Hypoexponential Distribution with Different Parameters

The Hypoexponential distribution is the distribution of the sum of n ≥ 2 independent Exponential random variables. This distribution is used in moduling multiple exponential stages in series. This distribution can be used in many domains of application. In this paper we consider the case of n exponential Random Variable having distinct parameters. Using convolution, some properties of Laplace transform and the moment generating function, we analyse this case and give new properties and identities. Moreover, we shall study particular cases when i  are arithmetic and geometric.


Introduction
The Random Variable (RV) plays an important role in modeling many events [1,2].In particular the sum of exponential random has important applications in the modeling in many domains such as communications and computer science [3,4], Markov process [5,6], insurance [7,8] and reliability and performance evaluation [4,5,9,10].Nadarajah [11], presented a review of some results on the sum of random variables.
Many processes in nature can be divided into sequential phases.If the time the process spends in each phase is independent and exponentially distributed, then the overall time is hypoexponentially distributed.The service times for input-output operations in a computer system often possess this distribution.The probability density function (pdf) and cummulative distribution function (cdf) of the hypoexponential with distinct parameters were presented by many authors [5,12,13].Moreover, in the domain of reliability and performance evaluation of systems and software many authors used the geometric and arithmetic parameters such as [10,14,15].
In this paper we study the hypoexponential distribution in the case of n independent exponential R. V. with distinct parameters We use in our work the properties of convolution, Laplace transform and moment generating function in finding the derivative of the pdf of this sum and the moment of this distribution of order k.In addition, we deduce some new equalities related to these parameters.Also we shall study the case when the parameters form an arithmetic and geometric sequence considered by [10,14,15] and find some new results.th k

Definitions and Notations
Let 1 2 , , , n X X X  be independent exponential random variables with different respective parameters . We define the random variable to be the Hypoexponential random variable with pa- The moment of order k of the RV X.  : product of all parameters. .

Applications on pdf and cdf Using Laplace
Th f the hypoexponential with distinct pa-

Transform
e pdf and cdf o rameters were presented by many authors [2,7,[11][12][13].We shall state in thoerem 1 and propostion 1 these results and provide another proof using Laplace transform.Next, we give some new properties of its pdf, where new identities are obtained.
and the Laplace transform of convolution of functions is the product of their Laplace transform, thus where [16], for However, by Heaviside Expansion Th distinct poles gives that On the other hand we have and we con- However, by Initial Value Theorem, we ave in the same manner till the derivative, we obtain the result.that derivative of the pdf of n S are zeros, which verifies the fact that the coefficient of variation of the hypoexponential distribution is less than one unlike the hyperexponential distribution that have the coefficient of variation greater than 1.
However, from Theorem 1, By Proposition 2, we obtain that with we obtain

Applications on pdf and cdf Using Moment Generating Function
In the previous section we saw the use of Laplace properties in the proofs of the theorems and propositions.
Thus we obtain the result.Next, we shall use the Proposition 3 and 4 to ind other identities on and higher orders for   Note that we may write where .
ies and , thus the above summation (3) shall be 1.
Proof.Let and .We have and using multinomial expansion formula, we obtain possibilit 0 0 Knowing that expectation is linear and i X , 1, 2, , i n   are independent with Since fro The following corollary is direct consequence of Proposition 5 and Equation ( 4), tak and 2 respectively.
Let Then
ultiplying in the num ator and denom nator by . Hence, we may write

Let and Then
Many authors used the identity  and proved it in many long and complicated m s Here we shall submit a more simple prove.In addition, w ties using the above results.
e shall find more related identi Next we shall find a more general equality using our previous results.
. Let Then Proposition 9 we obtain the first case and , k the case when here 1, and the Equation ( 5) gives that

The Main Results
Also Corollary 1 and Proposition 5 and 6 can be summarized in the following theorem.
Λ We recall Propostion 9 in the following corollary of Th eorem 3.
The study of reliability and p tion of systems and softwares use in dent exponential R.V. with distinct del of Jelinski and Moranda [14], con rameters changes in an arithmetic , when the parameters are arithmetic and geometric, and we present their pdf.

Case of Arithmetic Parameters
We first consider the case when , 1,2, i i 1 We may also note that the equalities obtaine for represent here a special case and worth mentioning s as

Conclusion
The pdf and cdf and some related properties of he hypoexponential distribution with distinct parameters w ng Also with the help of some known computational theorems as Heaviside expansion theorem and multinomial expansion formula the k th order der ative of t ere established.The proofs have been done by usi Laplace transform and moment generating function technique.iv n S f eslities.
i and the moment of this distribution of order k were tablished, in addition for some new related equa f for models when the parameters Eventually, the pd  are arithmetic and geometric were presented.However the other two cases for hypoexponential distribution when the parameters are equal or not all equal can be studied and observed for future studies.It may be checked if they have the same properties as in this paper.