Стационарные характеристики многоканальной неоднородной системы с FCFS орбитой и пороговым управлением

In the paper we deal with a Markovian queueing system with heterogeneous servers and constant retrial rate. The system operates under a threshold policy. The system is described by quasi-birth-and-death process with infinitesimal matrix depending on the threshold levels. Using a matrix-geometric approach we perform a stationary analysis of the system, derive expressions for the mean performance measures and formulas for optimal threshold levels.В работе рассматривается марковская система массового обслуживания с несколькими неоднородными приборами. Включение приборов производится в соответствии с пороговой политикой управления. Заявки, которые не получили обслуживания, направляются на орбиту, где они через случайное время повторяют попытку занять прибор. Заявки на орбите формируют очередь с FCFS дисциплиной обслуживания. Система описывается обобщённым процессом размножения и гибели с большим числом пограничных состояний. В работе проводится анализ системы в стационарном режиме, выводятся матричные выражения для вычисления средних характеристик производительности системы, а также формулы оптимальных пороговых уровней

Распределения времени ожидания в системе с FCFS орбитой и пороговым управлением

We perform a waiting time analysis of a retrial queue with several heterogeneous servers and threshold control policy. The blocked customers are dispatched to the infinitely large orbit with FCFS service discipline. The system is analyzed in steady-state by deriving expressions for the Laplace transforms of the waiting time, the probability generating functions for the number of retrials made by a customer and for various moments of corresponding quantities. Some illustrative numerical examples and their discussion are presented.Проводится анализ времени ожидания в марковской системе с повторными заявками, несколькими неоднородными приборами и пороговой политикой управления. Блокированные заявки ожидают своей очереди на орбите бесконечной ёмкости с FCFS дисциплиной обслуживания. Система исследуется в стационарном режиме. Выводятся выражения для преобразований Лапласа времени ожидания, производящих функций числа повторных попыток до момента поступления на прибор, а также для моментов рассматриваемых величин. Приводятся численные примеры с иллюстрациями и их обсуждение

To analysis of a two-buffer queuing system with cross-type service and additional penalties

The concept of cloud computing was created to better preserve user privacy and data storage security. However, the resources allocated for processing this data must be optimally allocated. The problem of optimal resource management in the loud computing environment is described in many scientific publications. To solve the problems of optimality of the distribution of resources of systems, you can use the construction and analysis of QS. We conduct an analysis of two-buffer queuing system with cross-type service and additional penalties, based on the literature reviewed in the article. This allows us to assess how suitable the model presented in the article is for application to cloud computing. For a given system different options for selecting applications from queues are possible, queue numbers, therefore, the intensities of transitions between the states of the system will change. For this, the system has a choice policy that allows the system to decide how to behave depending on its state. There are four components of such selection management models, which is a stationary policy for selecting a queue number to service a ticket on a vacated virtual machine each time immediately before service ends. A simulation model was built for numerical analysis. The results obtained indicate that requests are practically not delayed in the queue of the presented QS, and therefore the policy for a given model can be considered optimal. Although Poisson flow is the simplest for simulation, it is quite acceptable for performance evaluation. In the future, it is planned to conduct several more experiments for different values of the intensity of requests and various types of incoming flows.Концепция облачных вычислений была создана для улучшения конфиденциальности пользователей и безопасности хранения данных. Однако ресурсы, выделяемые для обработки этих данных, должны быть правильно распределены. Проблема оптимального управления ресурсами в среде облачных вычислений описана во многих научных публикациях. Для решения задач оптимальности распределения ресурсов систем можно использовать построение и анализ характеристик СМО. Авторами проведён анализ системы массового обслуживания с двумя очередями с кросс-типом обслуживания и дополнительными штрафами, который основывается на литературных источниках, рассмотренных в статье. Это позволяет нам оценить, насколько модель, представленная в статье, подходит для применения в облачных вычислениях. Данная система предполагает разные варианты выбора заявок из очередей, номеров очередей, следовательно, интенсивности переходов между состояниями системы будут меняться. Для этого предлагается политика выбора, которая позволяет системе решать, как себя вести в зависимости от своего состояния. Используются четыре компоненты модели управления выбором, которые представляют собой стационарную политику для определения номера очереди, из которой будет взята заявка на обслуживание. Данный выбор происходит каждый раз непосредственно перед окончанием обслуживания. Для численного анализа построена имитационная модель

Analysis of the busy period in threshold control system

Consideration was given to the controllable Markov queuing system with nonhomogeneous servers and threshold policy of server activation depending on the queue length. By the busy period is meant the time interval between the instant of customer arrival to the empty system and the instant of service completion when the system again becomes free. The system busy period and the number of serviced customers were analyzed. Recurrent relations for the distribution density in terms of the Laplace transform and generating function, as well as formulas for calculation of the corresponding arbitrary-order moments, were obtained. © 2010 Pleiades Publishing, Ltd

Analysis of the busy period in threshold control system

Consideration was given to the controllable Markov queuing system with nonhomogeneous servers and threshold policy of server activation depending on the queue length. By the busy period is meant the time interval between the instant of customer arrival to the empty system and the instant of service completion when the system again becomes free. The system busy period and the number of serviced customers were analyzed. Recurrent relations for the distribution density in terms of the Laplace transform and generating function, as well as formulas for calculation of the corresponding arbitrary-order moments, were obtained. © 2010 Pleiades Publishing, Ltd

Numerical analysis of optimal control for system with nonuniform units

For construction of the optimal service disciplines, different numerical methods of the Howard iterative algorithm type are often used. However because of usually high dimensionality of real problems, direct application of such the calculation approaches is often not efficient. With the help of the known Howard algorithm, numerical study is carried out of optimal service disciplines by a system with several nonuniform devices with respect to the minimization criterion of average stationary demands number in the system

Numerical analysis of optimal control for system with nonuniform units

For construction of the optimal service disciplines, different numerical methods of the Howard iterative algorithm type are often used. However because of usually high dimensionality of real problems, direct application of such the calculation approaches is often not efficient. With the help of the known Howard algorithm, numerical study is carried out of optimal service disciplines by a system with several nonuniform devices with respect to the minimization criterion of average stationary demands number in the system

Numerical Study of the Optimal Control of a System with Heterogeneous Servers

For a system with several heterogeneous servers, the well-known Howard algorithm was used to study numerically whether the optimal servicing disciplines satisfy the criterion for minimum stationary number of customers in system

On the slow server problem

In this paper the problem of optimal control over a Markov queueing system with heterogeneous servers and a joint queue is considered, which is also known in the literature as "the slow server problem." The classical model is generalized here to the case with delay and call serving penalties. It is proved here that the optimal control policy for servers' activating is of monotonic and threshold nature. © 2009 Pleiades Publishing, Ltd

On the slow server problem

In this paper the problem of optimal control over a Markov queueing system with heterogeneous servers and a joint queue is considered, which is also known in the literature as "the slow server problem." The classical model is generalized here to the case with delay and call serving penalties. It is proved here that the optimal control policy for servers' activating is of monotonic and threshold nature. © 2009 Pleiades Publishing, Ltd