Beyond the Poisson renewal process: A tutorial survey

Journal of Computational and Applied Mathematics

Volumn 205, Issue 2, 2007, Pages 725-735

  Mainardi, Francesco ^a   Gorenflo, Rudolf ^b   Vivoli, Alessandro ^a  

^a UNIVERSITY OF BOLOGNA   (Italy)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Fractional derivative; Mittag Leffler function; Poisson process; Power laws; Renewal theory; Stable distributions; Waiting times; Wright function

Indexed keywords

COMPUTATION THEORY; RANDOM PROCESSES;

FRACTIONAL DERIVATIVE; MITTAG-LEFFLER FUNCTION; POISSON PROCESS; POWER LAWS; STABLE DISTRIBUTIONS; WAITING TIMES; WRIGHT FUNCTION;

POISSON DISTRIBUTION;

EID: 34248215647     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.04.060     Document Type: Article

References (24)

1. Abramowitz M., and Stegun I.A. Handbook of Mathematical Functions (1965), Dover, New York
2. Barkai E. CTRW pathways to the fractional diffusion equation. Chem. Phys. 284 (2002) 13-27
3. Cox D.R. Renewal Theory. second ed. (1967), Methuen, London
4. Erdélyi A., Magnus W., Oberhettinger F., and Tricomi F.G. Higher Transcendental Functions vol. 3 (1955), McGraw-Hill, New York
5. W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, second ed., Wiley, New York, 1971 (1-st edn. 1966).
6. Gnedenko B.V., and Kovalenko I.N. Introduction to Queuing Theory (1968), Israel Program for Scientific Translations, Jerusalem
7. Gorenflo R., Luchko Yu., and Mainardi F. Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Comput. Appl. Math. 118 (2000) 175-191
8. Gorenflo R., and Mainardi F. Fractional calculus: integral and differential equations of fractional order. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer, Wien, New York 223-276. http://www.fracalmo.org 〈http://www.fracalmo.org〉
9. R. Gorenflo, F. Mainardi, A. Vivoli, Continuous time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 2006, to appear.
10. Hilfer R., and Anton L. Fractional master equations and fractal time random walks. Phys. Rev. E 51 (1995) R848-R851
11. Huillet T. Fractional Lévy motion and related processes. J. Phys. A 32 (1999) 7225-7248
12. Jayakamur K. Mittag-Leffler process. Math. Comput. Modelling 37 (2003) 1427-1434
13. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations (2006), North-Holland, Amsterdam
14. Lin G.D. On the Mittag-Leffler distributions. J. Statist. Plann. Inference 74 (1998) 1-9
15. Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer, Wien, New York 291-348. http://www.fracalmo.org 〈http://www.fracalmo.org〉
16. Mainardi F., and Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118 (2000) 283-299
17. Mainardi F., Gorenflo R., and Scalas E. A fractional generalization of the Poisson processes. Vietnam J. Math. 32 SI (2004) 53-64
18. Mainardi F., Gorenflo R., and Vivoli A. Renewal processes of Mittag-Leffler and Wright type. Fractional Calculus Appl. Anal. 8 (2005) 7-38
19. Mainardi F., and Pagnini G. The Wright functions as solutions of the time-fractional diffusion equations. Appl. Math. Comput. 141 (2003) 51-62
20. Mainardi F., Raberto M., Gorenflo R., and Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287 (2000) 468-481
21. Pillai R.N. On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42 (1990) 157-161
22. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
23. Ross S.M. Introduction to Probability Models. sixth ed. (1997), Academic Press, New York
24. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon & Breach, New York