12 research outputs found
Conditions for stability and instability of retrial queueing systems with general retrial times
We study the stability of single server retrial queues under general
distribution for retrial times and stationary ergodic service times, for three
main retrial policies studied in the literature: classical linear, constant and
control policies. The approach used is the renovating events approach to obtain
sufficient stability conditions by strong coupling convergence of the process
modeling the dynamics of the system to a unique stationary ergodic regime. We
also obtain instability conditions by convergence in distribution to improper
limiting sequences
Stability and partial instability of multi-class retrial queues
International audienc
Analysis of Multiserver Retrial Queueing System: A Martingale Approach and an Algorithm of Solution
The paper studies a multiserver retrial queueing system with servers.
Arrival process is a point process with strictly stationary and ergodic
increments. A customer arriving to the system occupies one of the free servers.
If upon arrival all servers are busy, then the customer goes to the secondary
queue, orbit, and after some random time retries more and more to occupy a
server. A service time of each customer is exponentially distributed random
variable with parameter . A time between retrials is exponentially
distributed with parameter for each customer. Using a martingale
approach the paper provides an analysis of this system. The paper establishes
the stability condition and studies a behavior of the limiting queue-length
distributions as increases to infinity. As , the paper
also proves the convergence of appropriate queue-length distributions to those
of the associated `usual' multiserver queueing system without retrials. An
algorithm for numerical solution of the equations, associated with the limiting
queue-length distribution of retrial systems, is provided.Comment: To appear in "Annals of Operations Research" 141 (2006) 19-52.
Replacement corrects a small number of misprint
Stability condition of multiclass classical retrials: a revised regenerative proof
We consider a multiclass retrial system with classical retrials, and present a new short proof of the sufficient stability (positive recurrence) condition of the system. The proof is based on the analysis of the departures from the system and a balance equation between the arrived and departed work. Moreover, we apply the asymptotic results from the theory of renewal and regenerative processes. This analysis is then extended to the system with the outgoing calls. A few numerical examples illustrate theoretical analysis
Stability condition of multiclass classical retrials: a revised regenerative proof
We consider a multiclass retrial system with classical retrials,
and present a new short proof of the sufficient stability (positive recurrence)
condition of the system. The proof is based on the analysis of the departures
from the system and a balance equation between the arrived and departed
work. Moreover, we apply the asymptotic results from the theory of renewal
and regenerative processes. This analysis is then extended to the system with
the outgoing calls. A few numerical examples illustrate theoretical analysis
Single server retrial queueing models.
Most retrial queueing research assumes that each retrial customer has its own orbit, and the retrial customers retry to enter service independently of each other. A small selection of papers assume that the retrial customers themselves form a queue, and only one customer from the retrial queue can attempt to enter at any given time. Retrial queues with exponential retrial times have been extensively studied, but little attention has been paid to retrial queues with general retrial times. In this thesis, we consider four retrial queueing models of the type in which the retrial customers form their own queue. Model I is a type of M/G/1 retrial queue with general retrial times and server subject to breakdowns and repairs. In addition, we allow the customer in service to leave the service position and keep retrying for service until the server has been repaired. After repair, the server is not allowed to begin service on other customers until the current customer (in service) returns from its temporary absence. We say that the server is in reserved mode, when the current customer is absent and the server has already been repaired. We define the server to be blocked if the server is busy, under repair or in reserved mode. In Model II, we consider a single unreliable server retrial queue with general retrial times and balking customers. If an arriving primary customer finds the server blocked, the customer either enters a retrial queue with probability p or leaves the system with probability 1 - p. An unsuccessful arriving customer from the retrial queue either returns to its position at the head of the retrial queue with probability q or leaves the system with the probability 1 - q. If the server fails, the customer in service either remains in service with probability r or enters a retrial service orbit with probability 1 - r and keeps returning until the server is repaired. We give a formal description for these two retrial queueing models, with examples. The stability of the system is analyzed by using an embedded Markov chain. We get a necessary and sufficient condition for the ergodicity of the embedded Markov chain. By employing the method of supplementary variables, we describe the state of the system at each point in time. A system of partial differential equations related to the models is derived from a stochastic analysis of the model. The steady state distribution of the system is obtained by means of probability generating functions. In steady state, some performance measures of the system are reported, the distribution of some important performance characteristics in the waiting process are investigated, and the busy period is discussed. In addition, some numerical results are given. Model III consists of a single-server retrial queue with two primary sources and both a retrial queue and retrial orbits. Some results are obtained using matrix analytic methods. Also simulation results are obtained. Model IV consists of a single server system in which the retrial customers form a queue. The service times are discrete. A stability condition and performance measures are presented.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2005 .W87. Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3883. Thesis (Ph.D.)--University of Windsor (Canada), 2006
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, 21Â25 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