The Mittag-Leffler function in the thinning theory for renewal processes

Authors:
R. Gorenflo and F. Mainardi

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **98** (2018).

Journal:
Theor. Probability and Math. Statist. **98** (2019), 105-113

MSC (2010):
Primary 33E12, 60K05, 60K25; Secondary 26A33, 76R50

DOI:
https://doi.org/10.1090/tpms/1065

Published electronically:
August 19, 2019

MathSciNet review:
3824681

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law-type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous time random walk and consequently of fractional diffusion in a plenary lecture by the late Professor Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic theory of renewal processes including the standard and the fractional Poisson processes, here we have revised the original approach by Gnedenko and Kovalenko for the convenience of the experts of stochastic processes who are not aware of the relevance of the Mittag-Leffler functions.

- [1]
Luisa
Beghin and Enzo
Orsingher,
*Iterated elastic Brownian motions and fractional diffusion equations*, Stochastic Process. Appl.**119**(2009), no. 6, 1975–2003. MR**2519353**, https://doi.org/10.1016/j.spa.2008.10.001 - [2]
Frank
Beichelt,
*Stochastic processes in science, engineering and finance*, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR**2207217** - [3]
M. Caputo,
*Linear models of dissipation whose is almost frequency independent, Part II*, Geophys. J. R. Astr. Soc.**13**(1967), 529-539. - [4]
M. Caputo,
*Elasticità e Dissipazione*, Bologna, Zanichelli, 1969. - [5]
M. Caputo and F. Mainardi,
*Linear models of in anelastic solids*, Riv. Nuovo Cimento (Ser. II)**1**(1971), 161-198. - [6]
D.
R. Cox,
*Renewal theory*, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962. MR**0153061** - [7]
Luc
Devroye,
*A note on Linnik’s distribution*, Statist. Probab. Lett.**9**(1990), no. 4, 305–306. MR**1047827**, https://doi.org/10.1016/0167-7152(90)90136-U - [8]
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Higher Transcendental Functions*, Bateman Project, McGraw-Hill, New York, 1955, vol. 3, pp. 206-227. - [9]
Richard
Cook,
*Book Review: An Introduction to Probability Theory and Its Applications by William Feller*, Math. Mag.**32**(1958), no. 1, 43–44. MR**1570945** - [10]
I.
M. Gel’fand and G.
E. Shilov,
*Generalized functions. Vol. I: Properties and operations*, Translated by Eugene Saletan, Academic Press, New York-London, 1964. MR**0166596** - [11]
B.
V. Gnedenko and I.
N. Kovalenko,
*Introduction to queueing theory*, Translated from Russian by R. Kondor. Translation edited by D. Louvish, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., Hartford, Conn., 1968. MR**0240884** - [12]
Rudolf
Gorenflo, Anatoly
A. Kilbas, Francesco
Mainardi, and Sergei
V. Rogosin,
*Mittag-Leffler functions, related topics and applications*, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. MR**3244285** - [13]
R.
Gorenflo and F.
Mainardi,
*Fractional calculus: integral and differential equations of fractional order*, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 223–276. MR**1611585** - [14]
R. Gorenflo and F. Mainardi,
*Anomalous Transport: Foundations and Applications*, Ch 4: Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects (R. Klages, G. Radons, I. M. Sololov, eds.), Wiley-VCH, Weinheim, Germany, 2008, pp. 93-127. - [15]
Rudolf
Gorenflo and Francesco
Mainardi,
*Parametric subordination in fractional diffusion processes*, Fractional dynamics, World Sci. Publ., Hackensack, NJ, 2012, pp. 229–263. MR**2932609** - [16]
A.
Y. Khintchine,
*Mathematical methods in the theory of queueing*, Translated by D. M. Andrews and M. H. Quenouille. Griffin’s Statistical Monographs & Courses, No. 7, Hafner Publishing Co., New York, 1960. MR**0115224** - [17]
Nick
Laskin,
*Fractional Poisson process*, Commun. Nonlinear Sci. Numer. Simul.**8**(2003), no. 3-4, 201–213. Chaotic transport and complexity in classical and quantum dynamics. MR**2007003**, https://doi.org/10.1016/S1007-5704(03)00037-6 - [18]
Francesco
Mainardi and Rudolf
Gorenflo,
*Time-fractional derivatives in relaxation processes: a tutorial survey*, Fract. Calc. Appl. Anal.**10**(2007), no. 3, 269–308. MR**2382782** - [19]
Francesco
Mainardi, Rudolf
Gorenflo, and Enrico
Scalas,
*A fractional generalization of the Poisson processes*, Vietnam J. Math.**32**(2004), no. Special Issue, 53–64. MR**2120631** - [20]
Kosto
V. Mitov and Edward
Omey,
*Renewal processes*, SpringerBriefs in Statistics, Springer, Cham, 2014. MR**3308384** - [21]
R.
N. Pillai,
*On Mittag-Leffler functions and related distributions*, Ann. Inst. Statist. Math.**42**(1990), no. 1, 157–161. MR**1054728**, https://doi.org/10.1007/BF00050786 - [22]
Igor
Podlubny,
*Fractional differential equations*, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR**1658022** - [23]
A. Renyi,
*A characteristic of the Poisson stream*, Proc. Math. Inst. Hungarica Acad. Sci.**1**(1956), no. 4, 563-570. (In Hungarian) - [24]
Sheldon
M. Ross,
*Introduction to probability models*, Academic Press, New York-London, 1972. Probability and Mathematical Statistics, Vol. 10. MR**0328973** - [25]
T.
Szántai,
*On limiting distributions for the sums of random number of random variables concerning the rarefaction of recurrent process*, Studia Sci. Math. Hungar.**6**(1971), 443–452. MR**297030** - [26]
T.
Szántai,
*On an invariance problem related to different rarefactions of recurrent processes*, Studia Sci. Math. Hungar.**6**(1971), 453–456. MR**293736** - [27]
David
Vernon Widder,
*The Laplace Transform*, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR**0005923**

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Additional Information

**R. Gorenflo**

Affiliation:
Department of Mathematics and Informatics, Freie Universität, Berlin, Germany

**F. Mainardi**

Affiliation:
Department of Physics and Astronomy, University of Bologna, Italy

Email:
francesco.mainardi@bo.infn.it (Corresponding Author)

DOI:
https://doi.org/10.1090/tpms/1065

Keywords:
Mittag-Leffler functions,
thinning (rarefaction),
renewal processes,
queuing theory,
Poisson process

Received by editor(s):
March 14, 2018

Published electronically:
August 19, 2019

Additional Notes:
The work of the second author was carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM)

Dedicated:
This note is dedicated to the memory of the late Professor Rudolf Gorenflo, who passed away on 20 October 2017 at the age of 87.

Article copyright:
© Copyright 2019
American Mathematical Society