A novel approach in Multi-hop networks technology with the ratio distribution of two Hyper-Erlang random variables

The distribution of ratio of two random variables has been studied by several authors especially when the two random variables are independent and come from the same family. In this paper, the exact distribution of the ratio of two independent Hyper-Erlang distribution is derived. However, closed expressions of the probability density, cumulative distribution function, reliability function, hazard function, moment generating function and the rth moment are found for this ratio distribution and proved to be a linear combination of the Generalized-F distribution. Finally, we will apply our results to real life application in analyzing the performance of wireless communication systems.


Introduction
The distributions of ratio of random variables are of interest in many areas of the sciences, engineering, physics, number theory, order statistics, economics, biology, genetics, medicine, hydrology, psychology, classification, and ranking and selection, see [1,2,3,4].Examples include safety factor in engineering, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, inventory ratios in economics and Mendelian inheritance ratios in genetics, see [1,2].Also ratio distribution involving two Gaussian random variables are used in computing error and outage probabilities, see [5].It has many applications especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures and the aging of concrete pressure vessels, see [6,7].An important example of ratios of random variables is the stress strength model in the context of reliability.It describes the life of a component which has a random strength Y and is subjected to random stress X.The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever Y > X.Thus, P r(X < Y ) is a measure of component reliability see [6,7].
The ratio distribution X/Y have been studied by several authors especially when X and Y are independent random variables and come from the same family.For a historical review, see the papers by Marsaglia [8] and Korhonen and Narula [9] for the normal family, Press [10] for Student's t family, Basu and Lochner [11] for the Weibull family, Hawkins and Han [12] for the non-central chi-squared family, Provost [13] for the gamma family,

The Hyper-Erlang distribution
The Hyper-Erlang distribution is the mixture of m mutually independent Erlang distributions or parallel m-phase Erlang distribution weighted with the probabilities p i and each i th Erlang stage E i has the shape parameter k i and rate parameter α i , for + are rate parameters, and where , see [18,19,20].
The particular case of this distribution is when ).This case is the Hyper-Exponential distribution where the m-phases are Exponential distribution, then This particular case can be written as

Generalized-F distribution
Let X be a random variable that has the Generalized-F distribution or called the generalized beta-prime distribution with three positive parameters v 1 , v 2 and γ.The PDF of X is where is the usual Beta function, see [7].Taking γ = β α , where α > 0 and β > 0, we can write the PDF as The Generalized-F distribution is related to the F distribution with two parameters, where γX is a F distribution with 2α and 2β degrees of freedom, [7] and [21].Thus, the distribution of Generalized-F distribution can be obtained from this relation, see [21,22,23].As a result we have the cumulative distribution function of X is where where else diverges, thus the expectation is The particular case of the Generalized-F distribution is when v 1 = v 2 = 1, which is the log-logistic distribution with parameter 1/γ and 1, we write when In this case we have the PDF of X is

Ratio of Hyper-Erlang Distribution
In this section, we examine the ratio of the Hyper-Erlang distribution.A closed expression of the PDF is derived and written as a linear combination of the Generalized-F distribution.As a consequence, the expressions CDF, reliability function, hazard function, moment generating function and moments of order r for this distribution are determined.
We suppose that X and Y are two independent Hyper-Erlang distribution.Then we take and and respectively, where

Theorem 1
Let X and Y be two independent Hyper-Erlang distributed according to (11).Then the PDF of X/Y is given by where f Wi,j (t) is the PDF of the Generalized-F distribution W i,j ∼ GF (k i , l j , βj αi ).

Proof
The PDF of ratio distribution X/Y is given by However the PDF of X and Y , are given in Eq (11), Thus we obtain that ) dy.The product in this integral is a multiplication of the PDF of two Erlang distribution corresponding to their parameters which is given as −y(t+ Then, for t ≥ 0, −y(t+ ) dy To evaluate the above integral, we use the integral ∫ ∞ 0 x a e −xb dx = a! b a+1 for b > 0 and any positive integer a.Thus ∫ ∞ 0 y ki+lj −1 e −y(t+ ) dy = (ki+lj −1)! (t+ ) k i +l j .We obtain that However, (ki−1)!(lj −1)! (ki+lj −1)! is the particular case of the beta function B(k i , l j ) in Eq (3), where k i and l j are integers.Therefore, we obtain that Referring to Eq (4), the expression in the above summation the PDF of the Generalized-F distribution, GF (k i , l j , βj αi ).Writing W i,j ∼ GF (k i , l j , βj αi ), we obtain the result.
Theorem 1 showed that the PDF of the ratio distribution of two independent Hyper-Erlang distribution is a linear combination of the Generalized-F distribution.From this linear form, we can find other related functions for X/Y , such as the CDF, reliability function, hazard function, moment generating function and moment of order r.As a consequence of Theorem 1, we prove in the next corollaries that these related functions are also a linear combination of the those of the Generalized-F distribution.

Corollary 1
Let X and Y be two independent Hyper-Erlang distributed according to (11).Then the CDF of X/Y is where I x (a, b) is the regularized incomplete beta function.
In the next corollaries, we give the expression of the reliability function and hazard function of X/Y.

Corollary 2
Let X and Y be two independent Hyper-Erlang distributed according to (11).Then the reliability function of X/Y is and the hazard function of X/Y is where I x (a, b) is the regularized incomplete beta function and

Proof
The reliability function of X/Y is related to the CDF as R X/Y (x) = 1 − F X/Y (x).We use the expression of F X/Y (x) in Corollary 1 as However, from [24], we have Then we have Thus, the reliability is Substituting the expressions of f X/Y (t) in Eq (12) and R X/Y (t),we obtain the result.

Corollary 3
Let X and Y be two independent Hyper-Erlang distributed according to (11).Then the moment generating function of X/Y is where ) and U (a, b, z) is the confluent hypergeometric function of the second kind.

Proposition 1
Let X and Y be two independent Hyper-Erlang distributed according to (11).Then the moment of Z = X/Y of order r is for r < min {l j }, 1 ≤ j ≤ n and W i,j ∼ GF (k i , l j , βj αi ).Proof From Eq (13) the moment generating function of Z is and the r th derivative is given as d r ΦZ (t) . Now, since the moment of order r of Z is the , where E[W r i,j ] is the moment of order r of the Generalized-F distribution W i,j ∼ GF (k i , l j , βj αi ).On the other hand, Eq (7) gives that Thus, we obtain the required results.
From proposition 1, we conclude some particular moments.The expectation, for r = 1,

Ratio of Hyper-Exponential Distribution
Next, we consider the particular case of the Hyper-Erlang distribution, the Hyper-Exponential distribution.Let X and Y be two independent Hyper-Exponential random variables each distribution of different stages.Thus, X ∼ ∑ n j=1 q j = 1, and We note again that in this case . Then referring to our results in section 3 and by taking the particular cases when − → k = (1, 1, ..., 1) and − → l = (1, 1, ..., 1), we obtain the results of the ratio of two independent Hyper-Exponential distribution.We state them in the following corollaries.

Corollary 4 Let
for t > 0 and f X/Y (t) = 0, for t ≤ 0. The CDF is for x > 0 and F X/Y (x) = 0, for x ≤ 0. The reliability function is for t > 0 and R X/Y (t) = 0, for t ≤ 0. The hazard function is for t > 0 and h X/Y (t) = 0, for t ≤ 0.

Proof
The proof is directly obtained by taking the particular case of the ratio of two independent Hyper-Erlang distribution given in Theorem 1 and Corollaries 1 and 2. In this particular case we have from Equation (8), 1) and knowing that PDF and CDF of W i,j are given in Eq (9) and Eq (10) as respectively.Thus, the results are obtained.

Application
In this section we will apply our result to real life application in analyzing the performance of wireless communication systems.Ratios of random variables have been widely utilized in reporting results in the biological, physical and social sciences.Examples include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology and inventory ratios in economics.The distributions of ratios of random variables are also of interest in performance analysis of wireless communication systems.Radio wave propagation through wireless channels is a complicated phenomenon characterized by fading which is the result of multipath propagation due to reflection, refraction and scattering of radio waves by buildings and other structures.When a received signal experiences fading during transmission, its envelope fluctuates over time.Desired signal in wireless communication systems is also subjected to cochannel interference (CCI) due to reuse of radio frequencies which is essential in increasing system capacity.Multi-hop relaying communication has a number of advantages over traditional direct-link transmission in the areas of connectivity, deployment, power saving and channel capacity.Relaying techniques enable network connectivity where traditional architectures are impractical due to location constraints.Multi-hop networks technology is a promising solution for future cellular, wireless local area networks (WLANs) and hybrid networks.It is reason why dual and multi-hop wireless communication systems operating in fading channels have been an important field of research in the past few years.Motivated by the preceding, we will find the closed form, using our methodology, of the probability distribution of the ratio X/Y in multi-hop relaying communication (figure 1, 2) system.Where X represents signal envelope which suffers from fading, while Y represents envelope of CCI simultaneously affected by both fading and shadowing.Therefore, random variable X/Y presents signal-to-interference ratio (SIR), hence its probability distribution can model different transmission environments.X and Y are two Hyper-Erlang random  for t > 0 and f X/Y (t) = 0 for t ≤ 0. Figure 3 shows the curve of f X/Y (t) and Figures 4, 5 and 6 show the CDF, reliability and hazard function of X/Y respectively, obtained in Corollaries 1 and 2.

Conclusion
The closed expressions of the PDF, CDF, reliability function, hazard function, moment generating function and the r th moment of the ratio of two independent Hyper-Erlang distribution are given.The expressions are proved to be a linear combination of the Generalized-F distribution.Next, we observe the particular case, the ratio of two

Figure 3 Figure 5 .
Figure 3. PDF Figure 4. CDF By applying Theorem 1, we obtain that the PDF of X/Y is