Joint Statistics of Partial Sums of Ordered i.n.d. Gamma Random Variables

Abstract

From the perspective of wireless communication, as communication systems become more complex, order statistics have gained increasing importance, particularly in evaluating the performance of advanced technologies in fading channels. However, existing analytical methods are often too complex for practical use. In this research paper, we introduce innovative statistical findings concerning the sum of ordered gamma-distributed random variables. We examine various channel scenarios where these variables are independent but not-identically distributed. To demonstrate the practical applicability of our results, we provide a comprehensive closed-form expression for the statistics of the signal-to-interference-plus-noise ratio in a multiuser scheduling system. We also present numerical examples to illustrate the effectiveness of our approach. To ensure the accuracy of our analysis, we validate our analytical results through Monte Carlo simulations.

1. Introduction

Analyzing the theoretical performance of emerging technologies is crucial for informed design decisions in real-world systems. Among the mathematical and statistical tools employed for evaluating digital wireless communication systems over fading channels, order statistics is a significant branch of statistics theory [1,2]. As wireless communication systems and algorithms become more complex, the statistical analysis of ordered random variables (RVs) in specific fading channel conditions has gained importance for timely performance assessment. Order statistics primarily focuses on the characteristics and distributions of ordered RVs and their statistical properties.

The application of order statistics is prevalent in various statistical fields, including life testing, quality control, signal and image processing, and the Internet of Things [3,4,5,6,7,8]. However, effectively working with order statistics presents challenges, especially in comparison to conventional communication systems. This difficulty arises because ordered RVs become dependent due to their inherent inequality relationships, making the analysis of joint statistics more complex. Typically, in order-based statistical analysis, the conventional approach involves intricate multiple integrations, and this complexity escalates as more RVs are considered for ordering. Additionally, obtaining specific statistical results, like probability density functions (PDFs) or moment generating functions (MGFs) for partial sums of ordered RVs, is significantly more challenging than for the sum of all ordered RVs or the best-ordered RV. Consequently, approximate or simulation methods are frequently employed to address these complexities [9,10,11,12].

In this research, we showcase the practical application of advanced findings in order statistics, leading to elegant closed-form solutions. We employ mathematical formalism to demonstrate this, focusing on a specific application: calculating the joint statistics of partial sums of ordered gamma-distributed RVs that are independent but non-identically distributed (i.n.d.).

To illustrate our approach and contributions, we focus on a state-of-the-art technology example, a multiuser scheduling scheme [13,14,15]. In conventional parallel multiuser scheduling, a scheduled user’s signal is detected by correlating signals from all scheduled users, leading to multiple-user interference (MUI). This MUI significantly impacts the capacity and performance of multiuser systems, particularly as the number of scheduled users increases. To address MUI’s effect, we use the signal-to-interference-plus-noise ratio (SINR), representing the ratio of useful power to noise and interference from other scheduled users. Given this context, we aim to investigate interference’s impact on scheduled users’ performance, including throughput, within a selection-based parallel multiuser scheduling scheme. Our study aims to determine the total average sum rate capacity and average spectral efficiency (ASE) based on the SINR of scheduled users. A significant challenge in this investigation is calculating SINR statistics for a specific scheduled user. Our study tackles this by deriving the SINR statistics based on the joint PDF of the signal-to-noise ratio (SNR) of the desired scheduled user and the sum of SNRs from other interfering scheduled users across i.n.d. gamma fading channels.

While various efforts have been made to address statistical issues of this nature, prior analytical findings have been limited and heavily reliant on asymptotic analysis, even in cases of independent and identically distributed (i.i.d.) channel settings. To the authors’ knowledge, the existing closed-form analysis of the collective statistics pertaining to the partial sums of ordered i.n.d. gamma RVs has not sufficiently established a comprehensive mathematical framework.

1.1. Related Works

Several approaches have been explored to obtain precise closed-form expressions for order statistics of interest. These methods include transforming dependent ordered RVs into independent unordered RVs using techniques like the spacing method [16], conditional PDF [17], and successive conditioning [18]. However, it’s important to note that these techniques are primarily suitable for straightforward scenarios. In [19,20,21], the papers discussed statistical outcomes for specific types of partial sums involving all RVs but did not address the partial sums related to the best RVs.

Fortunately, through the approach presented in [22,23], a unified analytical framework has been established to calculate the necessary joint statistics for the partial sums of ordered i.i.d. and i.n.d. RVs. This extension builds upon earlier work found in [19,20,21]. As a result, we are now able to systematically obtain joint statistics for various scenarios involving ordered statistics by utilizing the MGF and PDF. These equations play a crucial role in the analysis of performance within communication theory.

The authors in [24] have introduced new statistical results that expand the scope of performance analysis in wireless communication beyond the Rayleigh distribution. These results are particularly relevant for analyzing wireless communication over i.i.d. gamma fading channels. It’s worth noting that the gamma distribution is closely related to the Nakagami-m distribution, both of which can be seen as extensions of the chi-squared distribution [25]. The Nakagami-m distribution represents a chi-squared distribution with $2m$ degrees of freedom [26], and it is a versatile fading model used to characterize multipath fading in various environments, including urban [27] and indoor [28] settings, with the fading parameter m determining the severity of fading. Given the significance of the Nakagami-m (or gamma) distribution in performance analysis, extensive research is ongoing in this field [29,30,31,32,33,34,35].

In the context of Nakagami-m fading analysis, prior studies such as those referenced in [36,37,38,39] primarily yield basic one-dimensional or straightforward two-dimensional joint statistical findings. As a consequence, these findings often fall short in facilitating advanced performance assessments.

1.2. Contributions and Novelty

Performance analysis over i.i.d. gamma fading channels often necessitates order statistics. However, due to the intricacies of this analysis, most studies resort to simulations or experiments rather than precise theoretical outcomes. While joint statistical findings are crucial in communication theory, previous results for i.i.d. gamma RVs do not directly apply to our i.n.d. RVs. These results merely offer a framework, lacking the specific statistical outcomes associated with the relevant statistical distribution. Consequently, the challenges of formulating essential equations for particular fading scenarios persist.

In this research, we address unresolved issues through statistical analysis using a previously proposed mathematical framework applied to i.n.d. gamma RVs. Instead of the original complex multiple-integral form, we provide results in the form of either general closed-form expressions or, at the very least, a single integral form. When dealing with a large number of RVs, even a single integral form is computationally advantageous compared to multiple integrations. We employ an analysis framework introduced in previous works in [22,23] and build on prior results from [24,29] to yield novel statistical outcomes for the analysis of wireless communication performance over i.n.d. gamma fading channels.

The innovation in our findings lies in the derivation of common key functions that can be readily applied to assess the statistical performance of communication systems or algorithms relying on partial sums of ordered RVs over i.n.d. gamma fading channels. Utilizing these core equations, we illustrate how to obtain the desired statistical results. Our results do not introduce a new performance enhancement scheme but are validated by demonstrating their consistency with simulation results. Thus, our outcomes offer a means for precise performance analysis without the necessity for simulations or approximations.

2. System and Channel Model: Statistics of Selected User

Following the system model used in [13], we assume that $M_1$ users among M potential users are scheduled for simultaneous transmission per scheduling period. Instead of the i.i.d. fading channel assumption in [13], we consider more practical and realistic environments such that the potential M users experience non-homogeneous fading channels, i.e., the received SNRs for the j-th user, denoted by ${\left\{{\gamma }_j\right\}}_{j=1}^M$, are i.n.d. RVs. Furthermore, we apply a versatile Nakagami (or gamma) distribution to the RVs, $\left\{{\gamma }_j\right\}$.

We adopt the parallel multiuser scheduling scheme used in [40]. The base station schedules $M_1$ users that have the best channel quality for transmission. If we let $u_m(m=1,2,\cdots ,M_1)$ be the SNR of the m-th desired scheduled user among $M_1$ scheduled users such that $u_1\ge u_2\ge \cdots \ge u_{M_1}$, then the users with SNR $u_m$ are scheduled in each scheduling period. We also assume that there is some interference among the selected best $M_1$ users after the scheduling process. Assuming that inter-user interferences occur among $M_1$ scheduled users, we can write the SINR of the m-th user, $u_{{\mathrm{SINR}}_m}$, as

$$u_{{\mathrm{SINR}}_m}=\frac{X}{1+\alpha Y},$$

(1)

where $X=u_m$, $Y={\displaystyle \sum _{\genfrac{}{}{0pt}{}{n=1}{n\ne m}}^{M_1}}{u}_n$ which is the sum of the SNRs of the interfering $M_1-1$ scheduled users, and $\alpha (0\le \alpha \le 1)$ is a multiuser interference cancellation coefficient. It should be noted that, when $\alpha =0$, the interference is completely canceled, and the SINR in (1) reverts to the SNR, i.e., $u_{{\mathrm{SINR}}_m}=u_m$. However, when $\alpha =1$, the interference is fully present (a precise definition of the values of $\alpha$ falls outside the study’s scope). The cumulative distribution function (CDF) of $u_{{\mathrm{SINR}}_m}$, $F_{u_{{\mathrm{SINR}}_m}}\left(·\right)$, can be calculated in terms of the two-dimensional joint PDF, $f_{X,Y}\left(·,·\right)$, as

$$F_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)={\int }_0^{\infty }{\int }_0^{\gamma \left(1+\alpha y\right)}{f}_{X,Y}\left(x,y\right)dxdy.$$

(2)

On taking the derivative of (2) with respect to $\gamma$, we can express the PDF of $u_{{\mathrm{SINR}}_m}$, $f_{u_{{\mathrm{SINR}}_m}}\left(·\right)$, as

$$f_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)=\frac{\partial }{\partial \gamma }{F}_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)={\int }_0^{\infty }\left(1+\alpha y\right)·f_{X,Y}\left(\gamma \left(1+\alpha y\right),y\right)dy.$$

(3)

To obtain the target statistics of $u_{{\mathrm{SINR}}_m}$ such as $F_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$ and $f_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$ in (2) and (3), respectively, it is essential to derive the joint PDF, $f_{X,Y}\left(·,·\right)$.

3. Methodology: Statistical Analysis of Joint PDF ${\mathit{f}}_{\mathit{X},\mathit{Y}}\left(·,·\right)$

To consider the most general case, we assume that $1<m<M_1-1$. To follow the approach in [22], we let $Z_1={\sum }_{n=1}^{m-1}{u}_n$, $Z_2=u_m$, $Z_3={\sum }_{n=m+1}^{M_1-1}{u}_n$, and $Z_4=u_{M_1}$. We can then express the two-dimensional joint PDF of $W=[X,Y]$, $f_W\left(x,y\right)$, from the four-dimensional joint PDF of $Z=[Z_1,Z_2,Z_3,Z_4]$, $f_Z(z_1,z_2,z_3,z_4)$, as shown in [22] (Equations (35) and (36))

$$\begin{array}{cc}\hfill f_W(x,y)& ={\int }_0^x{\int }_{(m-1)x}^{y-(M_1-m)z_4}{f}_Z\left(z_1,x,y-z_1-z_4,z_4\right)d{z}_{1}d{z}_4\hfill \\ \hfill & \stackrel{\mathrm{or}}{=}{\int }_0^x{\int }_{(M_1-m-1)z_4}^{(M_1-m-1)x}{f}_Z\left(y-z_3-z_4,x,z_3,z_4\right)d{z}_{3}d{z}_4.\hfill \end{array}$$

(4)

Typically, the traditional method that relies on MGF for calculating the sum of RVs can also be extended to obtain statistical outcomes for ordered RVs. Nevertheless, when dealing with a two-dimensional statistical result for variables X and Y, employing the conventional MGF-based approach introduces significant computational complexity. This complexity arises from the need to work with intricate statistical results, particularly when the MGF expression involves multiple integrals, often represented as M-fold integrals, as illustrated in the following equation:

$$\begin{array}{ccc}\hfill & & {\mathrm{MGF}}_Z\left({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right)=E\left\{exp\left({\lambda }_1{Z}_1+{\lambda }_2{Z}_2+{\lambda }_3{Z}_3+{\lambda }_4{Z}_4\right)\right\}\hfill \\ \hfill & & =\sum _{\begin{array}{c}{i}_1,i_2,\cdots ,i_M\\ i_1\ne i_2\ne \cdots \ne i_M\end{array}}^{1,2,\cdots ,M}\underset{0}{\overset{\infty }{\int }}d{u}_1{f}_{i_1}\left(u_1\right)exp\left({\lambda }_1{u}_1\right)\cdots \underset{0}{\overset{u_{m-2}}{\int }}d{u}_{m-1}{f}_{i_{m-1}}\left(u_{m-1}\right)exp\left({\lambda }_1{u}_{m-1}\right)\hfill \\ \hfill & & \times \underset{0}{\overset{u_{m-1}}{\int }}d{u}_m{f}_{i_m}\left(u_m\right)exp\left({\lambda }_2{u}_m\right)\hfill \\ \hfill & & \times \underset{0}{\overset{u_m}{\int }}d{u}_{m+1}{f}_{i_{m+1}}\left(u_{m+1}\right)exp\left({\lambda }_3{u}_{m+1}\right)\cdots \underset{0}{\overset{u_{M_1-2}}{\int }}d{u}_{M_1-1}{f}_{i_{M_1-1}}\left(u_{M_1-1}\right)exp\left({\lambda }_3{u}_{M_1-1}\right)\hfill \\ \hfill & & \times \underset{0}{\overset{u_{M_1-1}}{\int }}d{u}_{M_1}{f}_{i_{M_1}}\left(u_{M_1}\right)exp\left({\lambda }_4{u}_{M_1}\right)\prod _{j=M_1+1}^M{F}_{i_j}\left(u_{M_1}\right).\hfill \end{array}$$

(5)

Fortunately, using the integral solution presented in [41] (Appendix V), we can simplify the above M-fold integral form as

$$\begin{array}{cc}\hfill & {\mathrm{MGF}}_Z\left({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right)\hfill \\ & =\sum _{\genfrac{}{}{0pt}{}{i_{M_1},\cdots ,i_M}{i_{M_1}\ne ,\cdots ,\ne i_M}}^{1,2,\cdots ,M}{\int }_0^{\infty }d{u}_{M_1}{f}_{i_{M_1}}\left(u_{M_1}\right)exp\left({\lambda }_4{u}_{M_1}\right)\prod _{\genfrac{}{}{0pt}{}{j=M_1+1}{\{i_{M_1+1},\cdots ,i_M\}}}^M{F}_{i_j}\left(u_{M_1}\right)\hfill \\ & \times \sum _{\genfrac{}{}{0pt}{}{i_{m=1}}{i_m\ne i_{M_1},\cdots ,i_M}}^M{\int }_{u_{M_1}}^{\infty }d{u}_m{f}_{i_m}\left(u_m\right)exp\left({\lambda }_2{u}_m\right)\hfill \\ & \times \sum _{\{i_{m+1},\cdots ,i_{M_1-1}\}\in {\mathrm{P}}_{M_1-m-1}(I_M-\left\{i_m\right\}-\{i_{M_1},\cdots ,i_M\})}\prod _{\genfrac{}{}{0pt}{}{k=m+1}{\{i_{m+1},\cdots ,i_{M_1-1}\}}}^{M_1-1}{\mu }_{i_k}\left(u_{M_1},u_m,{\lambda }_3\right)\hfill \\ & \times \sum _{\{i_1,\cdots ,i_{m-1}\}\in {\mathrm{P}}_{m-1}(I_M-\left\{i_m\right\}-\{i_{M_1},\cdots ,i_M\}-\{i_{m+1},\cdots ,i_{M_1-1}\})}\prod _{\genfrac{}{}{0pt}{}{l=1}{\{i_1,\cdots ,i_{m-1}\}}}^{m-1}{e}_{i_l}\left(u_m,{\lambda }_1\right),\hfill \end{array}$$

(6)

where $e_{i_l}(·,·)$ is a mixture of an exceedance distribution function (EDF) and an MGF [23] (Equation (4)), and ${\mu }_{i_k}(·,·,·)$ is an interval MGF [23] (Equation (5)) defined as

$$e_{i_l}({\gamma }_1,\lambda )={\int }_{{\gamma }_1}^{\infty }{f}_{i_l}\left(x\right)exp\left(\lambda x\right)dx$$

(7)

and

$$\begin{array}{cc}\hfill {\mu }_{i_l}({\gamma }_1,{\gamma }_2,\lambda )& ={\int }_{{\gamma }_1}^{{\gamma }_2}{f}_{i_l}\left(x\right)exp\left(\lambda x\right)dx\hfill \\ & \stackrel{\mathrm{or}}{=}{c}_{i_l}({\gamma }_2,\lambda )-c_{i_l}({\gamma }_1,\lambda )\hfill \\ & \stackrel{\mathrm{or}}{=}{e}_{i_l}({\gamma }_1,\lambda )-e_{i_l}({\gamma }_2,\lambda ),\hfill \end{array}$$

(8)

respectively, where $c_{i_l}(·,·)$ is a mixture of a CDF and an MGF [23] (Equation (3)) defined as

$$c_{i_l}({\gamma }_1,\lambda )={\int }_0^{{\gamma }_1}{f}_{i_l}\left(x\right)exp\left(\lambda x\right)dx.$$

(9)

Note that we can also call (7) as an upper incomplete MGF (IMGF) [42] (Equations (3) and (9)) as a lower IMGF [42] (Equation (1)). In (6), to simplify the complicated summation notation, we use the power set definition ${\mathrm{P}}_m\left(I_M\right)$, which denotes the subset of the index set $I_M$ ($I_M=\{i_1,i_2,\cdots ,i_M\}$), with $m(m\le M)$ elements [23]. The remaining indices are grouped into the set $I_M-{\mathrm{P}}_m\left(I_M\right)$. Based on this definition, for example, ${\mathrm{P}}_{M_1-m-1}(I_M-\{i_1,i_2,\cdots ,i_{m-1}\})$ denotes all possible subsets of the index set $I_M$ excluding the subset $\{i_1,i_2,\cdots ,i_{m-1}\}$ with $M_1-m-1$ elements. On applying inverse Laplace transforms to the MGF expressions given above, we arrive at the following joint PDF:

$$\begin{array}{cc}\hfill & f_Z(z_1,z_2,z_3,z_4)\hfill \\ & ={\mathcal{L}}_{S_1,S_2,S_3,S_4}^{-1}\left\{{\mathrm{MGF}}_Z(-S_1,-S_2,-S_3,-S_4)\right\}\hfill \\ & =\sum _{\genfrac{}{}{0pt}{}{i_{M_1},\cdots ,i_M}{i_{M_1}\ne ,\cdots ,\ne i_M}}^{1,2,\cdots ,M}{\int }_0^{\infty }d{u}_{M_1}{f}_{i_{M_1}}\left(u_{M_1}\right){\mathcal{L}}_{S_4}^{-1}\{exp\left(-S_4{u}_{M_1}\right)\}\prod _{\genfrac{}{}{0pt}{}{j=M_1+1}{\{i_{M_1+1},\cdots ,i_M\}}}^M{F}_{i_j}\left(u_{M_1}\right)\hfill \\ & \times \sum _{\genfrac{}{}{0pt}{}{i_{m=1}}{i_m\ne i_{M_1},\cdots ,i_M}}^M{\int }_{u_{M_1}}^{\infty }d{u}_m{f}_{i_m}\left(u_m\right){\mathcal{L}}_{S_2}^{-1}\{exp\left(-S_2{u}_m\right)\}\hfill \\ & \times \sum _{\{i_{m+1},\cdots ,i_{M_1-1}\}\in f_{M_1-m-1}(I_M-\left\{i_m\right\}-\{i_{M_1},\cdots ,i_M\})}{\mathcal{L}}_{S_3}^{-1}\left\{\prod _{\genfrac{}{}{0pt}{}{l=m+1}{\{i_{m+1},\cdots ,i_{M_1-1}\}}}^{M_1-1}{\mu }_{i_l}\left(u_{M_1},u_m,-S_3\right)\right\}\hfill \\ & \times \sum _{\{i_1,\cdots ,i_{m-1}\}\in f_{m-1}(I_M-\left\{i_m\right\}-\{i_{M_1},\cdots ,i_M\}-\{i_{m+1},\cdots ,i_{M_1-1}\})}{\mathcal{L}}_{S_1}^{-1}\left\{\prod _{\genfrac{}{}{0pt}{}{l=1}{\{i_1,\cdots ,i_{m-1}\}}}^{m-1}{e}_{i_l}\left(u_m,-S_1\right)\right\}\hfill \end{array}$$

(10)

for $z_4<z_2$, $z_1>(m-1)z_2$ and $(M_1-m-1)z_4<z_3<(M_1-m-1)z_2$.

On substituting (10) into (4), we obtain the desired joint PDF $f_W(x,y)$. It should be noted that (4) involves only the finite integrations of the joint PDFs. Hence, while obtaining a universal closed-form expression may not be feasible, it is possible to readily and numerically assess the desired joint PDF through the use of integral tables [43,44] or standard mathematical packages such as Mathematica or MATLAB. Note that generic joint statistics typically involves the products of a mixture of an EDF and an MGF, $e_{i_l}(·,·)$, and an interval MGF, ${\mu }_{i_l}(·,·,·)$. Herein, we focus on the derivations of the closed-form expression of these products for the given fading conditions.

4. Closed-Form Results of Key Functions with i.n.d. Chi-Squared Distribution

It should be noted that, although we assume that $u_m$ is the SNR of the m-th desired scheduled user among $M_1$ selected users, i.e., $m=1,2,\cdots ,M_1$, $u_m$ is the m-th order statistic of the M SNRs, such that $u_1\ge u_2\ge \cdots \ge u_M$, on arranging M nonnegative i.n.d. RVs, ${\left\{{\gamma }_{i_l}\right\}}_{i_l=1}^M$, the PDF of which is given by

$$f_{{\gamma }_{i_l}}\left(x\right)=\frac{x^{N_{i_l}-1}exp\left(-\frac{x}{a_{i_l}}\right)}{\Gamma \left(N_{i_l}\right){a_{i_l}}^{N_{i_l}}},$$

(11)

which is a chi-squared distribution with $2N_{i_l}$ degrees of freedom and a scale factor $a_{i_l}$, wherein $\Gamma (·)$ is the complete gamma function [44] (Sec. (8.310)). As we are considering the i.n.d. case, $N_{i_l}$ and $a_{i_l}$ are dependent on $i_l$. For the sake of mathematical convenience, in the upcoming discussion, we presume that the degrees of freedom are integers.

Even if the MGF-based systematic framework proposed in [22,23] can be adopted, to obtain the target PDF required for realizing a performance analysis based on the statistical distribution under consideration, the closed-form results for the following key common functions are required, such as the indefinite and definite products of $c_{i_l}(·,·)$, $e_{i_l}(·,·)$, and ${\mu }_{i_l}(·,·,·)$.

4.1. Product of a Mixture of a CDF and an MGF, ${\prod }_{l=n_1}^{n_2}{c}_{i_l}({\gamma }_1,\lambda )$

Substituting (11) into (9), we obtain

$$\begin{array}{c}\hfill c_{i_l}({\gamma }_1,\lambda )=\frac{1}{\Gamma \left(N_{i_l}\right){a_{i_l}}^{N_{i_l}}}{\int }_0^{{\gamma }_1}{x}^{N_{i_l}-1}exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right)x\right)dx,\end{array}$$

(12)

when $\lambda <\frac{1}{a_{i_l}}$. Based on [44] (Sec. (3.381)), the closed-form result of (12) and its product term can be obtained as

$$\begin{array}{cc}\hfill c_{i_l}({\gamma }_1,\lambda )& ={\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}\left[1-exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\right]\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & \prod _{l=n_1}^{n_2}{c}_{i_l}({\gamma }_1,\lambda )\hfill \\ & =\prod _{l=n_1}^{n_2}{\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}\prod _{l=n_1}^{n_2}\left[1-exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\right],\hfill \end{array}$$

(14)

respectively. To apply this result to the MGF-based approach, it is necessary to transform the product terms on the right-hand side of (14) into the desired formulae suitable for integration. In specific, the multiplication form is required to be replaced with the summation form to apply an inverse Laplace transform. Hence, based on the derivation presented in Appendix A, we can reform the first product term on the right-hand side (RHS) of (14) as

$$\begin{array}{c}\hfill \prod _{l=n_1}^{n_2}{\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}=\frac{1}{{\displaystyle \prod _{l=n_1}^{n_2}}{\left(-a_{i_l}\right)}^{N_{i_l}}}\sum _{l=n_1}^{n_2}\sum _{k=0}^{N_{i_l}-1}\frac{C_{l,k}}{{\left(\lambda -\frac{1}{a_{i_l}}\right)}^{N_{i_l}-k}},\end{array}$$

(15)

where

$$\begin{array}{c}\hfill C_{l,k}=\frac{1}{k!}·\frac{d^k}{d{\lambda }^k}{\left[\frac{1}{{\displaystyle \prod _{\genfrac{}{}{0pt}{}{l^{″}=n_1\hfill }{l^{″}\ne l\hfill }}^{n_2}}{\left(\lambda -\frac{1}{a_{i_{l^{\prime \prime }}}}\right)}^{N_{i_{l^{\prime \prime }}}}}\right]}_{\lambda =\frac{1}{a_{i_l}}},\end{array}$$

(16)

and the second product term can be expressed as

$$\begin{array}{cc}\hfill & \prod _{l=n_1}^{n_2}\left[1-exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\right]\hfill \\ & =1+\sum _{l=1}^{n_2-n_1+1}exp\left(l{\gamma }_1\lambda \right)\hfill \\ & \times [{(-1)}^l\sum _{j_1=j_0+n_1}^{n_2-l+1}\cdots \sum _{j_l=j_{l-1}+n_1}^{n_2}exp\left(-\sum _{m=1}^l\frac{{\gamma }_1}{a_{i_{j_m}}}\right)\sum _{k_{i_{j_1}}=0}^{N_{i_{j_1}}-1}\cdots \sum _{k_{i_{j_l}}=0}^{N_{i_{j_l}}-1}\frac{{(-1)}^{{\sum }_{m=1}^l{k}_{i_{j_m}}}{\gamma }_1^{{\sum }_{m=1}^l{k}_{i_{j_m}}}}{{\prod }_{m=1}^l{k}_{i_{j_m}}!}\hfill \\ & \times \sum _{p_{i_{j_1}}=0}^{k_{i_{j_1}}}\cdots \sum _{p_{i_{j_l}}=0}^{k_{i_{j_l}}}\left(\genfrac{}{}{0pt}{}{k_{i_{j_1}}}{p_{i_{j_1}}}\right)\cdots \left(\genfrac{}{}{0pt}{}{k_{i_{j_l}}}{p_{i_{j_l}}}\right){(-1)}^{{\sum }_{m=1}^l\left(k_{i_{j_m}}+p_{i_{j_m}}\right)}\left[\prod _{m=1}^l{\left(\frac{1}{a_{i_{j_m}}}\right)}^{k_{i_{j_m}}-p_{i_{j_m}}}\right]{\lambda }^{{\sum }_{m=1}^l{p}_{i_{j_m}}}].\hfill \end{array}$$

(17)

4.2. Product of a Mixture of an EDF and an MGF, ${\prod }_{l=n_1}^{n_2}{e}_{i_l}({\gamma }_1,\lambda )$

On substituting (11) into (7) and using [44] (Sec. (3.381)), the closed-form result of (7) and its product term can be written as

$$\begin{array}{cc}\hfill e_{i_l}({\gamma }_1,\lambda )& ={\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!},\hfill \end{array}$$

(18)

for $\lambda <\frac{1}{a_{i_l}}$ and

$$\begin{array}{cc}\hfill & \prod _{l=n_1}^{n_2}{e}_{i_l}({\gamma }_1,\lambda )\hfill \\ & =\prod _{l=n_1}^{n_2}{\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}\prod _{l=n_1}^{n_2}\left[exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\right],\hfill \end{array}$$

(19)

respectively. The first product term on the RHS of (19) is given in (15). Using an approach similar to that used in Section 4.1, we can obtain the second product term as

$$\begin{array}{cc}\hfill & \prod _{l=n_1}^{n_2}\left[exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\right]\hfill \\ & =\left[\prod _{l=n_1}^{n_2}exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\right]\left[\prod _{l=n_1}^{n_2}\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\right]\hfill \\ & =exp\left((n_2-n_1+1){\gamma }_1\lambda \right)exp\left(-\sum _{l=n_1}^{n_2}\frac{{\gamma }_1}{a_{i_l}}\right)\hfill \\ & \times \sum _{k_{i_{n_1}}=0}^{N_{i_{n_1}}-1}\cdots \sum _{k_{i_{n_2}}=0}^{N_{i_{n_2}}-1}\frac{{(-1)}^{{\sum }_{l=n_1}^{n_2}{k}_{i_l}}{\prod }_{l=n_1}^{n_2}{\left(\lambda -\frac{1}{a_{i_l}}\right)}^{k_{i_l}}{\gamma }_1^{{\sum }_{l=n_1}^{n_2}{k}_{i_l}}}{{\prod }_{l=n_1}^{n_2}{k}_{i_l}!},\hfill \end{array}$$

(20)

where ${\prod }_{l=n_1}^{n_2}{\left(\lambda -\frac{1}{a_{i_l}}\right)}^{k_{i_l}}$ is defined in (A13).

4.3. Product of an Interval MGF, ${\prod }_{l=n_1}^{n_2}{\mu }_{i_l}({\gamma }_1,{\gamma }_2,\lambda )$

On substituting (11) into (8), we obtain the closed-form result of (8) and its product term as follows:

$$\begin{array}{cc}\hfill {\mu }_{i_l}({\gamma }_1,{\gamma }_2,\lambda )& ={\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}[exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\hfill \\ & -exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_2\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_2^{k_{i_l}}}{k_{i_l}!}]\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill & \prod _{l=n_1}^{n_2}{\mu }_{i_l}({\gamma }_1,{\gamma }_2,\lambda )\hfill \\ \hfill & =\prod _{l=n_1}^{n_2}{\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}\prod _{l=n_1}^{n_2}[exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_1\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_1^{k_{i_l}}}{k_{i_l}!}\hfill \\ & -exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_2\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_2^{k_{i_l}}}{k_{i_l}!}]\hfill \\ & =\prod _{l=n_1}^{n_2}{\left(1-a_{i_l}\lambda \right)}^{-N_{i_l}}\prod _{l=n_1}^{n_2}\sum _{j_l=1}^2{(-1)}^{j_l+1}exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_{j_l}\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_{j_l}^{k_{i_l}}}{k_{i_l}!},\hfill \end{array}$$

(22)

respectively. The first product term on the RHS of (22) is given by (15). On applying the Cauchy product property and certain manipulations, we can rewrite the second product term as

$$\begin{array}{cc}\hfill & \prod _{l=n_1}^{n_2}\sum _{j_l=1}^2{(-1)}^{j_l+1}exp\left(-\left(\frac{1}{a_{i_l}}-\lambda \right){\gamma }_{j_l}\right)\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_{j_l}^{k_{i_l}}}{k_{i_l}!}\hfill \\ & =\sum _{j_{n_1}=1}^2\cdots \sum _{j_{n_2}=1}^2{(-1)}^{{\sum }_{l=n_1}^{n_2}{j}_l+n_2-n_1+1}exp\left((n_2-n_1+1){\gamma }_{j_l}\lambda \right)exp\left(-\sum _{l=n_1}^{n_2}\frac{{\gamma }_{j_l}}{a_{i_l}}\right)\hfill \\ & \times \prod _{l=n_1}^{n_2}\sum _{k_{i_l}=0}^{N_{i_l}-1}\frac{{\left(\frac{1}{a_{i_l}}-\lambda \right)}^{k_{i_l}}{\gamma }_{j_l}^{k_{i_l}}}{k_{i_l}!}.\hfill \end{array}$$

(23)

Applying the results presented in Section 4.1 with Appendix A, we can also obtain the desired form of (23). Following a sequence of operations, we reach the desired form (10) for i.n.d. gamma RVs, as described in Appendix B.

5. Numerical Results and Discussion

In this section, we apply the closed-form results from the previous section to the statistics of $u_{{\mathrm{SINR}}_m}$ given in (1). It should be noted that the PDF of ${\gamma }_{i_l}$ in (11) follows the central chi-squared distribution with $2N_{i_l}$ degrees of freedom. Therefore, the average SNR of ${\gamma }_{i_l}$, ${\overline{\gamma }}_{i_l}$, can be obtained as $N_{i_l}{a}_{i_l}$. To observe the effect of nonidentically distributed RVs on the statistics, instead of the uniform power delay profile (PDP), we consider an exponentially decaying PDP, for which ${\overline{\gamma }}_{i_l}={\overline{\gamma }}_1{e}^{\delta (i_l-1)}$, where ${\overline{\gamma }}_{i_l}(1\le i_l\le M)$ is the average SNR of the $i_l$-th RV of the total available M RVs, and $\delta$ is the average fading power decaying factor. Note that $\delta =0$ indicates identically distributed RVs, which means the i.i.d. fading channels.

Figure 1 presents the statistics of $u_{{\mathrm{SINR}}_m}$, $F_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$ and $f_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$, for various values of m when $M=6$, $N_{i_l}=1$, and ${\overline{\gamma }}_1=1$. The analytical findings presented in this research were validated through Monte Carlo simulations, demonstrating a precise alignment when $m=2$, for instance. In simulations, using MATLAB, we formulate and develop a channel model by generating $50\times {10}^6$ i.n.d. gamma RVs per each path and follow the mode of operation in consideration. It should be noted that, because $N_{i_l}$ is equal to 1, Figure 1 is actually dealing with the exponential (Rayleigh) distribution, which is one of the special cases of the gamma distribution. Figure 1a,b present a special case of $\alpha =0$ corresponding to the case of no interference, i.e., $u_{{\mathrm{SINR}}_m}=u_m$, and Figure 1a,c show that all the RVs are i.i.d., i.e., $\delta =0$. It can be further confirmed that, when $m=1$, the results in Figure 1a,b are consistent with the well-known selection combining (SC) scheme in the i.i.d. and i.n.d. Rayleigh fading environments, respectively [45,46]. Moreover, for arbitrary values of m in Figure 1a,b, our results show the statistics of the generalized selection combining scheme [16,36]. Finally, Figure 1d presents a general case for the given conditions.

Figure 2 presents the effect of $N_{i_l}$ (number of multipath components of the $i_l$-th channel) on the statistics of $u_{{\mathrm{SINR}}_m}$, $F_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$ and $f_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$, when $M=6$, $m=1$, and ${\overline{\gamma }}_1=1$. Likewise, all the analytical findings obtained in this research were subjected to validation through Monte Carlo simulations, e.g., refer to the exact match when $N_{i_l}=2$. It should be noted that, because m is equal to one, Figure 2 shows the results of the SC scheme. As mentioned in Figure 1, Figure 2a,b present a special case of $\alpha =0$ corresponding to the case of no interference, i.e., $u_{{\mathrm{SINR}}_m}=u_m$, and Figure 2a,c show that all the RVs are i.i.d., i.e., $\delta =0$. Figure 2 also includes a special case of $N_{i_l}=1$, which corresponds to an exponential (Rayleigh) distribution.

In Figure 3, we show the effect of i.n.d. RVs on the statistics of $u_{{\mathrm{SINR}}_m}$, $F_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$ and $f_{u_{{\mathrm{SINR}}_m}}\left(\gamma \right)$, when $M=6$, $N_{i_l}=2$, $m=2$, and ${\overline{\gamma }}_1=1$. All the analytical results derived in this study were also compared and verified using Monte Carlo simulations, for example, by referring to the exact match when $\delta =2$. Specifically, Figure 3a shows the case where $\alpha =0$ corresponding to the case of no interference, that is, $u_{{\mathrm{SINR}}_m}=u_m$. From this Figure, we can observe that the statistics of $u_{{\mathrm{SINR}}_m}$ are more susceptible to non identical RVs when interference exists.

In this study, we addressed the challenge of obtaining common functions or core equations to facilitate the final results through the proposed approach. One notable limitation is that while we consider a versatile fading model like i.n.d. gamma RVs, our results may not encompass a wide range of modern fading channels, thereby limiting the applicability of our methodology. This limitation is a significant aspect of our study, and it’s important to emphasize that even with other existing methods, deriving closed-form results can be challenging if core equations are difficult to derive.

Nevertheless, our approach offers substantially simpler solutions compared to conventional methods involving complex multifold integration. Furthermore, the process employed to derive the resulting statistics and core functions in this study can be applied to various research problems within wireless communication systems, particularly those dealing with partial sums of ordered RVs.

The outcomes of this study extend the scope of analysis from previously confined scenarios to more versatile i.n.d. gamma-fading environments. Consequently, this expansion enables sophisticated analysis of various research topics, including advanced diversity combining [16,17], channel adaptive transmission [18], multiuser scheduling [13], and advanced RAKE receivers [47,48,49,50], all within the framework of the gamma distribution. This enhancement broadens the potential applications of the mathematical framework introduced in this work, allowing for comprehensive analysis of wireless communication systems in diverse environments.

6. Conclusions and Recommendations

In the realm of wireless communication systems employing advanced methods over fading channels, the importance of order statistics in performance analysis has grown significantly. These advanced techniques demand fresh and more intricate order statistical findings. Nevertheless, the current analytical methods prove impractical due to their high complexity and limited analytical value. For example, to determine the SINR statistics for a certain scheduled user in an MUI situation to investigate the total average sum rate capacity and ASE, it is necessary to analyze the joint statistics of the ordered RVs. Nonetheless, prior analytical findings mostly depended on asymptotic analysis, even in scenarios with i.i.d. channel conditions. In our research, we have generated fresh and enlightening statistical outcomes for the partial sums of ordered gamma-distributed RVs by deriving essential common functions required for this analysis. Additionally, to address unexplored aspects, we conducted thorough mathematical transformations and introduced novel representations to attain comprehensive results for the i.n.d. gamma distribution in a concise format. We demonstrated that our results are consistent with the simulation results and thus accurate and useful for obtaining the required target analytical results. In the future, we will analyze the non-independent case to consider more general and versatile fading environments. It will be very meaningful to assess the effect of non-identically distributed RVs with correlation on the performance. The main challenge in generalizing the results of this study to non-independent fading cases is that the joint PDF of non-independent ordered RVs are considerably more complicated than independent ordered RVs.