5 research outputs found
A review on symmetry properties of birth-death processes
In this paper we review some results on time-homogeneous birth-death
processes. Specifically, for truncated birth-death processes with two absorbing
or two reflecting endpoints, we recall the necessary and sufficient conditions
on the transition rates such that the transition probabilities satisfy a
spatial symmetry relation. The latter leads to simple expressions for
first-passage-time densities and avoiding transition probabilities. This
approach is thus thoroughly extended to the case of bilateral birth-death
processes, even in the presence of catastrophes, and to the case of a
two-dimensional birth-death process with constant rates.Comment: 16 pages, 4 figure
On the Three Methods for Bounding the Rate of Convergence for some Continuous-time Markov Chains
Consideration is given to the three different analytical methods for the
computation of upper bounds for the rate of convergence to the limiting regime
of one specific class of (in)homogeneous continuous-time Markov chains. This
class is particularly suited to describe evolutions of the total number of
customers in (in)homogeneous queueing systems with possibly
state-dependent arrival and service intensities, batch arrivals and services.
One of the methods is based on the logarithmic norm of a linear operator
function; the other two rely on Lyapunov functions and differential
inequalities, respectively. Less restrictive conditions (compared to those
known from the literature) under which the methods are applicable, are being
formulated. Two numerical examples are given. It is also shown that for
homogeneous birth-death Markov processes defined on a finite state space with
all transition rates being positive, all methods yield the same sharp upper
bound
On some time non-homogeneous queueing systems with catastrophes
Non-stationary queueing systems subject to catastrophes occurring with time varying intensity are considered.
The effect of a catastrophe is to make the queue instantly empty.
The transition probabilities, the related moments and the first visit time density to zero state are analyzed. Particular attention is dedicated
to queueing systems in the presence of catastrophes with periodic intensity function. Various applications are provided, including the
non-stationary birth-death process with immigration, the queueing systems M(t)/M(t)/1
and M(t)/M(t)/∞
Stability Problems for Stochastic Models: Theory and Applications II
Most papers published in this Special Issue of Mathematics are written by the participants of the XXXVI International Seminar on Stability Problems for Stochastic Models, 2125 June, 2021, Petrozavodsk, Russia. The scope of the seminar embraces the following topics: Limit theorems and stability problems; Asymptotic theory of stochastic processes; Stable distributions and processes; Asymptotic statistics; Discrete probability models; Characterization of probability distributions; Insurance and financial mathematics; Applied statistics; Queueing theory; and other fields. This Special Issue contains 12 papers by specialists who represent 6 countries: Belarus, France, Hungary, India, Italy, and Russia