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 ofLaplacetransform 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.
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
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
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
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
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,
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 kth 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.
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doi:10.1109/LCOMM.2011.011011.102143K. S. Trivedi, “Probability and Statistics with Reliability, Queuing and Computer Science Applications,” 2nd Edi tion, John Wiley & Sons, Hoboken, 2002.W. Kordecki, “Reliability Bounds for Multistage Structure with Independent Components,” Statistics & Probability Letters, Vol. 34, No. 1, 1997, pp. 43-51.
doi:10.1016/S0167-7152(96)00164-2A. M. Mathai, “Storage Capacity of a Dam with Gamma Type Inputs,” Annals of the Institute of Statistical Mathe matics, Vol. 34, No. 1, 1982, pp. 591-597.
doi:10.1007/BF02481056L. D. Minkova, “Insurance Risk Theory,” Lecture Notes, TEMPUS Project SEE Doctoral Studies in Mathematical Sciences, 2010.G. E. Willmot and J. K. Woo, “On the Class of Erlang Mixtures with Risk Theoretic Applications,” North Ame rican Actuarial Journal, Vol. 11, No. 2, 2007, pp. 99-115.
doi:10.1080/10920277.2007.10597450G. Bolch, S. Greiner, H. Meer and K. Trivedi, “Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications,” 2nd Edition, Wiley-Interscience, New York, 2006.O. Gaudoin and J. Ledoux, “Modélisation Aléatoire en Fiabilité des Logiciels,” Hermès Science Publications, Paris, 2007.S. Nadarajah, “A Review of Results on Sums of Random Variables,” Acta Applicandae Mathematicae, Vol. 103, No. 2, 2008, pp. 131-140. doi:10.1007/s10440-008-9224-4M. Akkouchi, “On the Convolution of Exponential Dis tributions,” Chungcheong Mathematical Society, Vol. 21, No. 4, 2008, pp. 501-510.S. V. Amari and R. B. Misra, “Closed-Form Expression for Distribution of the Sum of Independent Exponential Random Variables,” IEEE Transactions on Reliability, Vol. 46, No. 4, 1997, pp. 519-522. doi:10.1109/24.693785Z. Jelinski and P. B. Moranda, “Software Reliability Re search,” Statistical Computer Performance Evaluation, Academic Press, New York, 1972, pp. 465-484.P. B. Moranda, “Event-Altered Rate Models for General Reliability Analysis,” IEEE Transactions on Reliability, Vol. R-28, No. 5, 1979, pp. 376-381.
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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.
be independent exponential random variables with different respective parameters
, 
, written as
. We define the random variable
, 
, written as
: 
: 
: 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.
: 
: 
:
.
and 
Then
.
where 
for
. Since 
are independent then 
is the convolutions of
, 
written as
However, by Heaviside Expansion Theorem [
.
. Thus
.
then 
and we conclude that
. 
derivative of 
and many equalities are obtained concerning 
form and some similar forms.
. Here, we shall state another simple proof using Laplace transform.
. Then
. But from Theorem 1,
. For
Therefore,
. 
Then
we have from Equation (1)
.
we have
derivative, we obtain the result. 
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.
Then
, we have from Lemma 1,
and from Initial Value Theorem, we have
. Then
. Then the 
derivative of 
is
.
then
with 
we obtain the result. 
and the moment of 
of order k is given. Moreover, we deduce more new related equalities concerning 
and higher order derivatives of pdf of
.
Then
.
then
. 
and
. Then
.
. But
. Thus we obtain the result. 
and higher orders for
. We start by noting that 
and by taking 
in Proposition 3, we again obtain the result in Proposition 1that is
.
and
. Then
.
and 
are equivalent representing a set of combination with repetition having 
, thus the above summation (3) shall be 1.
and
. We have
.
, 
are independent with
then
.
. 
and 2 respectively.
. Then 1) 
and 
and
.
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 
and
. Then
and
Then by Theorem 1, the pdf of 
is
and
.
in terms of
. We have
we obtain 
where 
. Hence, we may write
.
Proposition 5 gives that
.
and
. Then
Then,
Then
. By Corollary 1, taking 
we have 
then
.
. Then
. Then,
. We have from Corollary 1,
with 
we obtain the first case and the case when 
where
.
. By Proposition 6,
.
by 
we obtain the last case when
. 
Then
and
. Then 1) 
. Then
. Moreover, Moranda [
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.
form an arithmetic sequence of common difference
.
form an arithmetic sequence of common difference
. Then 
We have
.
.
.
Replace 
by
, we obtain
Then
where
where 
and by the Lemmas 2 and 3 we obtain the result. 
form a geometric sequence of common ratio
.
Then
.
.
form geometric sequence of common ratio
. Then 
and
. 
represent here a special case and worth mentioning such as
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.