63 research outputs found
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
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
EUROPEAN CONFERENCE ON QUEUEING THEORY 2016
International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the TakĂĄcs Award for outstanding PhD thesis on "Queueing Theory and its Applications"
Two extensions of Kingman's GI/G/1 bound
A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have an uncountable state-space. Both extensions are facilitated by a necessary and sufficient ordinary differential equation (ODE) condition for MAPs to admit continuous martingale transforms. Simulations show that the bounds on waiting time distributions are almost exact in heavy-traffic, including the cases of 1) heterogeneous input, e.g., mixing Weibull and Erlang-k classes and 2) Generalized Markovian Arrival Processes, a new class extending the Batch Markovian Arrival Processes to continuous batch sizes
Markovian arrivals in stochastic modelling: a survey and some new results
This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs),
which constitute a rich class of point processes used extensively in stochastic modelling. Our
starting point is the versatile process introduced by Neuts (1979) which, under some simplified
notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general
point process can be approximated by appropriate MAPs and, on the other hand, the MAPs
provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian
formalism. While a number of well-known arrival processes are subsumed under a BMAP as
special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous
settings or even spatial arrivals. We survey on the main aspects of the BMAP,
discuss on some of its variants and generalizations, and give a few new results in the context of a
recent state-dependent extension.Peer Reviewe
Markovian arrivals in stochastic modelling : a survey and some new results
This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension
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