PROBABILITY DENSITY FUNCTION OF M/G/1 QUEUES UNDER (0,K) CONTROL POLICIES: A SPECIAL CASE

In this paper we present probability density function of vacation period of M/G/1 queueing process that operates under (0,k) vacation policy, wherein the server goes on the vacation when the system becomes empty and re-opens for service immediately at the arrival of the kth customer. The number of lattice paths when last arrival is an arrival has also been derived. The transient analysis is based on approximating the general service time distribution by Coxian two-phase distribution and representing the corresponding queueing process as a lattice path. Finally the lattice path combinatorics is used to present the number of lattice paths

Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism

summary:Non-stationary behavior of departure process in a finite-buffer $M^{X}/G/1/K$-type queueing model with batch arrivals, in which a threshold-type waking up $N$-policy is implemented, is studied. According to this policy, after each idle time a new busy period is being started with the $N$th message occurrence, where the threshold value $N$ is fixed. Using the analytical approach based on the idea of an embedded Markov chain, integral equations, continuous total probability law, renewal theory and linear algebra, a compact-form representation for the mixed double transform (probability generating function of the Laplace transform) of the probability distribution of the number of messages completely served up to fixed time $t$ is obtained. The considered queueing system has potential applications in modeling nodes of wireless sensor networks (WSNs) with battery saving mechanism based on threshold-type waking up of the radio. An illustrating simulational and numerical study is attached

Fluctuation Analysis in a Queue with (L,N)-Policy and Secondary Maintenance. Discrete Time Parameter Process

This paper generalizes numerous classes of queues with vacationing servers. In our model, a server not only leaves the system, but he services packets of jobs at a secondary facility up until the total number of single jobs exceeds a specific threshold. The strategy of server processes is represented for different state of queue.We use various techniques (including fluctuation analysis) to deliver explicit formulas for the queueing processwith discrete time parameters.We also utilize some game-theoretic principles (namely sequential games) to efficiently construct our model.Обобщены многочисленные классы очередей с простаивающими серверами. В предлагаемой модели сервер не просто выходит из системы, а обрабатывает пакеты задач в фоновом режиме до тех пор, пока общее число одиночных заданий не превысит определенный порог. Представлена стратегия работы сервера при различных состояниях очереди. Использованы различные методики получения явных формул (включая флуктуационный анализ) для процесса массового обслуживания с дискретным временем, а также некоторые подходы теории игр (последовательные игры) для эффективного конструирования модели.Узагальнено багатoчисельні класи черг серверів що простоюють. В запропонованій моделі сервер не просто виходить з системи, а обробляє пакети задач у фоновому режимі до тих пір, поки загальне число поодиноких завдань не перевищить означений поріг. Представлено стратегію роботи сервера в умовах різного стану черги. Використано різні методики отримання явних формул (включаючи флуктуаційний аналіз) для процесу масового обслуговування з дискретним часом, а також деякі підходи теорії гри (послідовні ігри) для ефективного конструювання моделі

Analysis of some batch arrival queueing systems with balking, reneging, random breakdowns, fluctuating modes of service and Bernoulli schedulled server vacations.

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonThe purpose of this research is to investigate and analyse some batch arrival queueing systems with Bernoulli scheduled vacation process and single server providing service. The study aims to explore and extend the work done on vacation and unreliable queues with a combination of assumptions like balking and re-service, reneging during vacations, time homogeneous random breakdowns and fluctuating modes of service. We study the steady state properties, and also transient behaviour of such queueing systems. Due to vacations the arriving units already in the system may abandon the system without receiving any service (reneging). Customers may decide not to join the queue when the server is in either working or vacation state (balking). We study this phenomenon in the framework of two models; a single server with two types of parallel services and two stages of service. The model is further extended with re-service offered instantaneously. Units which join the queue but leave without service upon the absence of the server; especially due to vacation is quite a natural phenomenon. We study this reneging behaviour in a queueing process with a single server in the context of Markovian and non-Markovian service time distribution. Arrivals are in batches while each customer can take the decision to renege independently. The non-Markovian model is further extended considering service time to follow a Gamma distribution and arrivals are due to Geometric distribution. The closed-form solutions are derived in all the cases. Among other causes of service interruptions, one prime cause is breakdowns. We consider breakdowns to occur both in idle and working state of the server. In this queueing system the transient and steady state analysis are both investigated. Applying the supplementary variable technique, we obtain the probability generating function of queue size at random epoch for the different states of the system and also derive some performance measures like probability of server‟s idle time, utilization factor, mean queue length and mean waiting time. The effect of the parameters on some of the main performance measures is illustrated by numerical examples to validate the analytical results obtained in the study. The Mathematica 10 software has been used to provide the numerical results and presentation of the effects of some performance measures through plots and graphs

Stochastic Processes with Applications

Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included

Fluctuation Analysis in a Queue with (L,N)-Policy and Secondary Maintenance. Continuous Time Parameter Process

Investigation results on a queueing system with an auxiliary maintenance process launched during server’s vacancy period and initiated by the exhausted queue are presented. The server operation strategy at different queue conditions is proposed. Time-sensitive analysis to investigate the queueing process at arbitrary periods of time is used. The results are obtained in explicit forms for several related models. Computational examples illustrate their analytical tractability. Various performance measures (the buffer load, switchover rate, and the number of jobs processed per unit time) are introduced and optimization problems are discussed.Представлены результаты исследований систем очередей с дополнительным процессом обслуживания, который запускается в периоды простоя сервера и инициируется заполненной очередью. Предложена стратегия работы сервера при различных состояниях очереди. Использован времязависимый анализ для исследования процессов в системах очередей в произвольные периоды времени. Результаты получены в явной форме для нескольких моделей. Приведены примеры расчета, свидетельствующие о возможности их аналитической трактовки. Введены различные критерии производительности (загрузка буферов, скорость переключения и число заданий, обрабатываемых в единицу времени) и рассмотрены проблемы оптимизации.Наведено результати досліджень систем черг з доповняльним процесом обслуговування, який запускається в періоди простоїв сервера та ініціюється заповненою чергою. Запропоновано стратегію роботи сервера при різних станах черги. Використано часозалежний аналіз для дослідження процесів у системах черг в довільні періоди часу. Результати отримано в явній формі для декількох моделей. Наведено приклади розрахунків, які свідчать про можливість їхньої аналітичної трактовки. Введено різні критерії продуктивності (загрузка буферів, швидкість переключання та число завдань, оброблювальних за одиницею часу) і розглянуто проблеми оптимізації

Dependence-driven techniques in system design

Burstiness in workloads is often found in multi-tier architectures, storage systems, and communication networks. This feature is extremely important in system design because it can significantly degrade system performance and availability. This dissertation focuses on how to use knowledge of burstiness to develop new techniques and tools for performance prediction, scheduling, and resource allocation under bursty workload conditions.;For multi-tier enterprise systems, burstiness in the service times is catastrophic for performance. Via detailed experimentation, we identify the cause of performance degradation on the persistent bottleneck switch among various servers. This results in an unstable behavior that cannot be captured by existing capacity planning models. In this dissertation, beyond identifying the cause and effects of bottleneck switch in multi-tier systems, we also propose modifications to the classic TPC-W benchmark to emulate bursty arrivals in multi-tier systems.;This dissertation also demonstrates how burstiness can be used to improve system performance. Two dependence-driven scheduling policies, SWAP and ALoC, are developed. These general scheduling policies counteract burstiness in workloads and maintain high availability by delaying selected requests that contribute to burstiness. Extensive experiments show that both SWAP and ALoC achieve good estimates of service times based on the knowledge of burstiness in the service process. as a result, SWAP successfully approximates the shortest job first (SJF) scheduling without requiring a priori information of job service times. ALoC adaptively controls system load by infinitely delaying only a small fraction of the incoming requests.;The knowledge of burstiness can also be used to forecast the length of idle intervals in storage systems. In practice, background activities are scheduled during system idle times. The scheduling of background jobs is crucial in terms of the performance degradation of foreground jobs and the utilization of idle times. In this dissertation, new background scheduling schemes are designed to determine when and for how long idle times can be used for serving background jobs, without violating predefined performance targets of foreground jobs. Extensive trace-driven simulation results illustrate that the proposed schemes are effective and robust in a wide range of system conditions. Furthermore, if there is burstiness within idle times, then maintenance features like disk scrubbing and intra-disk data redundancy can be successfully scheduled as background activities during idle times