Bivariate gamma type distributions for modeling wireless performance metrics

In this paper a bivariate gamma type distribution, its noncentral counterpart, and a linked bivariate Weibullised gamma type distribution, following an elliptical assumption, are proposed and studied. The adaptability of this contribution is illustrated with the outage probability performance metric, where the proposed bivariate gamma type distributions may act as alternatives to existing fading models in wireless communications.


Introduction
Bivariate probability distributions have received significant attention in literature (see [1]), and even more so within the context of wireless systems (see [14], [18], [17], [5], [25], [6], [22], [11], [2]). Bivariate gamma distributions are used to describe the fading of signal channels between either two transmitters or two receivers -specifically in the modeling of dual-antenna wireless systems operating over correlated branches. Bivariate gamma distributions in particular have been of interest due to its pliable-and computable mathematical nature, exhibiting satisfactory fits to measured data subjected to multipath/shadowing fading (see [17], [20]). This paper proposes a new class of (correlated) bivariate gamma distributions emanating from the elliptical arena, thereby creating versatile bivariate distributions. These bivariate gamma type distributions (called as such due to the elliptical origin, justifying the word "type") are studied via their probability density function (pdf), cumulative distribution function (cdf), and product moment. The bivariate Nakagami distribution is also meaningful within the wireless communications framework as it models the envelope, or amplitude, of a received wireless signal. It is obtained as a transformation from a bivariate gamma distributed variable; thus, to this effect, a bivariate Weibullised gamma type distribution is also proposed with a bivariate Nakagami type distribution as special case. Via these contributions we don't only gain valuable insight into the distributional structure of these distributions, but also expand the knowledge base of alternative candidates for modeling within the wireless communications domain; and the results in this paper may be used for analyzing dual-antenna systems.
The mathematical representation of the distributions in this paper, which originates from the elliptical platform, is noteworthy (see [2]). The representation allows for flexible and alternative choices of underlying process distributions for the practitioner; usually chosen according to measurements taken in practice (such as power of wireless signals). This paper investigates the usual underlying normal assumption, as well as an underlying t assumption; due to the well-known statistical relevance of these two candidates. [5] discussed some early thoughts then there exists a scalar weight function W(·) on R + such that: (1) where f N (µ,σ 2 t −1 ) (x|t) is the pdf of a normal distribution with mean µ and variance σ 2 t −1 . Two well known members of the elliptical distribution is the normal distribution with weight: known as the dirac delta function (with property ∫ ∞ 0 f (x) δ (x) dx = f (0)); and the t distribution with weight: with v > 0 degrees of freedom (see [4]) and where Γ (·) denotes the gamma function.
Following the representation in (1), the pdf of R, where X and Y are independent elliptically distributed random variables, with mean µ 1 and µ 2 , and common covariance σ 2 t −1 is given by: where r > 0, s 2 = µ 2 1 + µ 2 2 , and I 0 (·) is the 0 th order Bessel function of the 1 st kind. This distribution is referred to as a Rician type distribution. By choosing W (t) as (2), (4) simplifies to A. BEKKER, J.T. FERREIRA 337 for r > 0, known as the Rician distribution (see [20], pp. 201, and [13]). When µ 1 = µ 2 = 0, (4) simplifies to: and referred to as a Rayleigh type distribution. This is used as a departure point from where we systematically construct bivariate gamma type distributions with their origins following this elliptical assumption.
This paper's contribution can be summarised as follows: The paper is laid out in the following way. To set the platform, Section 2 presents the description of methodology and construction of univariate gamma type distributions from the elliptical distribution. Section 3 proposes a bivariate gamma type distribution with some statistical characteristics as well as a bivariate noncentral gamma type distribution. Section 4 pays attention to the bivariate Weibullised gamma type that follows naturally from the proposed bivariate gamma type, with a bivariate Nakagami type distribution as proposed alternative model for the envelope. Section 5 discusses a possible application in terms of outage probability of a wireless fading channel subject to the developed models, and final thoughts are contained in the Section 6. The Appendix contains simplified results of interest to this paper.

Methodology and univariate construction
Suppose that X follows a Rayleigh type distribution as in (6) results in Y following an exponential type distribution with parameter Ω > 0 and pdf: ). This pdf in (7) forms the basic element of the composition of the proposed bivariate distributions in this paper. Using [8], p. 346, eq. 3.381.4, the Laplace transform of Y is given by: The pdf of this gamma type distribution follows via an inverse Laplace transform as: which is referred to as a Weibullised gamma type distribution with the Nakagami type distribution as a special case when β = 2.
We are interested in the bivariate distribution of U 1 and U 2 , with Laplace transform of (U 1 , U 2 ) given as Note that the Laplace transform of (U 1 , U 2 |t) can be decomposed as: since U 1 and U 2a is independent of U 2b .

BIVARIATE GAMMA TYPE DISTRIBUTIONS
Substituting (17) into (15), and subsequently (15) into (16), and then applying an inverse Laplace transform on (16), the pdf of the bivariate gamma type distribution is obtained as tρ For weights (2) and (3)     In Figure 1 the effect of increasing ρ on the pdfs are observable ((a) and (b)), particularly indicating increased concentration (or correlation) between the variables. Figure 1 (c) indicate the effect on the tails for increasing values of v.
Using (38) and (39), the Pearson correlation coefficient is illustrated for m 2 = 12, Ω 1 = 2.5, Ω 2 = 1.5, ρ = 0.5, and v = 10, 30.  In particular it is observed that the correlation under the t distribution assumption approaches the correlation under the normal assumption for increased values of v. Note the restriction of m 1 ≤ m 2 . Furthermore, it is seen that for varying m 1 that the correlation increases, as well as ρ U1,U2 = 0 when m 1 = 0. This is seemingly evident because the bivariate nature of the bivariate gamma type distribution collapses when m 1 = 0, due to the construction discussed at the start of Section 3.
Within a wireless communications framework, bivariate distributions with a LOS assumption are often employed to model land mobile satellite experiments. In particular, [11] emphasize that more sophisticated probability distributions are recalled to fit the behaviour of fading signals in more intricate scenarios. From a statistical viewpoint and origin, [16] showed that the noncentral chi-square distribution with n degrees of freedom and noncentrality parameter θ can be represented as a weighted sum of univariate chi-square probabilities with weights equal to the probabilities of a Poisson distribution with expected value θ 2 . The genesis of the extension to the noncentral counterpart of the bivariate gamma type distribution with pdf (18) proposed here originated from the papers by [24], [23], and [7].
The proposed methodology, amended from the methodology for obtaining bivariate noncentral chi-square distributions by [7], to obtain bivariate noncentral gamma distributions is given as follows: A bivariate noncentral gamma pdf can be obtained from a conditional bivariate central gamma distribution with pdf f (u 1 , u 2 |k 1 , k 2 ) in the following manner: , i = 1, 2 are Poisson probabilities where θi 2 > 0 denotes the noncentrality parameters, and f (u 1 , u 2 |k 1 , k 2 ) the pdf of a bivariate gamma distribution with conditions on the shape parameters respectively. In this regard, the Laplace transform of (U 1 , U 2 ) in the noncentral case can be written as: where L U1,U2 (s 1 , s 2 |k 1 , k 2 ) denotes the conditional Laplace transform of (U 1 , U 2 |k 1 , k 2 ) with pdf f (u 1 , u 2 |k 1 , k 2 ).
For weights (2) and (3) respectively, the results are given in Appendix equations (44) and (45). The contourplots of these specific cases are illustrated for m 1 = 10, m 2 = 12, Ω 1 = 3.5, Ω 2 = 1.5, β 1 = 2, ρ = 0.5, and arbitrary choices for other parameters.   The Laplace transform of the variables under which a fading channel is operating can be used to evaluate certain attributes of a wireless system, such as the average bit error rate (see [20], [21]). Thus the Laplace transform of (W 1 , W 2 ) with pdf (28) is given as: ..., a p b 1 , ..., b p ) denotes Meijer's G function (see [12]). (2) and (3) of (29) are given in the Appendix as equations (46) and (47) respectively, and particular values are evaluated; the machine runtime is listed as well. Furthermore, the integral representation (definition) of the Laplace transform was also calculated numerically for the normal-and t case ((29) for weights in (2) and (3)), together with its machine runtime. Similar as in [19], the calculated values of (46), (47), and integral definition for both normal-and t cases for select values of s 1 and s 2 are provided. The runtime is computed using Mathematica 11 on an Intel i7 processor. Whilst the runtime is affected by various factors of an individual processor, it gives some indication about the speed with which these expressions may be evaluated. The parameters for which these values are calculated are m 1 = 10, m 2 = 12, Ω 1 = 2.5, Ω 2 = 1.5, ρ = 0.5, v = 5, and β 1 = β 2 = 2. For completeness' sake, The product moment of the bivariate Weibullised gamma type distribution with pdf (28) is

Illustrative application
The maximum of a bivariate variable following a bivariate gamma type distribution such as (18) can be used to model the performance of a MIMO system subject to Rayleigh type fading (6) (see [17], [25]). The outage probability at a certain threshold is obtained as (see [21]): Using (22), the cdf of max(U 1 , U 2 ) is: By choosing W (t) as (2), the cdf is given by:  (3), the cdf is given by: respectively. These analytic expressions for the cdf of the maximum act as the expressions for the outage probability of the signal-to-noise ratio (SNR) for a MIMO system based on fading characteristics illustrated by the distribution with pdf (18). It is illustrated for arbitrary parameters, specifically, m 1 = 10, m 2 = 12, ρ = 0.5, Ω 1 = 2.5, Ω 2 = 1.5, and v = 5.   5 illustrates that (32) exhibits a lower outage probability for small outage thresholds than (33). However, for the scenario where the outage threshold is large, (33) exhibits a lower outage probability. This observation provides significant insight to the theoretical contribution of the candidacy of the underlying t distribution in comparison to the usual underlying normal case.

Conclusions
This paper proposes new bivariate distributions emanating from the elliptical distribution. In particular, a new bivariate gamma type distribution has been derived, along with its noncentral counterpart. Furthermore, a bivariate Weibullised gamma type distribution, stemming from the bivariate gamma type distribution, has also been derived and studied. Special cases, in particular for underlying normal-and underlying t distribution, has been presented in the Appendix for derived results. Possible application has been demonstrated in terms of applying the newly derived results within the wireless communications field. In particular, the outage probability evaluation for wireless systems provides significant insight into the behaviour of the bivariate gamma type distribution, specifically with an underlying t distribution, in such scenarios. Subsequent work would include investigating different members of the elliptical distributions as viable underlying distributions in wireless systems settings.