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Table Of Integrals Series And Products

Tables of integrals

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

We consider an extended birth-death-immigration process defined on a lattice formed by the integers of $d$ semiaxes joined at the origin. When the process reaches the origin, then it may jumps toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth-death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process, which involves a diffusion process with linear drift and infinitesimal variance on each ray. It possesses a gamma-type transient density admitting a stationary limit. As a byproduct of our study, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function expressed in terms of the Gauss hypergeometric function.

A linear birth-death process on a star graph and its diffusion approximation

Probabilistic mean value theorems for conditioned random variables with applications

Aiming to construct a simple stochastic model able to describe systems alternating due to state-dependent dichotomous noise, we consider a generalized telegraph process whose sample-paths fluctuates around the zero state. Indeed, the latter process describes the motion of a particle on the real line, which is characterized by constant velocities and state-dependent intensities that vanish when the motion is toward the origin. This assumption allows to adopt an approach based on renewal theory to obtain formal expressions of the forward and backward transition densities of the process. The special case when certain random times of the motion possess gamma distribution leads to closed-form expressions of the transition densities, given in terms of the generalized Mittag-Leffler function. We also analyze a first-passage-time problem for the considered process in the presence of two constant boundaries.

Probabilistic analysis of systems alternating for state-dependent dichotomous noise.

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of the approximation is finally discussed.

Antonio Di Crescenzo

Papers

A linear birth-death process on a star graph and its diffusion approximation

Probabilistic mean value theorems for conditioned random variables with applications

Probabilistic analysis of systems alternating for state-dependent dichotomous noise.

A doubled-ended queue with catastrophes and repairs, and a jump-diffusion approximation (Abstract)

Vectorized simulations of normal processes and the first passage time probability problem