ABSTRACT

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 are arithmetic and geometric.

1. 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 for written as . 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.

2. Definitions and Notations

Let be independent exponential random variables with different respective parameters, , written as. We define the random variable

to be the Hypoexponential random variable with parameters, , written as

Some notations used throughout the paper.

:

:

: The pdf of the random variable X.

: The cdf of the random variable X.

: The derivative of the pdf.

: Laplace-Stieltjes Transform.

: Laplace Inverse.

: The moment generating function of X.

: The moment of order k of the RV X.

: product of all parameters.

:

:

:.

3. Applications on pdf and cdf Using Laplace Transform

The pdf and cdf of the hypoexponential with distinct parameters were presented by many authors [2,7,11-13]. 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.

Theorem 1. Let and Then

and

.

Proof. We have

where for. Since are independent then is the convolutions of, written as

and the Laplace transform of convolution of functions is the product of their Laplace transform, thus

(1)

where However, by Heaviside Expansion Theorem [16], for distinct poles gives that

where

.

Therefore,

But. Thus

.

On the other hand we have

But then and we conclude that

.        

Next we shall discuss the derivative of and many equalities are obtained concerning form and some similar forms.

We start by noting from the previous proof that

. Here, we shall state another simple proof using Laplace transform.

Proposition 1. Let. Then

Proof. We have from Equation (1),

where. But from Theorem 1,

and

Hence,. For

Therefore,.   

Lemma 1. Let Then

for

Proof. The proof is done by induction. For we have from Equation (1)

.

However, by Initial Value Theorem, we have

and for we have

Moreover

Continuing in the same manner till the derivative, we obtain the result.                     

In the following propostion we shall prove that the first derivative of the pdf of 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.

Proposition 2. Let Then

Proof. Let, we have from Lemma 1,

for and from Initial Value Theorem, we have

Corollary 1. Let. Then

Proof. We have. Then the derivative of is

.

However, from Theorem 1,

then

and

(2)

By Proposition 2, we obtain that

By replacing with we obtain the result. 

4. 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. In a similar manner, in this section we use the moment genrating function to obtain more new related results. A new form of the moment generating function of and the moment of of order k is given. Moreover, we deduce more new related equalities concerning and higher order derivatives of pdf of.

Proposition 3. Let Then

.

Proof. We have

and from Theorem 1,

then

. 

Proposition 4. Let and. Then

Proof. We have from Proposition 3,

.

Then

and

which gives. But. Thus we obtain the result.                       

Next, we shall use the Proposition 3 and 4 to find other identities on and higher orders for. We start by noting that and by taking in Proposition 3, we again obtain the result in Proposition 1that is.

Proposition 5. Let and. Then

where

.

Note that we may write

, (3)

where

However and are equivalent representing a set of combination with repetition having

possibilities and, thus the above summation (3) shall be 1.

Proof. Let and. We have

and using multinomial expansion formula, we obtain

.

Knowing that expectation is linear and, are independent with

then

(4)

Since from Proposition 4,

.

Therefore,

.         

The following corollary is direct consequence of Proposition 5 and Equation (4), taking and 2 respectively.

Corollary 2. Let. Then 1)

2) and

3) and.

In Proposition 2, we found the first derivative of at 0, However to find higher order derivaties we recall Equation (2), that shows a direct relation between the derivative and. Hence, in the next propostion we shall use Propostion 5, to find an equation for by finding a relation between and

Proposition 6. Let and. Then

Proof. Let and

Then by Theorem 1, the pdf of is

where and.

Next, we shall find in terms of. We have

multiplying in the numerator and denominator by

we obtain where . Hence, we may write

.

But, for Proposition 5 gives that

.

Therefore,

Proposition 7. Let and. Then

Proof. We have from Equation (2),

and from Proposition 6,

for Then,

Many authors used the identity

and proved it in many long and complicated methods. Here we shall submit a more simple prove. In addition, we shall find more related identities using the above results.

Proposition 8. Let Then

Proof. Let. By Corollary 1, taking we have then

.

However,

Therefore,

Next we shall find a more general equality using our previous results.

Proposition 9. Let. Then

Proof. Let. Then,

(5)

Suppose that. We have from Corollary 1,

and Equation (5) gives that

Replace with we obtain the first case and the case when where.

Now, suppose. By Proposition 6,

and the Equation (5) gives that

.

Also, replace by we obtain the last case when.                               

5. The Main Results

We summarize Proposition 2 and 7 in the following theorem.

Theorem 2. Let Then

Also Corollary 1 and Proposition 5 and 6 can be summarized in the following theorem.

Theorem 3. Let and. Then 1)

and 2)

We recall Propostion 9 in the following corollary of Theorem 3.

Corollary 3. Let. Then

6. Case of Arithmetic and Geometric Parameters

The study of reliability and performance evaluation of systems and softwares use in general sum of independent exponential R.V. with distinct parameters. The model of Jelinski and Moranda [14], considered that the parameters changes in an arithmetic sequence. Moreover, Moranda [15], considered the model when changes in an geometric sequence. In this section, we study the hypoexponential in these two cases when the parameters are arithmetic and geometric, and we present their pdf.

6.1. Case of Arithmetic Parameters

We first consider the case when form an arithmetic sequence of common difference.

Lemma 2. For all

Proof. Suppose that form an arithmetic sequence of common difference. Then We have

.

Hence,

However,

.

Then

Lemma 3. For all

.

Proof. We have from Lemma 2,

for all Replace by, we obtain

Thus we obtain the result.                     

Proposition 10. Let Then

where

for all

Proof. We have from Theorem 1

that can be written as

where and by the Lemmas 2 and 3 we obtain the result.                          

6.2. Case of Arithmetic Parameters

Next, we consider the case when form a geometric sequence of common ratio.

Proposition 11. Let Then

.

Proof. We have from Theorem 1,

.

Suppose now the parameter form geometric sequence of common ratio. Then and

.   

We may also note that the equalities obtained for represent here a special case and worth mentioning such as

7. Conclusion

The pdf and cdf and some related properties of the hypoexponential distribution with distinct parameters were established. The proofs have been done by using Laplace transform and moment generating function technique. Also with the help of some known computational theorems as Heaviside expansion theorem and multinomial expansion formula the k^th order derivative of and the moment of this distribution of order k were established, in addition for some new related equalities. Eventually, the pdf for models when the parameters 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.

REFERENCES