388 research outputs found
Some universal limits for nonhomogeneous birth and death processes
In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP XN. Finally we present some examples where these bounds are used in order to approximate the double mean
On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process
We consider a multidimensional inhomogeneous birth-death process (BDP) and
obtain bounds on the rate of convergence for the corresponding one-dimensional
processes
On the ergodicity bounds for a constant retrial rate queueing model
We consider a Markovian single-server retrial queueing system with a constant
retrial rate. Conditions of null ergodicity and exponential ergodicity for the
correspondent process, as well as bounds on the rate of convergence are
obtained
Two approaches to the construction of perturbation bounds for continuous-time Markov chains
The paper is largely of a review nature. It considers two main methods used
to study stability and obtain appropriate quantitative estimates of
perturbations of (inhomogeneous) Markov chains with continuous time and a
finite or countable state space. An approach is described to the construction
of perturbation estimates for the main five classes of such chains associated
with queuing models. Several specific models are considered for which the limit
characteristics and perturbation bounds for admissible "perturbed" processes
are calculated
Weak Ergodicity of Mt /Mt /N /N + R Queue
2000 Mathematics Subject Classification: 60J27, 60K25.We consider nonstationary birth and death processes on finite state space and study the bounds of the rate of convergence to the limit regime. We also obtain some bounds on the rate of convergence for the queue-length process of Mt/Mt/N/N + R queue.The research has been partially supported by RFBR, grant No. 11-01-12026
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
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