AIMS Mathematics, 2016, 1(3): 212-224. doi: 10.3934/Math.2016.3.212.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On a fractional alternating Poisson process

Department of Mathematics, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA),Italy

We propose a generalization of the alternating Poisson process from the point of view offractional calculus. We consider the system of differential equations governing the state probabilitiesof the alternating Poisson process and replace the ordinary derivative with the fractional derivative (inthe Caputo sense). This produces a fractional 2-state point process. We obtain the probability massfunction of this process in terms of the (two-parameter) Mittag-Leffler function. Then we show thatit can be recovered also by means of renewal theory. We study the limit state probability, and certainproportions involving the fractional moments of the sub-renewal periods of the process. In conclusion,in order to derive new Mittag-Leffler-like distributions related to the considered process, we exploit atransformation acting on pairs of stochastically ordered random variables, which is an extension of theequilibrium operator and deserves interest in the analysis of alternating stochastic processes.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Caputo derivative; Mittag-Leffler function; fractional process; alternating process; renewal process; renewal function

Citation: Antonio Di Crescenzo, Alessandra Meoli. On a fractional alternating Poisson process. AIMS Mathematics, 2016, 1(3): 212-224. doi: 10.3934/Math.2016.3.212

References

  • 1. L. Beghin, E. Orsingher, Poisson-type processes governed by fractional and higher-order recursivedifferential equations, Electron. J. Probab., 15 (2010), 684-709.
  • 2. D.O. Cahoy, F. Polito, Renewal processes based on generalized Mittag-Leffler waiting times, Commun.Nonlinear Sci. Numer. Simul., 18 (2013), 639-650.
  • 3. A. Di Crescenzo, A probabilistic analogue of the mean value theorem and its applications to reliabilitytheory, J. Appl. Probab., 36 (1999), 706-719.
  • 4. A. Di Crescenzo, B. Martinucci, and A. Meoli, A fractional counting process and its connectionwith the Poisson process, ALEA Lat. Am. J. Probab. Math. Stat., 13 (2016), 291-307.
  • 5. R. Gorenflo, A.A. Kilbas, F. Mainardi, and S.V. Rogosin, Mittag-Leffler Functions, Related Topicsand Applications, Springer, Berlin, 2014.
  • 6. R. Gorenflo, F. Mainardi, On the fractional Poisson process and the discretized stable subordinator,Axioms, 4 (2015), 321-344.
  • 7. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Eighth edition, Elsevier/Academic Press, Amsterdam, 2014.
  • 8. T. Lancaster, The Econometric Analysis of Transition Data, Econometric Society Monographs, 17,Cambridge University Press, Cambridge, 1990.
  • 9. N. Laskin, Fractional Poisson process, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 201-213.
  • 10. N. Laskin, Some applications of the fractional Poisson probability distribution, J. Math. Phys., 50(2009), 113513.
  • 11. G.D. Lin, On the Mittag-Leffler distributions, J. Stat. Plan. Infer., 74 (1998), 1-9.
  • 12. K.V. Mitov, N.M. Yanev, Limiting distributions for lifetimes in alternating renewal processes,Pliska Stud. Math. Bulg., 16 (2004), 137-145.
  • 13. R. Nelson, Probability, Stochastic Processes, and Queueing Theory: the Mathematics of ComputerPerformance Modeling, Springer-Verlag, New York, 1995.
  • 14. E. Orsingher, F. Polito, Fractional pure birth processes, Bernoulli, 16 (2010), 858-881.
  • 15. E. Orsingher, F. Polito, On a fractional linear birth-death process Bernoulli, 17 (2011), 114-137.
  • 16. E. Orsingher, F. Polito, and L. Sakhno, Fractional non-Linear, linear and sublinear death processes,J. Stat. Phys., 141 (2010), 68-93.
  • 17. R.N. Pillai, On Mittag-Leffler functions and related distributions, Ann. Inst. Statist. Math., 42(1990), 157-161.
  • 18. I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, FractionalDifferential Equations, to Methods of Their Solution and Some of Their Applications. Mathematicsin Science and Engineering, 198, Academic Press, San Diego, 1999.
  • 19. V.V. Uchaikin, D.O. Cahoy, and R.T. Sibatov, Fractional processes: from Poisson to branchingone, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2717-2725.
  • 20. V.V. Uchaikin, R.T. Sibatov, A fractional Poisson process in a model of dispersive charge transportin semiconductors, Russian J. Numer. Anal. Math. Modelling, 23 (2008), 283-297.
  • 21. S. Zacks, Distribution of the total time in a mode of an alternating renewal process with applications,Sequential Anal., 31 (2012), 397-408.

 

This article has been cited by

  • 1. Murugan Suvinthra, Krishnan Balachandran, Rajendran Mabel Lizzy, Large Deviations for Stochastic Fractional Integrodifferential Equations, AIMS Mathematics, 2017, 2, 2, 348, 10.3934/Math.2017.2.348
  • 2. Antonio Di Crescenzo, Alessandra Meoli, On a jump-telegraph process driven by an alternating fractional Poisson process, Journal of Applied Probability, 2018, 55, 01, 94, 10.1017/jpr.2018.8

Reader Comments

your name: *   your email: *  

Copyright Info: 2016, Antonio Di Crescenzo, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved