_{m}\) increases. It implies that the interference to the transmission of a typical covered member decreases as \(\bar {N}_{m}\) increases. Since \(\bar {N}_{m}\) is affected by p _{ d }(T _{ d }), it increases by decreasing T _{ d }. Thus, Theorem 1 indicates that τ ^{ d y n } decreases and the interference to the transmission of a typical covered member decreases as T _{ d } decreases in the dynamic framed SAAC scheme.
Since the locations of the covered members are jointly changed by T _{ d } and the locations of the leaders, they are not PPP. It is challenging to model it accurately, but PPP approximation could be utilized [3]. In our model, we approximate them as a PPP. Let \(\Phi _{m_{c}}\) and \(\Phi _{m_{o}}\) denote the locations of the covered members and the locations of the noncovered members. We calculate the pdf of the distance between a leader and a covered member from the following theorem.
Theorem 2
Proof
Plugging p _{ d }(r,T _{ d }) in (8) into (12), \( f_{R_{m_{c}}}(r)\) is derived. □
where r _{ d u,i } is the distance between a covered member i and a typical leader.
Lemma 2
where \(\zeta _{m}(T_{u})=T^{2/\alpha }_{u}\frac {2\pi /\alpha }{\sin (2\pi /\alpha)}\).
Proof
See Appendix A.2 Proof of Lemma 2. □
Lemma 2 shows that p _{ u }(r,τ,T _{ u }) decreases by increasing r, T _{ u }, and λ _{ m }. Since the received signal strength from the covered member decreases as r increases, p _{ u }(r,τ,T _{ u }) decreases. For a certain r, the SIR value for the signal may be hard to exceed T _{ u } for large T _{ u }. Since the increase of λ _{ m } incurs the increase of the interference, p _{ u }(r,τ,T _{ u }) decreases. The expression in Lemma 2 is a closedform when α=4, since \(\zeta _{m}(T_{u})=\frac {\pi }{2}\sqrt {T_{u}}\). The following lemma provides the DL/UL joint probability which quantifies the performance of the SAAC scheme with τ. We define the target member which is a member with target distance, r _{ tar }. The DL/UL joint probability denotes the probability that a member with r _{ tar } from its nearest leader is in the coverage of the leader, i.e., the DL SIR exceeds the DL threshold T _{ d }, and the nearest leader is in the coverage of the member transmitting the UL signal with τ, i.e., the UL SIR exceeds the UL threshold T _{ u }. The performance of the DL/UL joint probability for the target member shows the effect of τ for r _{ tar }.
Lemma 3
where 0≤τ≤1.
Proof
By letting r=r _{ tar } in (7) and (15), p _{ d,u }(r _{ tar },τ,T _{ d },T _{ u }) is derived. □
Since the DL/UL joint probability is affected by T _{ d }, T _{ u }, and r _{ tar }, it decreases by increasing them. However, if the SAAC scheme uses τ ^{ d y n } as τ, τ ^{ d y n } increases as T _{ d } increases. Thus, the DL/UL joint probability for the SAAC scheme is sensitive to T _{ d }. If α=4, the closedform expression of Theorem 3 is derived from the closedfrom expressions in Lemma 1 and Lemma 2. From Lemma 3, we expect that τ varies according to r _{ tar }. Since p _{ d,u }(r _{ tar },τ,T _{ d },T _{ u }) has the global extreme values for τ, the optimal τ can be derived (see Appendix A.3 Proof of Optimal τ ). We now derive the optimal TP that maximizes p _{ d,u }(r _{ tar },τ,T _{ d },T _{ u }).
Theorem 3
where 0≤τ ^{∗}≤1.
Proof
Since τ ^{∗}≤1 in a slot, τ ^{∗}=1 when \(r_{tar}<\sqrt {1/\pi \lambda _{m}p_{d}(T_{d})\zeta _{m}(T_{u})}\). □
From Theorem 3, we derive τ ^{∗} to maximize the DL/UL joint probability for \(\bar {r}_{max}\). \(\bar {r}_{max}\) is the average maximum distance between a leader and its covered member. In general, since r _{ tar } has to be estimated by the target member and reported to the leader, \(\bar {r}_{max}\) is hard to be known. However, if we assume that λ _{ m }, T _{ d }, and T _{ u } are given, τ ^{∗} can be obtained by the average number of the covered members as in the dynamic TP. The average number of the covered members can be known by using the number of associated members. Since the number of the covered members within the distance r _{ tar } is π(r _{ tar })^{2} λ _{ m } p _{ d }(T _{ d }), \(\bar {N}_{m}=\pi (\bar {r}_{max})^{2}\lambda _{m}p_{d}(T_{d})\). So \(\bar {r}_{max}=\sqrt {\frac {\bar {N}_{m}}{\pi \lambda _{m}p_{d} (T_{d})}}\) \(=\sqrt {\frac {1}{\pi \lambda _{l}}}\) by plugging \(\bar {N}_{m}\) in Theorem 1. Thus, \(\tau ^{*}=\min \left (\frac {1}{\bar {N}_{m}\zeta _{m}(T_{u})},1\right)\). It implies that τ ^{∗} for the average number of the covered members is the same as τ ^{∗} for \(\bar {r}_{max}\) in the proposed SAAC scheme.
We now derive the average achievable rate to measure the spectral efficiency performance of the SAAC schemes. The average achievable rate for a target member with r _{ tar } and the average achievable rate of the members are obtained.
Lemma 4
Proof
See Appendix A.4 Proof of Lemma 4. □
Both metrics are computed by averaging the UL coverage probability when the transmission of a covered member is governed by τ. The average achievable rate of a covered member is derived from the average achievable rate for a target member. Both metrics are not closedform expressions, and the numerical integration is required to compute them.
5 Numerical results
In this section, we show the numerical results for the analytical models and simulation results with DL threshold T _{ d } and UL threshold T _{ u }. We use the pathloss exponent α=4, the transmit power of a leader P _{ l }=1 W, and the transmit power of a member P _{ m }=1 W [20]. The transmission of both the leaders and the members in a single channel is done in multiple time slots. The leader intensity λ _{ l }=3 leaders/km^{2} and the member intensity λ _{ m }=20 members/km^{2} [29]. Both the leaders and the members are distributed with their intensity in a 25km^{2} area. And we focus on the sample area in a 1km^{2} area to evaluate the performance. Thus, the numerical results within the sample area are obtained in the simulation. In DL, each member computes the DL SIR and checks the DL SIR if it exceeds the DL threshold T _{ d }. If the DL SIR exceeds the DL threshold T _{ d }, the member becomes the covered member. In UL, the covered members transmit their signals according to the TP which is determined by the traditional SAAC schemes and the proposed SAAC scheme. The leader computes the UL SIR for each of the covered members and checks if the UL SIR exceeds the UL threshold T _{ u }. if the UL SIR exceeds the UL threshold T _{ u }, the DL/UL of the covered member is successful. We compare the performance of the SAAC scheme with τ ^{∗} to those of the SAAC schemes with τ ^{ d y n } as in Theorem 1 and τ ^{ f i x } as in [27].
Since r _{ tar } is the distance between the target member and its leader, the signal strength of the target member decreases as r _{ tar } increases. Since τ is the TP of the covered members in a slot, the number of the covered members participating in the transmission in a slot increases as τ increases. The signals of the covered members except the target member become the interference to the target member. Thus, for the target member with r _{ tar }, the SIR of the target member decreases as r _{ tar } increases or τ increases. Since the average achievable rate of a target member with r _{ tar } decreases as the SIR decreases, it also decreases as r _{ tar } increases or τ increases. If r _{ tar } decreases or τ decreases, the average achievable rate for a target member with r _{ tar } increases. Since the optimal τ decreases by (r _{ tar })^{2} as in Theorem 3, the interference decreases as r _{ tar } increases. However, the signals of the target member and its leader become weaker by α and the number of the transmissions decreases as the optimal τ decreases, then the average achievable rate of the covered member with the optimal τ decreases as r _{ tar } increases even if the interference decreases as the optimal τ decreases.
6 Conclusions
In this paper, we have proposed an SAAC scheme and developed an analytical model of the SAAC schemes for group communications network. The proposed analytical model is affected by the intensity of leaders, the intensity of members, and the thresholds for DL and UL which are the important factors of service quality. The proposed analytical model presents the optimal TP to maximize the DL/UL joint probability at a target distance. Since the importance of the DL/UL joint probability at a target distance varies with the type of service, the optimal TP is carefully determined for providing services efficiently. The proposed analytical model has been validated via simulations, and the performance of the SAAC schemes has been demonstrated. For group communications, the DL/UL joint probability at a target distance can be maximized by the proposed SAAC scheme, which is superior to other schemes for the target distance. As a result, the importance of considering DL/UL joint probability in the group communication network has been proven.
7 Appendix
7.1 A.1 Proof of Lemma 1
Lemma 1 and its derivation corresponds to the interference limited network case as in [4].
7.2 A.2 Proof of Lemma 2
7.3 A.3 Proof of Optimal τ
Since f(τ)≥0, f(0)=0 is the global minimum of f(τ).
Thus, \(f\left (\frac {1}{b}\right)\) is the global maximum of f(τ) when b≥1. Therefore an optimal τ exists for f(τ) in the interval of [0,1].
7.4 A.4 Proof of Lemma 4
where (a _{9}) utilizes the change of the variable z=r ^{2} and (9) then change the order of integrals. (a _{10}) comes from the approximation of \(1/\left (1+a\sqrt {e^{t}+1}\right)\) to \(1/\left (a\sqrt {e^{t}+1}\right)\) where a>0. Also, (a _{11}) stems from the change of variables as \(C = \frac {\pi \lambda _{m}p_{d}(T_{d})^{2}\tau }{2\lambda _{l}}\) and \(u = C\sqrt {e^{t}1}\).
Declarations
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (2015R1A2A2A01004067), and Basic Science Research Program through the NRF funded by the Ministry of Education (NRF20100020210).
Authors’ contributions
ML’s contribution is writing the paper and conducting performance analysis and simulations. YK’s contribution is partly writing the paper and conducting simulations. TJL’s contribution is writing and revising the paper and guiding the direction and organization of the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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