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Linear Combination of Independent Exponential Random Variables

  • Kim-Hung LiEmail author
  • Cheuk Ting Li
Article
  • 104 Downloads

Abstract

In this paper we prove a recursive identity for the cumulative distribution function of a linear combination of independent exponential random variables. The result is then extended to probability density function, expected value of functions of a linear combination of independent exponential random variables, and other functions. Our goal is on the exact and approximate calculation of the above mentioned functions and expected values. We study this computational problem from different views, namely as a Hermite interpolation problem, and as a matrix function evaluation problem. Examples are presented to illustrate the applicability and performance of the methods.

Keywords

Affine combination Erlang distribution Hypoexponential distribution Hermite interpolating polynomial Matrix function Recurrence relation 

Mathematics Subject Classification (2010)

65Q30 65C50 

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Notes

Acknowledgements

The work of the second author was done when he was with the Department of Electrical Engineering, Stanford University, USA.

References

  1. Amari SV, Misra RB (1997) Closed-form expressions for distribution of sum of exponential random variables. IEEE Trans Reliab 46:519–522CrossRefGoogle Scholar
  2. Anjum B, Perros H (2011) Adding percentiles of Erlangian distributions. IEEE Commun Lett 15:346–348CrossRefGoogle Scholar
  3. Bekker R, Koeleman PM (2011) Scheduling admissions and reducing variability in bed demand. Health Care Manag Sci 14:237–249CrossRefGoogle Scholar
  4. Buchholz P, Kriege J, Felko I (2014) Input modeling with phase-type distributions and markov models: theory and applications. Springer, ChamCrossRefzbMATHGoogle Scholar
  5. Cox DR (1962) Renewal Theory. Methuen and CoGoogle Scholar
  6. Davies PI, Higham NJ (2003) A Schur-Parlett algorithm for computing matrix functions. SIAM J Matrix Anal Appl 25:464–485MathSciNetCrossRefzbMATHGoogle Scholar
  7. Davis C (1973) Explicit functional calculus. Lin Alg Applic 6:193–199MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dehghan M, Hajarian M (2009) Determination of a matrix function using the divided difference method of Newton and the interpolation technique of Hermite. J Comput Appl Math 231:67–81MathSciNetCrossRefzbMATHGoogle Scholar
  9. Descloux J (1963) Bounds for the spectral norm of functions of matrices. Numer Math 5:185–190MathSciNetCrossRefzbMATHGoogle Scholar
  10. Favaro S, Walker SG (2010) On the distribution of sums of independent exponential random variables via Wilks’ integral representation. Acta Appl Math 109:1035–1042MathSciNetCrossRefzbMATHGoogle Scholar
  11. Feller W (1971) An introduction to probability theory and its applications, vol II. Wiley, New YorkzbMATHGoogle Scholar
  12. Gershinsky M, Levine DA (1964) Aitken-hermite interpolation. JACM 11:352–356MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gertsbakh I, Neuman E, Vaisman R (2015) Monte Carlo for estimating exponential convolution. Comm Statist Simulation Comput 44:2696–2704MathSciNetCrossRefGoogle Scholar
  14. Goulet V, Dutang C, Maechler M, Firth D, Shapira M, Stadelmann M, expm-developers@lists.R-forge.R-project.org (2015) expm: Matrix Exponential. R package version 0.999-0. https://CRAN.R-project.org/package=expm
  15. Higham NJ (1993) The accuracy of floating point summation. SIAM J Sci Comput 14:783–799MathSciNetCrossRefzbMATHGoogle Scholar
  16. Higham NJ (2008) Functions of matrices: theory and computation. SIAMGoogle Scholar
  17. Higham NJ, Al-Mohy AH (2010) Computing matrix functions. Acta Numerica 19:159–208.  https://doi.org/10.1017/S0962492910000036 MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jasiulewicz H, Kordecki W (2003) Convolutions of Erlang and of Pascal distributions with applications to reliability. Demonstratio Math 36:231–238MathSciNetzbMATHGoogle Scholar
  19. Kadri T, Smaili K (2014) The exact distribution of the ratio of two independent hypoexponential random variables. British J Math Computer Sci 4:2665–2675CrossRefGoogle Scholar
  20. Kadri T, Smaili K, Kadry S (2015) Markov modeling for reliability analysis using hypoexponential distribution. In: Kadry S, El Hami A (eds) Numerical methods for reliability and safety assessment: multiscale and multiphysics systems. Springer, Switzerland, pp 599–620Google Scholar
  21. Khuong HV, Kong H-Y (2006) General expression for pdf of a sum of independent exponential random variables. IEEE Commun Lett 10:159–161CrossRefGoogle Scholar
  22. Kordecki W (1997) Reliability bounds for multistage structures with independent components. Statist Probab Lett 34:43–51MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kotz S, Kozubowski TJ, Podgórski K (2001) The laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Birkhäuser, BostonGoogle Scholar
  24. Legros B, Jouini O (2015) A linear algebraic approach for the computation of sums of Erlang random variables. Appl Math Model 39:4971–4977MathSciNetCrossRefGoogle Scholar
  25. Macleod AJ (1982) A comparison of algorithms for polynomial interpolation. J Comput Appl Math 8:275–277CrossRefzbMATHGoogle Scholar
  26. Maechler M (2015) Rmpfr: R MPFR - multiple precision Floating-Point reliable. R package version 0.6–0. https://CRAN.R-project.org/package=Rmpfr
  27. Mathai AM (1982) Storage capacity of a dam with Gamma type inputs. Ann Inst Statist Math 34:591–597MathSciNetCrossRefzbMATHGoogle Scholar
  28. Moler C, Van Loan C (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev 45:3–49MathSciNetCrossRefzbMATHGoogle Scholar
  29. Ogita T, Rump SM, Oishi S (2005) Accurate sum and dot product. SIAM J Sci Comput 26:1955–1988MathSciNetCrossRefzbMATHGoogle Scholar
  30. Parlett BN (1976) A recurrence among the elements of functions of triangular matrices. Lin Alg Applic 14:117–121MathSciNetCrossRefzbMATHGoogle Scholar
  31. R Core Team (2016) R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. https://www.R-project.org/
  32. Ratanov N (2015) Hypo-exponential distributions and compound Poisson process with alternating parameters. Statist Probab Lett 107:71–78MathSciNetCrossRefzbMATHGoogle Scholar
  33. Ross SM (2014) Introduction to probability models, 11th edn. Academic Press, San DiegozbMATHGoogle Scholar
  34. Scheuer EM (1988) Reliability of an m-out-of-n system when component failure induces higher failure rates in survivors. IEEE Trans Reliab 37:73–74CrossRefzbMATHGoogle Scholar
  35. Shao H, Beaulieu NC (2012) An investigation of block coding for Laplacian noise. IEEE Trans Wireless Commun 11:2362–2372CrossRefGoogle Scholar
  36. Smaili K, Kadri T, Kadry S (2013) Hypoexponential distribution with different parameters. Appl Math 4:624–631CrossRefGoogle Scholar
  37. Smaili K, Kadri T, Kadry S (2014) A modified-form expressions for the hypoexponential distribution. British J Math Computer Sci 4:322–332CrossRefGoogle Scholar
  38. Van Loan CF (1975) A study of the matrix exponential. Numerical Analysis Report No. 10 Department of Mathematics. University of Manchester, EnglandGoogle Scholar
  39. Wen YZ, Yin CC (2014) A generalized Erlang(n) risk model with a hybrid dividend strategy (in Chinese). Sci Sin Math 44:1111–1122Google Scholar
  40. Yin M -L, Angus JE, Trivedi KS (2013) Optimal preventive maintenance rate for best availability with hypo-exponential failure distribution. IEEE Trans Reliab 62:351–361CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Asian Cities Research Centre Ltd.Win PlazaHong Kong
  2. 2.Department of Electrical Engineering and Computer SciencesUniversity of California, BerkeleyBerkeleyUSA

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