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The definition of generalized random processes in Gel'fand sense allows to extend well-known stochastic models, such as the fractional Brownian motion, and study the related fractional pde's, as well as stochastic differential equations in distributional sense. By analogy with the construction (in the infinite-dimensional white-noise space) of the latter, we introduce two processes defined by means of Hadamard-type fractional operators. When used to replace the time derivative in the governing p.d.e.'s, the Hadamard-type derivatives are usually associated with ultra-slow diffusions. On the other hand, in our construction, they directly determine the memory properties of the so-called Hadamard fractional Brownian motion (H-fBm) and its long-time behaviour. Still, for any finite time horizon, the H-fBm displays a standard diffusing feature. We then extend the definition of the H-fBm from the white noise space to an infinite dimensional grey-noise space built on the Le Roy measure, so that our model represents an alternative to the generalized grey Brownian motion. In this case, we prove that the one-dimensional distribution of the process satisfies a heat equation with non-constant coefficients and fractional Hadamard time-derivative. Finally, once proved the existence of the distributional derivative of the above defined processes and derived an integral formula for it, we construct an Ornstein-Uhlenbeck type process and evaluate its distribution., Comment: 30 pages

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time PDFs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barab\'asi-Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

We consider three kinds of discrete-time arrival processes: transient, intermediate and recurrent, characterized by a finite, possibly finite and infinite number of events, respectively. In this context, we study renewal processes which are stopped at the first event of a further independent renewal process whose inter-arrival time distribution can be defective. If this is the case, the resulting arrival process is of an intermediate nature. For non-defective absorbing times, the resulting arrival process is transient, i.e.\ stopped almost surely. For these processes we derive finite time and asymptotic properties. We apply these results to biased and unbiased random walks on the d-dimensional infinite lattice and as a special case on the two-dimensional triangular lattice. We study the spatial propagator of the walker and its large time asymptotics. In particular, we observe the emergence of a superdiffusive (ballistic) behavior in the case of biased walks. For geometrically distributed stopping times, the propagator converges to a stationary non-equilibrium steady state (NESS), which is universal in the sense that it is independent of the stopped process. In dimension one, for both light- and heavy-tailed step distributions, the NESS has an integral representation involving alpha-stable distributions.

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred this random walk to as squirrel random walk (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the `generalized Sibuya distribution' (GSD) is such that the moments are finite up to a certain order $r\leq m-1$ ($m \geq 1$) and diverging for orders $r \geq m$ capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case ($m=1$). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order $m$ of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.

We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at {\tt https://github.com/habyarimanacassien/Fractional-Hawkes}., Comment: 19 pages, 6 figures. This version includes the exact inversion of a Laplace transform that we previously inverted only numerically

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the `aging effect' as a hallmark of non-Markovianity where the discrete-time version of the `aging renewal process' comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a $t^2$-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (`narrow' with finite mean and variance) the walk exhibits normal diffusion and for `broad' waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Interspike intervals describe the output of neurons. Signal transmission in a neuronal network implies that the output of some neurons becomes the input of others. The output should reproduce the main features of the input to avoid a distortion when it becomes the input of other neurons, that is input and output should exhibit some sort of consistency. In this paper, we consider the question: how should we mathematically characterize the input in order to get a consistent output? Here we interpret the consistency by requiring the reproducibility of the input tail behaviour of the interspike intervals distributions in the output. Our answer refers to a system of interconnected neurons with stochastic perfect integrate and fire units. In particular, we show that the class of regularly-varying vectors is a possible choice to obtain such consistency. Some further necessary technical hypotheses are added.

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the `it generator process' of the walk. We refer the so defined walk to as `Asymmetric Discrete-Time Random Walk' (ADTRW). We highlight connections of the waiting-time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and analyze recurrent/transient features of simple ADTRWs (walks with unit jumps in both directions). We explore the connections of the recurrence/transience with the bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple ADTRWs are either unbiased in the large time limit or `strictly unbiased' at all times with symmetric Bernoulli generator process. In this analysis we highlight the connections of bias and light-tailed/fat-tailed features of the waiting time density in the generator process. As a prototypical example with fat-tailed feature we consider the ADTRW with Sibuya distributed waiting times. We also introduce time-changed versions: We subordinate the ADTRW to a continuous-time renewal process which is independent from the generator process and the jumps to define the new class of `Asymmetric Continuous Time Random Walk' (ACTRW). This new class - apart of some special cases - is not a Montroll--Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may open large interdisciplinary fields in anomalous transport, birth-death models and others.

In recent years a huge interdisciplinary field has emerged which is devoted to the complex dynamics of anomalous transport with long-time memory and non-markovian features. It was found that the framework of fractional calculus and its generalizations are able to capture these phenomena. Many of the classical models are based on continuous-time renewal processes and use the Montroll Weiss continuous time random walk (CTRW) approach. On the other hand their discrete time counterparts are rarely considered in the literature despite their importance in various applications. The goal of the present paper is to give a brief sketch of our recently introduced discrete-time Prabhakar generalization of the fractional Poisson process and the related discrete-time random walk (DTRW) model. We show that this counting process is connected with the continuous time Prabhakar renewal process by a (well scaled) continuous-time limit. We deduce the state probabilities and discrete time generalized fractional Kolmogorov-Feller equations governing the Prabhakar DTRW and discuss effects such as long time memory (nonmarkovianity) as a hallmark of the complexity of the process., Comment: 6 Pages, 1 Figure

This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the $n$ partitions of the interval $[0,W_n]$ are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process `{\it space-time Mittag-Leffler process}'. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a `well-scaled' diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the `state density kernel' solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with integer IID interarrival times which converge in well-scaled continuous-time limits to the Prabhakar-generalized fractional Poisson process. These processes exhibit non-Markovian features and long-time memory effects. We recover for special choices of parameters the discrete-time versions of classical cases, such as the fractional Bernoulli process and the standard Bernoulli process as discrete-time approximations of the fractional Poisson and the standard Poisson process, respectively. We derive difference equations of generalized fractional type that govern these discrete time-processes where in well-scaled continuous-time limits known evolution equations of generalized fractional Prabhakar type are recovered. We also develop in Montroll-Weiss fashion the `Prabhakar Discrete-time random walk (DTRW)' as a random walk on a graph time-changed with a discrete-time version of Prabhakar renewal process. We derive the generalized fractional discrete-time Kolmogorov-Feller difference equations governing the resulting stochastic motion. Prabhakar-discrete-time processes open a promising field capturing several aspects in the dynamics of complex systems.

The Mittag-Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function., Comment: 35 pages, 1 figure, Matches the version accepted in FCAA

Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD's) to allow both overdispersion and underdispersion. Similarly to Kemp's generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD's related to a generalization of Mittag--Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

We analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using, with probability $p$, an anti-preferential attachment mechanism or, with probability $1-p$, a preferential attachment mechanism. We derive the asymptotic degree distribution in the general case and study the asymptotic behaviour of the expected degree process in the general and that of the degree process in the pure anti-preferential attachment case. Degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter.

In the last years, many authors studied a class of continuous time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop the discrete-time version of such a theory. We show that a class of discrete-time semi-Markov chains can be seen as time-changed Markov chains and we obtain governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process., Comment: Published at https://doi.org/10.15559/18-VMSTA125 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)

In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.

We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton--Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model.

We prove that the Barab\'asi-Albert model converges weakly to a set of generalized Yule models via an appropriate scaling. To pursue this aim we superimpose to its graph structure a suitable set of processes that we call the planted model and we introduce an ad-hoc sampling procedure. The use of the obtained limit process represents an alternative and advantageous way of looking at some of the asymptotic properties of the Barab\'asi-Albert random graph.

With the aim of considering models with persistent memory we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macrovolution. Here the model is analyzed and interpreted in the framework of the development of networks such as the World Wide Web. Nonlinearity is introduced by replacing the linear birth process governing the growth of the in-links of each specific webpage with a fractional nonlinear birth process with completely general birth rates. Among the main results we derive the explicit distribution of the number of in-links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction. The mean value of the latter distribution is also calculated explicitly in the most general case. Furthermore, in order to show the usefulness of our results, we particularize them in the case of specific birth rates giving rise to a saturating behaviour, a property that is often observed in nature. The further specialization to the non-fractional case allows us to extend the Yule model accounting for a nonlinear growth.

In this paper we study some properties of the Prabhakar integrals and derivatives and of some of their extensions such as the regularized Prabhakar derivative or the Hilfer--Prabhakar derivative. Some Opial- and Hardy-type inequalities are derived. In the last section we point out on some relationships with probability theory.

We give a common description of Simon, Barab\'asi--Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barab\'asi--Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter $\alpha$) goes to infinity, a portion of them behave as a Yule model with parameters $(\lambda,\beta) = (1-\alpha,1)$, and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in \cite{Newman2005}. References to traditional and recent applications of the these models are also discussed.

This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.

This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators $L$ and converting their fractional power $L^{\alpha}$ into Erd\'elyi--Kober fractional integrals. We study also probabilistic properties of the r.v.'s whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed r.v.'s.

We congratulate the authors for the interesting paper. The reading has been really pleasant and instructive. We discuss briefly only some of the interesting results given in Devroye and James "On simulation and properties of the stable law", 2014 with particular attention to evolution problems. The contribution of the results collected in the paper is useful in a more wide class of applications in many areas of applied mathematics.

We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

In this paper we discuss some exact results related to the fractional Klein--Gordon equation involving fractional powers of the D'Alembert operator. By means of a space-time transformation, we reduce the fractional Klein--Gordon equation to a fractional hyper-Bessel-type equation. We find an exact analytic solution by using the McBride theory of fractional powers of hyper-Bessel operators. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein-Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein-Gordon equation with power law nonlinearities.

Real-world networks may exhibit detachment phenomenon determined by the cancelling of previously existing connections. We discuss a tractable extension of Yule model to account for this feature. Analytical results are derived and discussed both asymptotically and for a finite number of links. Comparison with the original model is performed in the supercritical case. The first-order asymptotic tail behavior of the two models is similar but differences arise in the second-order term. We explicitly refer to World Wide Web modeling and we show the agreement of the proposed model on very recent data. However, other possible network applications are also mentioned.

This paper presents finite-velocity random motions driven by fractional Klein-Gordon equations of order $\alpha \in (0,1]$. A key tool in the analysis is played by the McBride's theory which converts fractional hyper-Bessel operators into Erdelyi-Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein-Gordon equation. The distribution of the fractional telegraph process for $\alpha = 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein-Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,\alpha)$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,\alpha)$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of directions. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$-dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler-Poisson-Darboux equation to which our theory applies.

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.

We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional diffusion equation and the fractional telegraph equation. The stochastic solution is expressed as a L\'evy process time-changed with the inverse process to a linear combination of (possibly subordinated) independent stable subordinators of different indices. Furthermore a related SDE is derived and discussed.

If interspike intervals are dependent the instantaneous firing rate does not catch important features of spike trains. In this case the conditional instantaneous rate plays the role of the instantaneous firing rate for the case of samples of independent interspike intervals. If the conditional distribution of the interspikes intervals is unknown, it becomes difficult to evaluate the conditional firing rate. We propose a non parametric estimator for the conditional instantaneous firing rate for Markov, stationary and ergodic ISIs. An algorithm to check the reliability of the proposed estimator is introduced and its consistency properties are proved. The method is applied to data obtained from a stochastic two compartment model and to in vitro experimental data.

This paper is devoted to the study of a fractional version of non-linear $\mathpzc{M}^\nu(t)$, $t>0$, linear $M^\nu (t)$, $t>0$ and sublinear $\mathfrak{M}^\nu (t)$, $t>0$ 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 $T_{2 \nu} (t)$, $t>0$. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

In this paper we consider the Riemann--Liouville fractional integral $\mathcal{N}^{\alpha,\nu}(t)= \frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1}N^\nu(s) \, \mathrm ds $, where $N^\nu(t)$, $t \ge 0$, is a fractional Poisson process of order $\nu \in (0,1]$, and $\alpha > 0$. We give the explicit bivariate distribution $\Pr \{N^\nu(s)=k, N^\nu(t)=r \}$, for $t \ge s$, $r \ge k$, the mean $\mathbb{E}\, \mathcal{N}^{\alpha,\nu}(t)$ and the variance $\mathbb{V}\text{ar}\, \mathcal{N}^{\alpha,\nu}(t)$. We study the process $\mathcal{N}^{\alpha,1}(t)$ 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.

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}.

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.

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.

The classical binomial process has been studied by \citet{jakeman} (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes. More statistical properties are also derived.

The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{\nu_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<\nu_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{\nu_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $\nu_k$ differs from that constructed from the fractional state equations (in the case $\nu_k = \nu$, for all $k$, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally we consider the fractional birth process governed by equations with state-dependent fractionality.

We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian and Markovian properties, which naturally provide greater flexibility in modeling real queue systems than its classical counterpart. Algorithms to simulate M/M/1 queue process and the related linear birth-death process are provided. Closed-form expressions of the point and interval estimators of the parameters of these fractional stochastic models are also presented. These methods are necessary to make these models usable in practice. The proposed fractional M/M/1 queue model and the statistical methods are illustrated using S&P data.

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.