Performance statistics of broadcasting networks with receiver diversity and Fountain codes

ABSTRACT

The performance of broadcasting networks employing Fountain codes with receiver diversity techniques is investigated in the present work. Particularly, we derive the closed-form expressions of the cumulative distribution function (CDF), the probability mass function (PMF), and the raw moments of the number of the needed time slots to deliver a common message to all users under two diversity schemes, namely, maximal ratio combining (MRC) and selection combining (SC). Numerical results are supplied to verify the accuracy of the considered networks and highlight the behaviours of these metrics as a function of some vital parameters such as the number of receivers, and the number of received antennae. Additionally, we also confirm the advantages of the MRC scheme compared with the SC scheme in the broadcasting networks.

KEYWORDS:

• Broadcasting networks
• fountain codes
• receiver diversity
• performance analysis

1. Introduction

Improving spectral efficiency (SE) and/or energy efficiency (EE) in the contemporary ultra-dense wireless networks such as, cellular networks (Di Renzo et al., 2018; Di Renzo, Zappone, et al., 2019; Hossain, 2022), low power wide area networks (LPWAN) with LoRa (Nguyen et al., 2020; Tu et al., 2021; Tu, Bradai, et al., 2022), SigFox, and WiFi networks (Feng et al., 2022; Ragpot et al., 2019), is a challenging task owing to the strong interference. There are many approaches in the literature, for example, relaying networks (Nguyen and Kong, 2017; Pham et al., 2022), energy harvesting (Nguyen and Kong, 2016; N. T. Nguyen et al., 2019), full-duplex communications, satellite communications (N. T. Nguyen et al., in press) and unmanned aerial devices (UAV) (Motlagh et al., 2021), are proposed to figure out this problem. Nevertheless, these techniques fairly boost up the SE and the EE since it only focuses on point-to-point communications rather than considering either multi-cast or broadcast networks. Broadcasting networks, differently, simultaneously broadcast information to all users instead of transmitting consecutively to all users. As a consequence, ones can theoretically facilitate the SE and the EE of the networks. Nevertheless, it is an extremely difficult task, in reality, to successfully broadcast messages to all users under limited resources (frequency and/or time). The principal reason is that each user suffers from different fading levels and different transmission distances. Consequently, there is a high probability that some users are not able to decode the error-free message. The sender, as a result, needs to re-send these lost packets until all users successfully receive those messages (Hassan et al., 2019). The situation gets worse if the number of devices goes without bound. Thus, the broadcasting networks is not attracted many researchers unless the re-transmission problem is figured out. Fortunately, a powerful coding scheme namely Fountain code has been proposed recently which can overcome such issue (Indoonundon & Fowdur, 2021). In fact, with Fountain code, the message is divided into many equal-length packets and each packet is encoded by a random generator matrix which ensures that each novel packet is independent of the previous one. On the receiver side, it only requires to collect a sufficient number of packets to decode the message and is uncorrelated with other users. As a result, it overcomes the re-transmission issue. Nevertheless, in the 5G-Advanced era, employing broadcasting networks is insufficient to ensure reliability owing to the strong interference. Therefore, other techniques which significantly facilitate the system diversity are required along with the broadcasting networks to achieve a comprehensive performance. One of the outstanding technologies to boost reliability is to employ the receiver diversity techniques such as maximal ratio combining (MRC) and selection combining (SC) (Tu et al., 2018). Under this context, in the present paper, we investigate the performance of the broadcasting networks utilizing Fountain code and receiver diversity techniques.

Before highlighting main contributions and novelties of the paper. Let us review the state-of-the-art of Fountain codes and receiver diversity schemes. Particularly, the performance of Fountain codes was studied extensively in Dang et al. (2019), Danufane and Di Renzo (2021), Li et al. (2020), Lim et al. (2021), Tran et al. (2012), and Tran et al. (2021) while the performance of receiver diversity was given in Arya and Chung (2018), Liao et al. (2018), Phu et al. (2020), and Tsai et al. (2022). Particularly, Dang et al. (2019) derived the outage probability (OP) of the multiple-input multiple-output (MIMO) systems with and without employing non-orthogonal multiple access (NOMA). The results showed that with NOMA, ones can save a significant number of time slots to convey the message compared with non-NOMA systems. They, however, consider only the single-user scenario rather than multi-user cases. Numerical results unveiled that the power splitting protocol is better than the time switching protocol. They, nonetheless, do not take into account the receiver diversity scheme. The comparison between FC and random linear network coding (RLNC) in short packet communications (SPC) was conducted in Li et al. (2020). Their outcomes stated that the SPC FC scheme provides better performance. Particularly, it has a smaller decoding error probability compared with RLNC. Besides, the authors in Lim et al. (2021) proposed a novel Fountain code dedicated to the SPC. The results revealed that their proposed Fountain code attains a lower bit error rate and higher rate compared to the conventional FC scheme. The distribution of the average time slot to deliver a packet in cellular networks was studied in Danufane and Di Renzo (2021) via tools from stochastic geometry (SG) (Al Hajj et al., 2020). Duy and others derived the average time slots in multi-hop communications with FCs in the closed-form equation (Tran et al., 2012) and the performance of the cooperative cognitive radio networks (CCRNs) employing FCs was investigated in Tran et al. (2021). Authors in Tsai et al. (2022) focussed on the diversity techniques at the receiver. The application of the receiver diversity technique in ultraviolet (UV) communications was studied in Arya and Chung (2018). More precisely, the switch and stay combining (SSC) was applied to improve the outage probability and symbol error probability (SEP). The maximal ratio combining scheme was used in Liao et al. (2018) to improve the coverage area as well as the diversity gain. The performance of transmit antenna selection (TAS) combined with Fountain code was studied in Phu et al. (2020). They, however, consider point-to-point communications and transmit diversity which requires the perfect channel state information at the transmitter (CSIT) that is difficult to obtain in practice.

Different with the above-mentioned works, in the present paper, we study the performance of the broadcasting networks utilizing Fountain code and receiver diversity techniques. Additionally, we consider different metrics the statistics of the number of the needed time slot to broadcast packets to all users rather than the outage probability and/or the average rate as like works in the literature. The main contributions and novelties are summarized as follows:

• We derive the cumulative distribution function (CDF), the probability mass function (PMF), the raw moments of the needed time slot to deliver messages to an arbitrary number of end user.

• We consider two popular diversity techniques namely, the maximal ratio combining and the selection combining.

• We provide simulation results based on Monte-Carlo method to verify the correctness of the considered networks as well as to unveil the impact of some important parameters on the performance of all considered metrics.

The remainder of the manuscript is organized as follows: Section 2 provides the system model. The main derivations and trends are given in Section 3. Numerical results are provided in Section 4. Section 5 concludes the paper.

2. System model

Let us consider a wireless networks where an access point (AP) denoted by A sends a common message to $\mathcal{U}\in \mathbb{N}$ users associated with it as shown in Figure 1. AP employs Fountain code to encode the message. More precisely, AP splits the message into $\mathcal{P}$ equal-length packets and encodes each packet with a random generator matrix. At each time slot, AP broadcasts a novel packet and all users will collect these packets until successfully decode the message. After decoding the message, each user will send back to the AP an acknowledgment (ACK) message via a high accuracy feedback channel (N.-L. Nguyen et al., 2023; Tu, Nguyen, et al., 2022). The AP stops transmitting once receiving all ACK packets. For simplicity, we assume that the AP only sends a packet per time slot. The AP is equipped with a single antenna while users are equipped with an arbitrary number of antennae. Specifically, the uth user, $u\in \{1,\dots ,\mathcal{U}\}$, is equipped with $N_u\in \{1,\dots ,{\mathcal{N}}_{max}\}$ antennae, ${\mathcal{N}}_{max}\in \mathbb{N}$ is the maximal number of antennae can be installed on the user devices.

2.1. Channel modelling

We consider both large-scale path-loss and small-scale fading in the present work. The influence of shadowing is not taken into consideration since it is a popular case in the literature (Tu & Di Renzo, 2020).

Figure 1. The considered broadcasting networks.

2.1.1. Small-scale fading

Let us denote $c_{u,o}^{\omega }$ as the channel coefficient from AP to the oth antenna at the uth user at the $\omega \in \mathbb{N}$ time slot, it is modelled by a complex Gaussian random variable (RV) with zero mean and ${\tau }_n^{\omega }$ variance, i.e. $c_{u,o}^{\omega }\in \mathcal{C}\mathcal{N}(0,{\tau }_u^{\omega })$. The channel gain is then followed by an exponential random variable with scale parameter ${\tau }_{u,o}^{\omega }$. In this work, we assume that small-scale fading follows by a flat fading which is unchanged in a time slot and varies independently between time slots (N. T. Nguyen, Tran, Tran, et al., 2022).

2.1.2. Large-scale path-loss

Let us denote ${\mathrm{{\rm Y}}}_u$ as the large-scale path-loss from AP to the uth user, it is formulated as follows: (1) ${\mathrm{{\rm Y}}}_u={\mathcal{C}}_0{\left(1+d_u\right)}^{\upsilon },(1)$(1) where ${\mathcal{C}}_0=(\frac{4\pi }{\lambda }{)}^2$ is the path-loss constant, $\lambda =\frac{c}{f_c}$ is the wavelength, $c=3\times {10}^8$ is the speed of light, $f_c$ is the carrier frequency and $d_u$ is the Euclidean distance from the AP to the uth user. υ is the path-loss constant and its value relies on the transmission environment, i.e. urban, suburban, rural areas, etc.

Remark 2.1

It is noted that when the transmission distance approaches zero or the distance between transmitter and receiver reaches zero, the adopted path-loss model in (1(1) ${\mathrm{{\rm Y}}}_u={\mathcal{C}}_0{\left(1+d_u\right)}^{\upsilon },(1)$(1) ) does not go to infinity. It, as a result, tackles the popular issue of the unbounded path-loss model in the literature (Di Renzo, Tu, et al., 2019).

2.2. Receiver diversity

In this work, if the number of antennae at the receiver is greater than 2, it can then utilize diversity techniques such as the maximal ratio combining or selection combining to facilitate the transmission reliability (Tu et al., 2018). The former linearly combines the channel gain from all received antenna while the latter simply chooses the antenna having the largest channel gain. As a consequence, theoretically, the MRC scheme always outperforms the SC. From the implementation perspective, the MRC, however, is more challenging since it requires perfect synchronizations between antennae and the adder while the SC solely employs a comparator and a switch to realize this technique. In the present work, we assume that all users can deploy both schemes.

The received signals at the uth receiver and the ω time-slots under the MRC and SC schemes denoted by $y_u^{\omega ,\theta },\theta \in \{\mathrm{MRC},\mathrm{SC}\}$, are given as follows: (2) $\begin{array}{rl}(2)& y_u^{\omega ,\mathrm{MRC}}& =\sqrt{P_A{\mathrm{{\rm Y}}}_u^{-1}}\left(\sum _{o=1}^{{\mathcal{N}}_u}{c}_{u,o}^{\omega }\right)x^{\omega }+n_u,y_u^{\omega ,\mathrm{SC}}& =\sqrt{P_A{\mathrm{{\rm Y}}}_u^{-1}}\left(\underset{o\in \left\{1,\dots ,{\mathcal{N}}_u\right\}}{max}{c}_{u,o}^{\omega }\right)x^{\omega }+n_u.\end{array}(2)$(2) Here $P_A$ is the transmit power of the AP, and $x^{\omega }$ is the transmitted packet at ω time slot. We consider the unit power signal, i.e. $\mathbb{E}\{{|x^{\omega }|}^2\}=1,\mathrm{\forall }\omega$ and $\mathbb{E}\{\cdot \}$ is the expectation operator.

2.3. Signal-to-noise ratio (SNR)

From (2(2) $\begin{array}{rl}(2)& y_u^{\omega ,\mathrm{MRC}}& =\sqrt{P_A{\mathrm{{\rm Y}}}_u^{-1}}\left(\sum _{o=1}^{{\mathcal{N}}_u}{c}_{u,o}^{\omega }\right)x^{\omega }+n_u,y_u^{\omega ,\mathrm{SC}}& =\sqrt{P_A{\mathrm{{\rm Y}}}_u^{-1}}\left(\underset{o\in \left\{1,\dots ,{\mathcal{N}}_u\right\}}{max}{c}_{u,o}^{\omega }\right)x^{\omega }+n_u.\end{array}(2)$(2) ), the signal-to-noise ratio (SNR) at the uth receiver at the ω time slot under the $\theta \in \{\mathrm{MRC},\mathrm{SC}\}$ diversity scheme denoted by ${\gamma }_u^{\omega ,\theta }$ is formulated as follows: (3) ${\gamma }_u^{\omega ,\theta }=\frac{P_A{X}_u^{\omega ,\theta }{\left|x^{\omega }\right|}^2}{{\mathrm{{\rm Y}}}_u{\sigma }_u^2}.(3)$(3) Here ${\sigma }_u^2$ is the variance of the additive white Gaussian noise (AWGN) and computed as ${\sigma }_u^2={\sigma }^2=-174+\mathrm{NF}+10\log (\mathrm{Bw})\mathrm{\forall }u$, Bw (in [Hz]) is the transmission bandwidth, and NF (in [dB]) is the noise figure at the receiver. $X_u^{\omega ,\theta }$ is the channel gain at the receiver and counts on the adopted diversity scheme. Particularly, if the MRC scheme is employed, $X_u^{\omega ,\theta }$ is computed as follows (Tu et al., 2018): (4) $X_u^{\omega ,\mathrm{MRC}}=\sum _{o=1}^{{\mathcal{N}}_u}{\left|c_{u,o}^{\omega }\right|}^2,(4)$(4) and if SC is considered, $X_u^{\omega ,\theta }$ is rewritten as (Tu et al., 2018): (5) $X_u^{\omega ,\mathrm{SC}}=\underset{o\in \left\{1,{\mathcal{N}}_u\right\}}{max}\left\{{\left|c_{u,o}^{\omega }\right|}^2\right\},(5)$(5) where $max\{\cdot \}$ is the maximum function. As mentioned in Section 2.1.1, we consider the scenario where fading changes across time slots. Consequently, the superscript ω in (3(3) ${\gamma }_u^{\omega ,\theta }=\frac{P_A{X}_u^{\omega ,\theta }{\left|x^{\omega }\right|}^2}{{\mathrm{{\rm Y}}}_u{\sigma }_u^2}.(3)$(3) ) can be omitted and the SNR is rewritten as (6) ${\gamma }_u^{\theta }=\frac{P_A{X}_u^{\theta }{\left|x_u\right|}^2}{{\mathrm{{\rm Y}}}_u{\sigma }_u^2}.(6)$(6)

3. Performance analysis

In this section, we are going to address the performance of the number of the required time slot to dispatch the message to all users. More precisely, we study the cumulative distribution function, the probability mass function, and the raw moments of the number of the needed time slot to deliver the message. Before going to derive these metrics, let us begin with the probability of successfully sending a packet to a user denoted by $P_{\mathrm{del}}$. For the MRC scheme, it is denoted by $P_{\mathrm{del}}^{\mathrm{MRC}}$ and is computed as follows (N. T. Nguyen, Nguyen, et al., 2022): (7) $\begin{array}{rl}(7)& P_{\mathrm{del}}^{\mathrm{MRC}}\left(u\right)& =Pr\left\{{\log }_2\left(1+{\gamma }_u^{\mathrm{MRC}}\right)\ge \alpha \right\}=Pr\left\{{\gamma }_u^{\mathrm{MRC}}\ge \varphi \right\}& \stackrel{(a)}{=}1-F_{{\gamma }_u^{\mathrm{MRC}}}\left(\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A}\right),\end{array}(7)$(7) where $\varphi =2^{\alpha }-1$, ${\alpha }_u=\alpha \mathrm{\forall }u$ is the expected rate, $Pr\{\cdot \}$ is the probability operator, $F_{{\gamma }_u^{\mathrm{MRC}}}(x)$ is the CDF of RV ${\gamma }_u^{\mathrm{MRC}}$. To derive $F_{{\gamma }_u^{\mathrm{MRC}}}(x)$ we commence with the definition of the moment generating function (MGF) of $X_u^{\mathrm{MGF}}$ as follows: (8) $\begin{array}{rl}(8)& M_{{\gamma }_u^{\mathrm{MRC}}}\left(s\right)& =\mathbb{E}\left\{\exp \left(-s\sum _{o=1}^{{\mathcal{N}}_u}{\left|c_{u,o}\right|}^2\right)\right\}& ={\int }_{x_1=0}^{\mathrm{\infty }}\cdots {\int }_{x_{{\mathcal{N}}_u}=0}^{\mathrm{\infty }}\exp \left(-s\sum _{o=1}^{{\mathcal{N}}_u}{x}_o\right)f_{{\left|c_{u,1}\right|}^2,\dots ,{\left|c_{u,{\mathcal{N}}_u}\right|}^2}\left(x_1,\dots ,x_{{\mathcal{N}}_u}\right)\mathrm{d}{x}_1\dots \mathrm{d}{x}_{{\mathcal{N}}_u}\\ & \stackrel{(a)}{=}\prod _{o=1}^{{\mathcal{N}}_u}{M}_{{\left|c_{u,o}\right|}^2}\left(s\right)\stackrel{(b)}{=}\prod _{o=1}^{{\mathcal{N}}_u}\frac{1}{1+s{\tau }_{u,o}}={\left(1+s{\tau }_u\right)}^{-{\mathcal{N}}_u},\end{array}(8)$(8) where $M_{{|c_{u,o}|}^2}(s)$ is the MGF of the RV ${|c_{u,o}|}^2$, $f_{{|c_{u,1}|}^2,\dots ,{|c_{u,{\mathcal{N}}_u}|}^2}(x_1,\dots ,x_{{\mathcal{N}}_u})\mathrm{d}{x}_1\dots \mathrm{d}{x}_{{\mathcal{N}}_u}$ is the joint PDF of ${\mathcal{N}}_u$ RVs; (a) is held by yielding the uncorrelated property between RVs ${|c_{u,o}|}^2$, (b) is attained by using the MGF of the exponential RV, and the last equation is obtained as the channel gain between antennae is equal, i.e. ${\tau }_u={\tau }_{u,o},\mathrm{\forall }o$. Looking at (8(8) $\begin{array}{rl}(8)& M_{{\gamma }_u^{\mathrm{MRC}}}\left(s\right)& =\mathbb{E}\left\{\exp \left(-s\sum _{o=1}^{{\mathcal{N}}_u}{\left|c_{u,o}\right|}^2\right)\right\}& ={\int }_{x_1=0}^{\mathrm{\infty }}\cdots {\int }_{x_{{\mathcal{N}}_u}=0}^{\mathrm{\infty }}\exp \left(-s\sum _{o=1}^{{\mathcal{N}}_u}{x}_o\right)f_{{\left|c_{u,1}\right|}^2,\dots ,{\left|c_{u,{\mathcal{N}}_u}\right|}^2}\left(x_1,\dots ,x_{{\mathcal{N}}_u}\right)\mathrm{d}{x}_1\dots \mathrm{d}{x}_{{\mathcal{N}}_u}\\ & \stackrel{(a)}{=}\prod _{o=1}^{{\mathcal{N}}_u}{M}_{{\left|c_{u,o}\right|}^2}\left(s\right)\stackrel{(b)}{=}\prod _{o=1}^{{\mathcal{N}}_u}\frac{1}{1+s{\tau }_{u,o}}={\left(1+s{\tau }_u\right)}^{-{\mathcal{N}}_u},\end{array}(8)$(8) ), it is straightforward to realize that it is the MGF function of the Gamma distribution. As a result, the complement cumulative distribution function (CCDF) of ${\gamma }_u^{\mathrm{MRC}}$ in (7(7) $\begin{array}{rl}(7)& P_{\mathrm{del}}^{\mathrm{MRC}}\left(u\right)& =Pr\left\{{\log }_2\left(1+{\gamma }_u^{\mathrm{MRC}}\right)\ge \alpha \right\}=Pr\left\{{\gamma }_u^{\mathrm{MRC}}\ge \varphi \right\}& \stackrel{(a)}{=}1-F_{{\gamma }_u^{\mathrm{MRC}}}\left(\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A}\right),\end{array}(7)$(7) ) is rewritten as (9) $P_{\mathrm{del}}^{\mathrm{MRC}}\left(u\right)=\frac{\mathrm{\Gamma }\left(N_u,\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A{\tau }_u}\right)}{\mathrm{\Gamma }\left(N_u\right)}.(9)$(9) Here $\mathrm{\Gamma }(.)$ is the Gamma function (Gradshteyn & Ryzhik, 2007, Equation 8.310.1) and $\mathrm{\Gamma }(.,.)$ is the upper incomplete Gamma function (Gradshteyn & Ryzhik, 2007, Equation 350.2).

Next, the $P_{\mathrm{del}}$ with SC technique is defined and is given below (N. T. Nguyen, Nguyen, et al., 2022) (10) $\begin{array}{rl}(10)& P_{\mathrm{del}}^{\mathrm{SC}}\left(u\right)=& Pr\left\{{\log }_2\left(1+{\gamma }_u^{\mathrm{SC}}\right)\ge \alpha \right\}=Pr\left\{{\gamma }_u^{\mathrm{SC}}\ge \varphi \right\}& =1-Pr\left\{\underset{o\in \left\{1,{\mathcal{N}}_u\right\}}{max}\left\{{\left|c_{u,o}\right|}^2\right\}\le \frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A}\right\}=1-\prod _{o=1}^{{\mathcal{N}}_u}{F}_{{\left|c_{u,o}\right|}^2}\left(\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A}\right)\\ & \stackrel{(a)}{=}1-\prod _{o=1}^{{\mathcal{N}}_u}\left(1-\exp \left(-\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A{\tau }_{u,o}}\right)\right)=1-{\left(1-\exp \left(-\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A{\tau }_u}\right)\right)}^{{\mathcal{N}}_u},\end{array}(10)$(10) where (a) is attained by substituting the CDF of the RV ${|c_{u,o}|}^2$.

3.1. CDF of the needed time slot to deliver $\mathcal{P}$ packets

In the considered broadcasting networks with Fountain codes, the number of the required time slot to dispatch a common message to all users is equivalent to the number of the needed time slot of the worst user. Mathematical speaking, the CDF of the needed time slot to deliver $\mathcal{P}$ packets under the $\theta \in \{\mathrm{MRC},\mathrm{SC}\}$ scheme denoted by $F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB})$ is computed as follows (N.-L. Nguyen et al., 2023): (11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) where $F_{t_u}(\mathcal{P},\theta ,t_{FB})$ is the CDF of the required time slot of receiving $\mathcal{P}$ packets of the uth user under the θ scheme. (a) is held by the independence of receiving packets between users and (b) is held by utilizing the CDF of the negative binomial distribution, $(c.)$ is the binomial coefficient.

3.2. PMF of the needed time slot to deliver $\mathcal{P}$ packets

Having obtained the CDF, the PMF denoted by $p_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB})$ can be obtained straightforwardly and computed as follows: (12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12)

3.3. Raw moments of the needed time slot to deliver $\mathcal{P}$ packets

Moments are used to describe the characteristic of a distribution. Particularly, the first moment is the expected value of the distribution while the second moment measures how the distribution spreads out around the mean value. The third moment is called ‘Skewness’ which measures the symmetric of the distribution and the fourth moment is ‘kurtosis’ measuring whether the distribution is heavy tails or not. In this section, we address the raw moments of the needed time slot to dispatch the message to all $\mathcal{U}$ users. The kth raw moments of the needed time slot under the θ diversity scheme denoted by ${\mu }_k^{{}^{\prime }}(\mathcal{P},\theta )$ is calculated as follows (13) ${\mu }_k^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^k{p}_{FB}\left(\mathcal{P},\theta ,w\right).(13)$(13)

Corollary 3.1:

The first raw moment or the expectation of the considered RV can be directly obtained from (13(13) ${\mu }_k^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^k{p}_{FB}\left(\mathcal{P},\theta ,w\right).(13)$(13) ) by setting k = 1. Mathematical speaking, the expectation can be computed as (14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14)

Corollary 3.2:

The variance of the needed time slot to broadcast $\mathcal{P}$ packets to all users denoted by ${\mu }_2(\mathcal{P},\theta )$ is computed as follows: (15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15)

4. Numerical results

In this section, numerical results are given to confirm the accuracy of the proposed mathematical framework as well as to unveil the behaviours of these metrics as a function of some key parameters such as the number of received antennae, the number of users, and the number of packets, etc. Without loss of generality, following parameters are employed throughout this section: $\mathrm{NF}=6$ dB, $\mathrm{Bw}=2$ MHz, $\upsilon =3.25$, $f_c=2.1$ GHz,¹ $\alpha =1$ bits/s/Hz, $P_A=10$ dBm, ${\tau }_u=\tau =2,\mathrm{\forall }u$, $\mathcal{P}=4$ packets, ${\mathcal{N}}_{max}=4$. We consider 4 users, i.e. $\mathcal{U}=4$ and the number of received antennae of these users are $\{2,4,3,2\}$, respectively. The position of AP is at $(0,0)$ [m] and the locations of all users are $(200,0)$, $(150,-35)$, $(-100,25)$, and $(-60,-60)$, respectively. Figures 2 and 3 investigate the impact of the expected rate α on the performance of the CDF, the PMF, the expectation, and the variance of the number of the needed time slots to dispatch the message. Generally, we observe a good agreement between the derived mathematical framework and simulation results via the Monte-Carlo method. Figure 2(a) illustrates that the CDF is a monotonic decrease function regarding to α. This can be explained effortlessly from the definition of the success probability in (9(9) $P_{\mathrm{del}}^{\mathrm{MRC}}\left(u\right)=\frac{\mathrm{\Gamma }\left(N_u,\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A{\tau }_u}\right)}{\mathrm{\Gamma }\left(N_u\right)}.(9)$(9) ) and (10(10) $\begin{array}{rl}(10)& P_{\mathrm{del}}^{\mathrm{SC}}\left(u\right)=& Pr\left\{{\log }_2\left(1+{\gamma }_u^{\mathrm{SC}}\right)\ge \alpha \right\}=Pr\left\{{\gamma }_u^{\mathrm{SC}}\ge \varphi \right\}& =1-Pr\left\{\underset{o\in \left\{1,{\mathcal{N}}_u\right\}}{max}\left\{{\left|c_{u,o}\right|}^2\right\}\le \frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A}\right\}=1-\prod _{o=1}^{{\mathcal{N}}_u}{F}_{{\left|c_{u,o}\right|}^2}\left(\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A}\right)\\ & \stackrel{(a)}{=}1-\prod _{o=1}^{{\mathcal{N}}_u}\left(1-\exp \left(-\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A{\tau }_{u,o}}\right)\right)=1-{\left(1-\exp \left(-\frac{{\mathrm{{\rm Y}}}_u{\sigma }^2\varphi }{P_A{\tau }_u}\right)\right)}^{{\mathcal{N}}_u},\end{array}(10)$(10) ) that $P_{\mathrm{del}}^{\theta }$, $\theta \in \{\mathrm{MRC},\mathrm{SC}\}$ always decreases with α. As a consequence, the CDF in (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ) reduces too as it is a product of numbers between zero and one. Additionally, we also see that increasing $t_{FB}$ will scale up the CDF as it is a monotonic increasing function with $t_{FB}$. Besides, it is obvious that the MRC scheme outperforms the SC scheme for example when $\alpha =3$, the CDF of the SC scheme is solely under 0.1 for $t_{FB}=12$ while the MRC scheme is approximately 0.25. Figure 2(b) stretches the performance of the PMF versus the expected rate α. We notice that it is a unimodal function with respect to α. We see that the peak of the SC scheme is the left-hand side of the MRC scheme. Moreover, increasing the transmit power will shift the peak towards the right-hand side. Both expectation and variance are monotonically increasing with α as showed in Figure 3. The explanation is straightforward as the expected rate goes large, it becomes more difficult to receive packets without errors, and as a consequence, the expectation goes up without bound. Regarding the variance, we observe a similar trend as the expectation but it approaches infinity faster owing to the second order.

Figures 4 and 5 study the influence of the transmit power $P_A$ on the performance of all considered metrics. It is certain that raising the transmit power will consistently improve the CDF. Moreover, the larger the transmitted packets the lower the CDF. Similar to the influence of α, the PMF is also a unimodal function with respect to $P_A$. We see that the peak of MRC scheme is at the left-hand side of the SC scheme. Additionally, the peak of case $\mathcal{P}=8$ is higher than case $\mathcal{P}=2$. It can be explained that the probability of successfully delivering 2 packets required $t_{FB}=10$ is small compared to delivering 8 packets since it is likely to dispatch 2 packets with smaller $t_{FB}$.

Figure 2. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) CDF vs. α with various values of $t_{FB}$. and (b) PMF vs. α with various values of $P_A$.

Figure 3. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) Expectation vs. α. and (b) Variance vs. α with different values of $P_A$.

Figure 4. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) CDF vs. $P_A$ with various values of $t_{FB}$. and (b) PMF vs. $P_A$ with various values of $\mathcal{P}$.

Figure 5. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) Expectation vs. $P_A$ with various values of $\mathcal{P}$. and (b) Variance vs. $P_A$ with different values of $\mathcal{P}$.

We observe the contrary behaviours of mean with the CDF with respect to $P_A$. Particularly, we see that by increasing $P_A$ the mean will approach the lower bound, i.e. $\mathcal{P}$ packets. The rationale behind this phenomenon is following: when $P_A$ is small, there is a high probability that all users can not decode packets thus the expectation is very high. On the other hand, when $P_A\gg 1$, it is obvious that all users can receive packets without error and thus requiring exactly $\mathcal{P}$ time slots to deliver $\mathcal{P}$ packets. Besides, the variance approaches zero when $P_A$ goes to infinity. This can be explained directly from the definition of the variance that the subtraction of two tiny numbers is a very tiny number and it means that the system convergences.

Figures 6 and 7 address the performance of the number of received antennae on the performance of these metrics. We see that increasing $\mathcal{N}$ simply boosts the CDF and the larger the transmit power the sooner the CDF reaching one. More precisely, with $P_A=8$ dBm, it solely requires 3 antennae to reach one for both schemes while for $P_A=4$, it needs 4 antennae for the MRC scheme and larger than 8 antennae for the SC scheme. Again, there exists a peak for the PMF, the main reason is that when $\mathcal{N}$ is small it requires more effort to successfully deliver $\mathcal{P}$ packets in exactly $t_{FB}$ time slots. On the other hand, when $\mathcal{N}$ is large, the probability to broadcast $\mathcal{P}$ packets in the same $t_{FB}$ time slots will be zero because there is a high probability that the transmitter has already delivered all transmitted packets less than $t_{FB}$. Figure 7(a) stretches the performance of the expectation on the $\mathcal{N}$ with different values of $P_A$. We see that there also exists a lower bound which is equal to $\mathcal{P}=4$ packets for both schemes when $\mathcal{N}$ goes large. The explanation is similar to Figure 5(a). However, the pace of reaching its lower bound is different between schemes and transmit power. More particular, the larger the transmit power the sooner the expectation reaches its lower bound and the MRC scheme always approaches faster than the SC scheme regardless of the transmit power. As the expectation goes to a constant number, the variance then goes to zero. It can be justified immediately from the definition of the variance.

Figure 6. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) CDF vs. ${\mathcal{N}}_u=\mathcal{N},\mathrm{\forall }u$ with various values of $P_A$. and (b) PMF vs. ${\mathcal{N}}_u=\mathcal{N},\mathrm{\forall }u$ with various values of $t_{FB}$.

Figure 7. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) Expectation vs. ${\mathcal{N}}_u=\mathcal{N},\mathrm{\forall }u$ with various values of $P_A$. and (b) Variance vs. ${\mathcal{N}}_u=\mathcal{N},\mathrm{\forall }u$ with different values of $P_A$.

The impact of the number of transmitted packets on the performance of the considered broadcasting networks is given in Figures 8 and 9. Particularly, Figure 8(a) shows the behaviour of the CDF concerning on $\mathcal{P}$ with a constant number of $t_{FB}$. We see that under this scenario, the CDF monotonically decreases, and the smaller the transmit power the smaller the CDF. Additionally, we observe that the CDF of the MRC scheme is substantially better than the SC scheme. Also, fixing $t_{FB}$ the PMF in Figure 8(b) is also a unimodal function with respect to $\mathcal{P}$. On the other hand, both the expectation and the variance go up with $\mathcal{P}$ in Figure 9. The reason is obvious that broadcasting more packets, for sure, requires more time slots. However, the required time slots for MRC and SC schemes are not the same. The SC scheme, of course, needs more time slots than the MRC scheme to deliver the same $\mathcal{P}$ packets. For example, with $\mathcal{P}=6$ packets and $P_A=5$ dBm, SC needs approximately 12 time slots while the MRC scheme requires around 8 time slots. We observe the same trend for the variance that it simply increases with $\mathcal{P}$. Nevertheless, the increasing pace is different and relies on the transmit power. Particularly, the variance increases slowly with large $P_A$.

Figure 8. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) CDF vs. $\mathcal{P}$ with various values of $P_A$. and (b) PMF vs. $\mathcal{P}$ with various values of $P_A$.

Figure 9. Solid and dash-dot lines are plotted by employing (11(11) $\begin{array}{rl}(11)& F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)& =Pr\left\{\underset{u\in \left\{1,\dots ,\mathcal{U}\right\}}{max}\left\{t_u\right\}\le t_{FB}\right\}\stackrel{(a)}{=}\prod _{u=1}^{\mathcal{U}}{F}_{t_u}(\mathcal{P},\theta ,t_{FB}),F_{t_u}(\mathcal{P},\theta ,t_{FB})& =Pr\left\{t_u\le t_{FB}\right\}\stackrel{(b)}{=}\sum _{w=\mathcal{P}}^{t_{FB}}\left(\begin{array}{c}w-1\\ \mathcal{P}-1\end{array}\right){\left(P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{\mathcal{P}}{\left(1-P_{\mathrm{del}}^{\theta }\left(u\right)\right)}^{w-\mathcal{P}},\end{array}(11)$(11) ), (12(12) $p_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)=F_{\mathrm{FB}}\left(\mathcal{P},\theta ,t_{FB}\right)-F_{\mathrm{FB}}(\mathcal{P},\theta ,t_{FB}-1)(12)$(12) ), (14(14) ${\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}w{p}_{FB}\left(\mathcal{P},\theta ,w\right).(14)$(14) ), and (15(15) ${\mu }_2\left(\mathcal{P},\theta \right)=\sum _{w=\mathcal{P}}^{\mathrm{\infty }}{w}^2{p}_{FB}\left(\mathcal{P},\theta ,w\right)-{\left({\mu }_1^{{}^{\prime }}\left(\mathcal{P},\theta \right)\right)}^2.(15)$(15) ). Markers are Monte-Carlo simulations. (a) Expectation vs. $\mathcal{P}$ with various values of $P_A$. and (b) Variance vs. $\mathcal{P}$ with different values of $P_A$.

5. Conclusion

This paper investigated the performance of the broadcasting network employing Fountain code and receiver diversity techniques. Particularly, we derive the closed-form equations of the CDF, PMF, expectation, and variance of the needed time slots to dispatch an arbitrary packets to all users under both MRC and SC schemes. Numerical results confirmed that MRC scheme always outperformed the SC scheme and an increasing number of received antennae as well as the transmit power are beneficial for the systems. The paper can be extended in several directions such as employing relays (N. T. Nguyen, Tran, Trinh et al., 2022) and/or reconfigurable intelligent surfaces (RISs) to further improve the system performance (Trinh, Papazafeiropoulos, et al., 2021; Trinh, Tu, et al., 2021). On the other hand, the paper can also apply tools from stochastic geometry to model the randomness of the wireless nodes (Aravanis et al., 2019; Kumar & Tyagi, 2021) or apply the deep learning to further optimize the system by exploiting the power of data-driven approach (Tu et al., in press; Zappone et al., 2019). Furthermore, we can also combine with other techniques/systems such as energy harvesting, full-duplex, device-to-device communications, cognitive radio networks, etc.

Disclosure statement

The authors declare that there is no conflict of interest in this paper.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was financially supported by Ton Duc Thang University [grant number FOSTECT.2023.09].

Notes on contributors

Lam-Thanh Tu

Lam-Thanh TU (Email: [email protected]) received the B.Eng. degree in electronics and telecommunications engineering from the Ho Chi Minh City University of Technology, Vietnam, in 2009, the M.Sc. degree in telecommunications engineering from the Posts and Telecommunications Institute of Technology, Vietnam, in 2014, and the Ph.D. degree from the University of Paris Sud, Paris-Saclay University, France, in 2018. From 2015 to 2018, he was with the French National Center for Scientific Research (CNRS), Paris, as an Early Stage Researcher of the European-Funded Project H2020 ETN-5Gwireless. From 2019-2021, he was with the Xlim Research Institute, University of Poitiers, France as a postdoctoral research fellow. From 2022, he is with the Faculty of Electrical and Electronics Engineering, at Ton Duc Thang University, Vietnam. His research interests include stochastic geometry, LoRa networks, reconfigurable intelligent surfaces, and artificial intelligence applications for wireless communications.

Tan N. Nguyen

Tan N. Nguyen was born in 1986 in Nha Trang City, Vietnam. He received a BS degree in electronics in 2008 from Ho Chi Minh University of Natural Sciences and an MS degree in telecommunications engineering in 2012 from Vietnam National University. He received a Ph.D. in communications technologies in 2019 from the Faculty of Electrical Engineering and Computer Science at VSB - Technical University of Ostrava, Czech Republic. He joined the Faculty of Electrical and Electronics Engineering of Ton Duc Thang University, Vietnam, in 2013, and since then has been lecturing. His major interests are cooperative communications, cognitive radio, physical layer security and signal processing.

Phuong T. Tran

Phuong T. Tran received the B.Eng. and M.Eng. degrees in electrical engineering from the Ho Chi Minh City University of Technology, Ho Chi Minh City, in 2002 and 2005, respectively, and the M.S. degree in mathematics and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, USA, in 2013. In 2007, he became a Vietnam Education Foundation Fellow at Purdue University. In 2013, he joined the Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Vietnam, and he has been served as the Vice Dean of Faculty since October 2014. His major interests are in the area of wireless communications and network information theory.

Tran Trung Duy

Tran Trung Duy received the Ph.D. degree in electrical engineering from University of Ulsan, South Korea. In 2013, he joined Posts and Telecommunications Institute of Technology, Ho Chi Minh city campus (PTIT-HCM), as a lecturer. From 2022, he is an associate Professor of Wireless Communications at PTIT-HCM. His major research interests are cooperative communications, cooperative multi-hop, cognitive radio, physical-layer security, energy harvesting, hardware impairments and Fountain codes.

Quang-Sang Nguyen

Quang-Sang Nguyen received the B.E. degree from Ho Chi Minh City University of Transport, Ho Chi Minh City, Vietnam, in 2010, the M.E. degree from Ho Chi Minh City University of Technology, Ho Chi Minh City, in 2013, and the Ph.D. degree in electrical engineering from the University of Ulsan, Ulsan, South Korea, in 2017. From January 2017 to June 2017, he was a Postdoctoral Research Fellow with Queen's University Belfast, Belfast, U.K. From June 2017 until May 2021, he has been a Lecturer at Duy Tan University, Ho Chi Minh City. Since May 2021, He is currently the Dean of the Faculty of Electrical and Electronics Telecommunications Engineering at Ho Chi Minh City University of Transport. His major research interests are cooperative communication, cognitive radio network, physical layer security, energy harvesting, and non-orthogonal multiple access, Artificial intelligence.

Notes

1 Here, the carrier frequency is selected to align with the current 4G systems. However, the proposed mathematical framework can be applied to all sub-6 Ghz carrier frequencies (Phu et al., 2021).