A multispecies birth–death–immigration process and its diffusion approximation

Abstract

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.

Logistic Growth Described by Birth-Death and Diffusion Processes

“…Mathematical models to describe growth phenomena can be useful in many fields of interest such as biology, ecology, medicine or economics (see, for instance [5][6][7]). Frunzo et al [8] performed a generalization of Gompertz law via a Caputo-like definition of fractional derivative.…”

“…Moreover, the lag timeλ denotes the intercept with the t-axis of this tangent line. For the logistic model (4), if y < C/2, thanks to (7), the maximum specific growth rate is given by…”

Probabilistic mean value theorems for conditioned random variables with applications

“…Example Consider an extended birth‐death‐immigration process defined on a lattice formed by the integers of d semiaxes joined at the origin. A suitable limit procedure leads to a diffusion process with linear drift and infinitesimal variance on each ray (see the work of Di Crescenzo et al). Denoting by X ( τ ), τ > 0, the state of the diffusion process along the rays at time τ , this possesses a gamma‐type probability density with shape parameter γ > 0 and time‐varying rate parameter ψ ( τ ) = (e βτ − 1) −1 β / μ , for and μ > 0 (see Proposition 6.3 in the aforementioned work).…”

“…A suitable limit procedure leads to a diffusion process with linear drift and infinitesimal variance on each ray (see the work of Di Crescenzo et al). Denoting by X ( τ ), τ > 0, the state of the diffusion process along the rays at time τ , this possesses a gamma‐type probability density with shape parameter γ > 0 and time‐varying rate parameter ψ ( τ ) = (e βτ − 1) −1 β / μ , for and μ > 0 (see Proposition 6.3 in the aforementioned work). By direct calculations, since ψ ( t ) is decreasing in t > 0, we have that, for 0 < γ < 1, the hazard rate of X ( τ ), given by , is a strictly decreasing function for τ > 0 and t > 0.…”

Application of Mb/M/1 Bulk Arrival Queueing System to Evaluate Key-In System in Islamic University of Indonesia

“…Crescenzo et all [1] study of a birth-death process N(t) with a reflecting state at 0 we propose a method able to construct a new birth-death process M(t) defined on the same state-space. Under a suitable assumption we obtain the conditional probabilities, the mean of the process, and the Laplace transforms of the downward firstpassage-time densities of M(t).…”