2013/05/31 by Rudolf Gorenflo, Francesco Mainardi · 16 citations
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · Physics and Astronomy · #Branching process #Compound Poisson process #Counting process #Diffusion and Search Dynamics #Discretization #Erlang (programming language) #Erlang distribution #Fractional Differential Equations Solutions #Inverse #Laplace transform #Molecular Communication and Nanonetworks #Renewal theory #Subordinator #math-ph #math.MP #math.PR #msc:26A33 #msc:33E12 #msc:45K05 #msc:60G18 #msc:60G50 #msc:60G52 #msc:60K05 #msc:76R50
paper · pdf · doi:10.3390/axioms4030321
published in Axioms 4(3), 321-344 (Multidisciplinary Digital Publishing Institute) · 30 pages, 4 figures. A preliminary version of this paper was an invited talk given by R. Gorenflo at the Conference ICMS2011, held at the International Centre of Mathematical Sciences, Pala-Kerala (India) 3-5 January 2011, devoted to Prof Mathai on the occasion of his 75 birthday
openalex publication_date 2015/08/04 · arxiv created 2016/01/13 · arxiv updated 2016/01/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively.