Hypoexponential Distribution with Different Parameters

Affiliation(s)

Department of Applied Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon.

Department of Mathematics, Faculty of Sciences, Beirut Arab University, Beirut, Lebanon.

School of Engineering, American University of the Middle East, Eguaila, Kuwait.

Department of Applied Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon.

Department of Mathematics, Faculty of Sciences, Beirut Arab University, Beirut, Lebanon.

School of Engineering, American University of the Middle East, Eguaila, Kuwait.

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 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.

Cite this paper

K. Smaili, T. Kadri and S. Kadry, "Hypoexponential Distribution with Different Parameters,"*Applied Mathematics*, Vol. 4 No. 4, 2013, pp. 624-631. doi: 10.4236/am.2013.44087.

K. Smaili, T. Kadri and S. Kadry, "Hypoexponential Distribution with Different Parameters,"

References

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[12] M. Akkouchi, “On the Convolution of Exponential Dis tributions,” Chungcheong Mathematical Society, Vol. 21, No. 4, 2008, pp. 501-510.

[13] 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:10.1109/24.693785

[14] Z. Jelinski and P. B. Moranda, “Software Reliability Re search,” Statistical Computer Performance Evaluation, Academic Press, New York, 1972, pp. 465-484.

[15] P. B. Moranda, “Event-Altered Rate Models for General Reliability Analysis,” IEEE Transactions on Reliability, Vol. R-28, No. 5, 1979, pp. 376-381. doi:10.1109/TR.1979.5220648

[16] M. R. Spiegel, “Schaum’s Outline of Theory and Prob lems of Laplace Transforms,” Schaum, New York, 1965.

[1] W. Feller, “An Introduction to Probability Theory and Its Applications,” Vol. II, Wiley, New York, 1971.

[2] S. M. Ross, “Introduction to Probability Models,” 10th Edition, Academic Press, San Diego, 2011.

[3] B. Anjum and H. G. Perros, “Adding Percentiles of Erlangian Distributions,” IEEE Communications Letters, Vol. 15, No. 3, 2011, pp. 346-348. doi:10.1109/LCOMM.2011.011011.102143

[4] K. S. Trivedi, “Probability and Statistics with Reliability, Queuing and Computer Science Applications,” 2nd Edi tion, John Wiley & Sons, Hoboken, 2002.

[5] 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)00164-2

[6] A. 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/BF02481056

[7] L. D. Minkova, “Insurance Risk Theory,” Lecture Notes, TEMPUS Project SEE Doctoral Studies in Mathematical Sciences, 2010.

[8] 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:10.1080/10920277.2007.10597450

[9] G. 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.

[10] O. Gaudoin and J. Ledoux, “Modélisation Aléatoire en Fiabilité des Logiciels,” Hermès Science Publications, Paris, 2007.

[11] 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/s10440-008-9224-4

[12] M. Akkouchi, “On the Convolution of Exponential Dis tributions,” Chungcheong Mathematical Society, Vol. 21, No. 4, 2008, pp. 501-510.

[13] 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:10.1109/24.693785

[14] Z. Jelinski and P. B. Moranda, “Software Reliability Re search,” Statistical Computer Performance Evaluation, Academic Press, New York, 1972, pp. 465-484.

[15] P. B. Moranda, “Event-Altered Rate Models for General Reliability Analysis,” IEEE Transactions on Reliability, Vol. R-28, No. 5, 1979, pp. 376-381. doi:10.1109/TR.1979.5220648

[16] M. R. Spiegel, “Schaum’s Outline of Theory and Prob lems of Laplace Transforms,” Schaum, New York, 1965.