arXiv subjects

Federico Polito

Publications and source records attributed to Federico Polito.

53 records · Page 3Linked to original sources

arXiv2013

Superprocesses as models for information dissemination in the Future Internet

Future Internet will be composed by a tremendous number of potentially interconnected people and devices, offering a variety of services, applications and communication opportunities. In particular, short-range wireless communications, which are available on almost all portable devices, will enable the formation of the largest cloud of interconnected, smart computing devices mankind has ever dreamed about: the Proximate Internet. In this paper, we consider superprocesses, more specifically super Brownian motion, as a suitable mathematical model to analyse a basic problem of information dissemination arising in the context of Proximate Internet. The proposed model provides a promising analytical framework to both study theoretical properties related to the information dissemination process and to devise efficient and reliable simulation schemes for very large systems.

cs.NI↗

arXiv2013

Fractional Non-Linear, Linear and Sublinear Death Processes

This paper is devoted to the study of a fractional version of non-linear $\mathpzc{M}^ν(t)$, $t>0$, linear $M^ν(t)$, $t>0$ and sublinear $\mathfrak{M}^ν(t)$, $t>0$ death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan--Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process $T_{2 ν} (t)$, $t>0$. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

math.PR↗

arXiv2013

Simulation and estimation for the fractional Yule process

In this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution and the representations then yield algorithms on how to simulate sample paths of the fYp. We also attempt to estimate the model parameters in order for the fYp to be usable in practice. The estimation procedure is then tested using simulated data as well. We also illustrate some major characteristics of fYp which will be helpful for real applications.

math.PR↗

arXiv2013

Parameter estimation for fractional birth and fractional death processes

The fractional birth and the fractional death processes are more desirable in practice than their classical counterparts as they naturally provide greater flexibility in modeling growing and decreasing systems. In this paper, we propose formal parameter estimation procedures for the fractional Yule, the fractional linear death, and the fractional sublinear death processes. The methods use all available data possible, are computationally simple and asymptotically unbiased. The procedures exploited the natural structure of the random inter-birth and inter-death times that are known to be independent but are not identically distributed. We also showed how these methods can be applied to certain models with more general birth and death rates. The computational tests showed favorable results for our proposed methods even with relatively small sample sizes. The proposed methods are also illustrated using the branching times of the plethodontid salamanders data of \cite{hal79}.

math.ST↗

arXiv2013

Analytic solutions of fractional differential equations by operational methods

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional differentiation are shown and discussed in detail. Finally, a last example concerns the application of the method to the study of a fractional Poisson process.

math.AP↗

arXiv2013

Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity

In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case.

math.AP↗

arXiv2013

Renewal processes based on generalized Mittag--Leffler waiting times

The fractional Poisson process has recently attracted experts from several fields of study. Its natural generalization of the ordinary Poisson process made the model more appealing for real-world applications. In this paper, we generalized the standard and fractional Poisson processes through the waiting time distribution, and showed their relations to an integral operator with a generalized Mittag--Leffler function in the kernel. The waiting times of the proposed renewal processes have the generalized Mittag--Leffler and stretched-squashed Mittag--Leffler distributions. Note that the generalizations naturally provide greater flexibility in modeling real-life renewal processes. Algorithms to simulate sample paths and to estimate the model parameters are derived. Note also that these procedures are necessary to make these models more usable in practice. State probabilities and other qualitative or quantitative features of the models are also discussed.

math.PR↗

arXiv2013

On the Integral of Fractional Poisson Processes

In this paper we consider the Riemann--Liouville fractional integral $\mathcal{N}^{α,ν}(t)= \frac{1}{Γ(α)} \int_0^t (t-s)^{α-1}N^ν(s) \, \mathrm ds $, where $N^ν(t)$, $t \ge 0$, is a fractional Poisson process of order $ν\in (0,1]$, and $α> 0$. We give the explicit bivariate distribution $\Pr \{N^ν(s)=k, N^ν(t)=r \}$, for $t \ge s$, $r \ge k$, the mean $\mathbb{E}\, \mathcal{N}^{α,ν}(t)$ and the variance $\mathbb{V}\text{ar}\, \mathcal{N}^{α,ν}(t)$. We study the process $\mathcal{N}^{α,1}(t)$ for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on $\mathcal{N}^{1,1}(t)$ are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalised harmonic numbers is discussed.

math.PR↗

arXiv2013

Coupled systems of fractional equations related to sound propagation: analysis and discussion

In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive and also memoryless processes like classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations.

math.AP↗

arXiv2013

On Some Operators Involving Hadamard Derivatives

In this paper we introduce a novel Mittag--Leffler-type function and study its properties in relation to some integro-differential operators involving Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then the utility of these results to solve some integro-differential equations involving these operators by means of operational methods. We show the advantage of our approach through some examples. Among these, an application to a modified Lamb--Bateman integral equation is presented.

math.AP↗

arXiv2011

A note on fractional linear pure birth and pure death processes in epidemic models

In this note we highlight the role of fractional linear birth and linear death processes recently studied in \citet{sakhno} and \citet{pol}, in relation to epidemic models with empirical power law distribution of the events. Taking inspiration from a formal analogy between the equation of self consistency of the epidemic type aftershock sequences (ETAS) model, and the fractional differential equation describing the mean value of fractional linear growth processes, we show some interesting applications of fractional modelling to study \textit{ab initio} epidemic processes without the assumption of any empirical distribution. We also show that, in the frame of fractional modelling, subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes. Moreover we discuss a simple toy model to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth.

math.PR↗

arXiv2011

Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_α(t)$, $N_β(t)$, $t>0$, we show that $N_α(N_β(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_β(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form $N_α(τ_k^ν)$, $ν\in (0,1]$, where $τ_k^ν$ is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form $Θ(N(t))$, $t>0$, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.

math.PR↗

arXiv2011

The Space-Fractional Poisson Process

In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^α(t)$, $t>0$, $α\in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k(t) = -λ^α(1-B)p_k^α(t)$, where $(1-B)^α$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^α(t)$, 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.

math.PR↗

arXiv2011

Randomly Stopped Nonlinear Fractional Birth Processes

We present and analyse the nonlinear classical pure birth process $\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\mathpzc{N}^ν(t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $α$-stable subordinator $\mathpzc{S}^α(t)$, $α\in (0,1)$, and others. For all of them we derive the state probability distribution $\hat{p}_k (t)$, $k \geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for both the subordinated classical birth process $\hat{\mathpzc{N}} (t)$, $t>0$, and its fractional counterpart $\hat{\mathpzc{N}}^ν(t)$, $t>0$ in terms of classical birth processes with random rates evaluated on a stretched or squashed time scale. Various types of compositions of the fractional pure birth process $\mathpzc{N}^ν(t)$ have been examined in the last part of the paper. In particular, the processes $\mathpzc{N}^ν(T_t)$, $\mathpzc{N}^ν(\mathpzc{S}^α(t))$, $\mathpzc{N}^ν(T_{2ν}(t))$, have been analysed, where $T_{2ν}(t)$, $t>0$, is a process related to fractional diffusion equations. Also the related process $\mathpzc{N}(\mathpzc{S}^α({T_{2ν}(t)}))$ is investigated and compared with $\mathpzc{N}(T_{2ν}(\mathpzc{S}^α(t))) = \mathpzc{N}^ν(\mathpzc{S}^α(t))$. As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained.

math.PR↗

arXiv2011

On a fractional linear birth--death process

In this paper, we introduce and examine a fractional linear birth--death process $N_ν(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^ν(t)$, $t>0$, $k\geq0$. We present a subordination relationship connecting $N_ν(t)$, $t>0$, with the classical birth--death process $N(t)$, $t>0$, by means of the time process $T_{2ν}(t)$, $t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^ν(t)$ and the state probabilities $p_k^ν(t)$, $t>0$, $k\geq1$, in the three relevant cases $λ>μ$, $λ<μ$, $λ=μ$ (where $λ$ and $μ$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth--death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_ν(t)$ and $\operatorname {\mathbb{V}ar}N_ν(t)$ are derived and analyzed.

math.PR↗

arXiv2011

Fractional pure birth processes

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 difference-differential equations which govern the probability law of the process with the Dzherbashyan--Caputo fractional derivative. We derive the probability distribution of the number $\mathcal{N}_ν(t)$ 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 $\mathcal{N}_ν(t)=\mathcal{N}(T_{2ν}(t))$, where $\mathcal{N}(t)$ is the classical generalized birth process and $T_{2ν}(t)$ 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.

math.PR↗

arXiv2007

Flow of autonomous traffic on a single multi-lane street

We investigate the behaviour of an original traffic model. The model considers a single multi-lane street, populated by autonomous vehicles directed from either end to the other. Lanes have no intrinsic directionality, and the vehicles are inserted at random at either end and any lane. Collision avoidance is fully automatic and, to enhance the transport capacity of the street, vehicles form_trains_ in which they may travel at high speed quite close to the vehicle in front. We report on the transit times for vehicles under a wide variety of conditions: vehicle insertion probability & imbalance and their maximum speed distribution. We also outline an interesting feature of the model, that the complex interactions of many vehicles are considerably more powerful than a simple "keep left" directive which each vehicle should obey.

physics.comp-ph↗