Про системи з повторними викликами та керованим вхідним потоком

Дослiджуються марковськi системи з повторними викликами i керованою iнтенсивнiстю вхiдного потоку. Для моделей такого типу знайденi умови iснування стацiонарного режиму, одержанi явнi формули та ефективнi алгоритми розрахунку стацiонарних iмовiрностей. Розглянуто застосування одержаних результатiв до розв’язання оптимiзацiйних задач в класi багатопорогових стратегiй.Markov retrial queues with the controlled rate of the input flow are investigated. For such models, the existence conditions of a stationary regime are obtained. To calculate the stationary probabilities, explicit formulas and effective algorithms are constructed. The obtained results are used to solve optimization problems in a class of multithreshold strategies

Resource retrial queue with two orbits and negative customers

In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is limited in the system. The service time does not depend on the customer’s resource requirement and is exponentially distributed with parameters depending on the customer’s type. If there is not enough amount of resource for the arriving customer, the customer goes to one of the two orbits, according to his type. The duration of the customer delay in the orbit is exponentially distributed. A negative customer removes all the customers that are served during his arrival and leaves the system. The objects of the study are the number of customers in each orbit and the number of customers of each type being served in the stationary regime. The method of asymptotic analysis under the long delay of the customers in the orbits is applied for the study. Numerical analysis of the obtained results is performed to show the influence of the system parameters on its performance measure

Algorithmic analysis of the maximum level length in general-block two-dimensional Markov processes

Two-dimensional continuous-time Markov chains (CTMCs) are useful tools for studying stochastic models such as queueing, inventory, and production systems. Of particular interest in this paper is the distribution of the maximal level visited in a busy period because this descriptor provides an excellent measure of the system congestion. We present an algorithmic analysis for the computation of its distribution which is valid for Markov chains with general-block structure. For a multiserver batch arrival queue with retrials and negative arrivals, we exploit the underlying internal block structure and present numerical examples that reveal some interesting facts of the system

Diffusion approximation in overloaded switching queueing models

The asymptotic behavior of a queueing process in overloaded state-dependent queueing models (systems and networks) of a switching structure is investigated. A new approach to study fluid and diffusion approximation type theorems (without reflection) in transient and quasi-stationary regimes is suggested. The approach is based on functional limit theorems of averaging principle and diffusion approximation types for so-called Switching processes. Some classes of state-dependent Markov and non-Markov overloaded queueing systems and networks with different types of calls, batch arrival and service, unreliable servers, networks (MSM, Q/MSM, Q/1/∞ )r switched by a semi-Markov environment and state-dependent polling systems are considered

Analysis of Markov multiserver retrial queues with negative arrivals

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. In the paper we analyze multiserver retrial queues; i.e., any customer finding all servers busy upon arrival must leave the service area and re-apply for service after some random time. The control mechanism is such that, whenever the service facility is full occupied, an exponential timer is activated. If the timer expires and the service facility remains full, then a random batch of customers, which are stored at the retrial pool, are automatically removed. This model extends the existing literature, which only deals with a single server case and individual removals. Two different approaches are considered. For the stable case, the matrix–analytic formalism is used to study the joint distribution of the service facility and the retrial pool. The approximation by more simple infinite retrial model is also proved. In the overloading case we study the transient behaviour of the trajectory of the suitably normalized retrial queue and the long-run behaviour of the number of busy servers. The method of investigation in this case is based on the averaging principle for switching processes

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, 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

Queues: Flows, Systems, Networks

В сборнике излагаются новые результаты научных исследований в области разработки и оптимизации моделей процессов передачи информации в телекоммуникационных сетях с использованием аппарата теории систем и сетей массового обслуживания. Предназначен специалистам в области вероятностного анализа, случайных процессов, математического моделирования, и математической статистики, а также специалистам в области проектирования и эксплуатации сетей связи и компьютерных сетей