We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of fractional order. For this process we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time. The time argument is itself a random process whose distribution is related to the fractional diffusion equation.\ud We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process. For this model we obtain\ud the distributions of the random vector representing the position at time t, under the condition of a fixed number of events and in the unconditional case.\ud For some specific values of $\nu$∈ (0,1] we show that the random position has a Brownian behavior (for $\nu$= 1/2) or a cylindrical-wave structure (for $\nu$= 1)

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{{1/2^n}}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $\nu =\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{\nu}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.Comment: Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper we introduce the space-fractional Poisson process whose state probabilities pα is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions p α k (t), the probability generating functions G α (u, t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.

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In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X (t), t ≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP X N . Finally we present some examples where these bounds are used in order to approximate the double mean.

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We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. d dt p k (t) = −λ(p k (t) − p k−1 (t)), k ≥ 1, t > 0 by introducing fractional time-derivatives of order ν, 2ν, ..., nν . We show that the so-called "Generalized Mittag-Leffler functions" E k α,β (x) (introduced by Prabhakar [20]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t → ∞. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter ν varying in (0, 1] .For integer values of ν, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.analytical form, in terms of infinite series or successive derivatives of Mittag-Leffler functions. We recall the definition of the (two-parameters) Mittag-Leffler function:A different definition of Poisson fractional process has been proposed by Wang and Wen [30] and successively studied in [31]-[32]: in analogy to the well-known fractional Brownian motion, the new process is defined as a stochastic integral with respect to the Poisson measure. It displays properties similar to fractional Brownian motion, such as self-similarity (in the wide-sense) and long-range dependence.Another approach was followed by Repin and Saichev [23]: they start by generalizing, in a fractional sense, the distribution of the interarrival times U j between two Poisson events. This is expressed, in terms of Mittag-Leffler functions, as follows, for ν ∈ (0, 1] :and coincides with the solution to the fractional equationwhere δ(·) denotes the Dirac delta function and again the fractional derivative is intended in the Riemann-Liouville sense. For ν = 1 formula (1.2) reduces to the well-known density appearing in the case of a homogeneous Poisson process, N (t), t > 0, with intensity λ = 1, i.e. f (t) = e −t . The same approach is followed by Mainardi et al.[14]-[15]- [16], where a deep analysis of the related process is performed: it turns out to be a true renewal process, while it looses the Markovian property. Their first step is the study of the following fractional equation (instead of (1.3)) 4) with initial condition ψ(0 + ) = 1 and with fractional derivative defined in the Caputo sense. The solution ψ(t) = E ν,1 (−t ν ) to (1.4) represents the survival probability of the fractional Poisson process. As a consequence its probability distribution is expressed in terms of derivatives of Mittag-Leffler functions, while the density of the k-th event waiting time is a fractional generalization of the Erlang distribution and coincides with the k-fold convolution of (1.2). The analysis carried out by Beghin and Orsingher [2] starts, as in [11], from the generalization of the equation governing the Poisson process, where the time-derivative is substituted by the fractional derivat...

We consider a fractional version of the classical nonlinear birth process of which the Yule Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the differencedifferential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N(v)(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N(v)(t) = N(T(2v) (t)), where N(t) is the classical generalized birth process and T(2v) (t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed