Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

https://link.springer.com/content/pdf/10.1057%2Fjors.1984.224.pdf

Comparison Methods for Queues and Other Stochastic Models

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Loss Models From Data To Decisions

We consider a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whoseprobabilitiessatisfy a suitablesystemoffractionaldifference-differential equations. We obtain the moment generating function and the probability law of the result- ing process in terms of generalized Mittag-Leffler functions. We also discuss two equiv- alent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the countingprocess over their means tend to 1 in probability.

/pdf/a-fractional-counting-process-and-its-connection-with-the-49xu5nvbl7.pdf

A fractional counting process and its connection with the Poisson process

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms.

https://www.mdpi.com/1660-4601/17/6/2014/pdf

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles.

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

/pdf/limit-theorems-for-the-fractional-non-homogeneous-poisson-16d1mtim4z.pdf

Limit theorems for the fractional non-homogeneous Poisson process

Fractional generalized cumulative entropy and its dynamic version

The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D7166DDFD95CAE4502F5BF04816EB661/S0021900218000086a.pdf/div-class-title-on-a-jump-telegraph-process-driven-by-an-alternating-fractional-poisson-process-div.pdf

On a jump-telegraph process driven by an alternating fractional Poisson process

We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment g...

Certain functionals of squared telegraph processes

In order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor's theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion we discuss the probabilistic Taylor's theorem based on fractional Caputo derivatives.

Alessandra Meoli

Papers

A fractional counting process and its connection with the Poisson process

Fractional generalized cumulative entropy and its dynamic version

On a jump-telegraph process driven by an alternating fractional Poisson process

Certain functionals of squared telegraph processes

On the fractional probabilistic Taylor's and mean value theorems