Особенности программной реализации численно-аналитического метода расчёта моделей нестационарных систем обслуживания

A numerical-analytical method for non-stationary queueing systems models computation is presented. The solution of Chapman—Kolmogorov equations is found in the analytical form. The algorithm and its practical implementation with Java language are discussed. Computation time and results precision for the presented method and the Runge—Kutta type method used in Matlab are compared.В статье описывается численно-аналитический метод расчёта моделей нестационарных систем обслуживания. Находится решение системы уравнений Чепмена — Колмогорова в аналитическом виде. Приводится алгоритм построения решения и особенности его программной реализации на языке Java. Также приводятся результаты сравнения времени работы и точности выходных данных метода со временем работы и точностью выходных данных численного метода типа Рунге — Кутты, который используется в Matlab для решения аналогичных задач

Нейросетевая аппроксимация характеристик многоканальных немарковских систем массового обслуживания

It is proposed to use a neural network to calculate an approximation of the probabilistic-time characteristics of multichannel queuing systems (QS) with a "warm-up" and the unlimited capacity of the queue. From the results of numerical experiments, we observe a significant reduction in the complexity of computing probabilistic-time characteristics of the multi-channel QS with "warm-up" with minor errors of calculation of characteristics, compared with the numerical iterative algorithms. The advisability of the use of Bayesian regularization method for training a neural network and the best number of neurons are shown.Предлагается использование нейросетевой аппроксимации для расчета вероятностно-временных характеристик многоканальных систем массового обслуживания (СМО) и неограниченной емкостью очереди. Приводятся результаты численных экспериментов, показывающие, что по сравнению с численными итерационными алгоритмами достигается существенное снижение трудоемкости вычислений вероятностно-временных характеристик многоканальных СМО с «разогревом» при незначительной погрешности расчета характеристик. Обоснованы целесообразность применения метода Байесовской регуляризации для обучения нейросети и наилучшее число нейронов

Комплекс моделей нестационарных систем обслуживания с распределениями фазового типа

A complex of new models of non-stationary queuing systems with finite source is presented. In contrast to traditional models of queuing theory the proposed models allow to describe the processes of customers servicing in the specified time interval under general assumptions on the time distribution between customer arrival and service. The article presents the principles of such models development, their graphical interpretation and formulae for computation of probabilistic and time characteristics as well as Chapman—Kolmogorov differential equations systems.Представлен комплекс нового класса моделей нестационарных систем обслуживания с источником  конечного числа заявок. В отличие от традиционных моделей теории массового обслуживания они позволяют моделировать процессы обслуживания  на заданном (директивном) временном интервале при общих предположениях о законах распределения временных интервалов между поступлениями и обслуживаниями заявок. Определены принципы построения этих моделей, их графическая интерпретация, расчет вероятностно-временных характеристик, выведены системы дифференциальных уравнений Чепмена — Колмогорова

Efficient duration modelling in the hierarchical hidden semi-Markov models and their applications

Modeling patterns in temporal data has arisen as an important problem in engineering and science. This has led to the popularity of several dynamic models, in particular the renowned hidden Markov model (HMM) [Rabiner, 1989]. Despite its widespread success in many cases, the standard HMM often fails to model more complex data whose elements are correlated hierarchically or over a long period. Such problems are, however, frequently encountered in practice. Existing efforts to overcome this weakness often address either one of these two aspects separately, mainly due to computational intractability. Motivated by this modeling challenge in many real world problems, in particular, for video surveillance and segmentation, this thesis aims to develop tractable probabilistic models that can jointly model duration and hierarchical information in a unified framework. We believe that jointly exploiting statistical strength from both properties will lead to more accurate and robust models for the needed task. To tackle the modeling aspect, we base our work on an intersection between dynamic graphical models and statistics of lifetime modeling. Realizing that the key bottleneck found in the existing works lies in the choice of the distribution for a state, we have successfully integrated the discrete Coxian distribution [Cox, 1955], a special class of phase-type distributions, into the HMM to form a novel and powerful stochastic model termed as the Coxian Hidden Semi-Markov Model (CxHSMM). We show that this model can still be expressed as a dynamic Bayesian network, and inference and learning can be derived analytically.Most importantly, it has four superior features over existing semi-Markov modelling: the parameter space is compact, computation is fast (almost the same as the HMM), close-formed estimation can be derived, and the Coxian is flexible enough to approximate a large class of distributions. Next, we exploit hierarchical decomposition in the data by borrowing analogy from the hierarchical hidden Markov model in [Fine et al., 1998, Bui et al., 2004] and introduce a new type of shallow structured graphical model that combines both duration and hierarchical modelling into a unified framework, termed the Coxian Switching Hidden Semi-Markov Models (CxSHSMM). The top layer is a Markov sequence of switching variables, while the bottom layer is a sequence of concatenated CxHSMMs whose parameters are determined by the switching variable at the top. Again, we provide a thorough analysis along with inference and learning machinery. We also show that semi-Markov models with arbitrary depth structure can easily be developed. In all cases we further address two practical issues: missing observations to unstable tracking and the use of partially labelled data to improve training accuracy. Motivated by real-world problems, our application contribution is a framework to recognize complex activities of daily livings (ADLs) and detect anomalies to provide better intelligent caring services for the elderly.Coarser activities with self duration distributions are represented using the CxHSMM. Complex activities are made of a sequence of coarser activities and represented at the top level in the CxSHSMM. Intensive experiments are conducted to evaluate our solutions against existing methods. In many cases, the superiority of the joint modeling and the Coxian parameterization over traditional methods is confirmed. The robustness of our proposed models is further demonstrated in a series of more challenging experiments, in which the tracking is often lost and activities considerably overlap. Our final contribution is an application of the switching Coxian model to segment education-oriented videos into coherent topical units. Our results again demonstrate such segmentation processes can benefit greatly from the joint modeling of duration and hierarchy

Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields

This book covers a broad range of research results in the field of Markov and Semi-Markov chains, processes, systems and related emerging fields. The authors of the included research papers are well-known researchers in their field. The book presents the state-of-the-art and ideas for further research for theorists in the fields. Nonetheless, it also provides straightforwardly applicable results for diverse areas of practitioners

Entropy Maximisation and Queues With or Without Balking. An investigation into the impact of generalised maximum entropy solutions on the study of queues with or without arrival balking and their applications to congestion management in communication networks.

An investigation into the impact of generalised maximum entropy solutions on the study of queues with or without arrival balking and their applications to congestion management in communication networks Keywords: Queues, Balking, Maximum Entropy (ME) Principle, Global Balance (GB), Queue Length Distribution (QLD), Generalised Geometric (GGeo), Generalised Exponential (GE), Generalised Discrete Half Normal (GdHN), Congestion Management, Packet Dropping Policy (PDP) Generalisations to links between discrete least biased (i.e. maximum entropy (ME)) distribution inferences and Markov chains are conjectured towards the performance modelling, analysis and prediction of general, single server queues with or without arrival balking. New ME solutions, namely the generalised discrete Half Normal (GdHN) and truncated GdHN (GdHNT) distributions are characterised, subject to appropriate mean value constraints, for inferences of stationary discrete state probability distributions. Moreover, a closed form global balance (GB) solution is derived for the queue length distribution (QLD) of the M/GE/1/K queue subject to extended Morse balking, characterised by a Poisson prospective arrival process, i.i.d. generalised exponential (GE) service times and finite capacity, K. In this context, based on comprehensive numerical experimentation, the latter GB solution is conjectured to be a special case of the GdHNT ME distribution. ii Owing to the appropriate operational properties of the M/GE/1/K queue subject to extended Morse balking, this queueing system is applied as an ME performance model of Internet Protocol (IP)-based communication network nodes featuring static or dynamic packet dropping congestion management schemes. A performance evaluation study in terms of the model’s delay is carried out. Subsequently, the QLD’s of the GE/GE/1/K censored queue subject to extended Morse balking under three different composite batch balking and batch blocking policies are solved via the technique of GB. Following comprehensive numerical experimentation, the latter QLD’s are also conjectured to be special cases of the GdHNT. Limitations of this work and open problems which have arisen are included after the conclusion

Advanced Trends in Wireless Communications

Physical limitations on wireless communication channels impose huge challenges to reliable communication. Bandwidth limitations, propagation loss, noise and interference make the wireless channel a narrow pipe that does not readily accommodate rapid flow of data. Thus, researches aim to design systems that are suitable to operate in such channels, in order to have high performance quality of service. Also, the mobility of the communication systems requires further investigations to reduce the complexity and the power consumption of the receiver. This book aims to provide highlights of the current research in the field of wireless communications. The subjects discussed are very valuable to communication researchers rather than researchers in the wireless related areas. The book chapters cover a wide range of wireless communication topics

Modelling activities at a neurological rehabilitation unit

A queuing model is developed for the neurological rehabilitation unit at Rookwood Hospital in Cardiff. Arrivals at the queuing system are represented by patient referrals and service is represented by patient length of stay (typically five months). Since there are often delays to discharge, length of stay is partitioned into two parts: admission until date ready for discharge (modelled by Coxian phase-type distribution) and date ready for discharge until ultimate discharge (modelled by exponential distribution). The attributes of patients (such as age, gender, diagnosis etc) are taken into account since they affect these distributions. A computer program has been developed to solve this multi-server (21 bed) queuing system to produce steady-state probabilities and various performance measures. However, early on in the project it became apparent that the intensity of treatment received by patients has an effect on the time, from admission, until they are ready for discharge. That is, the service rates of the Coxian distribution are dependent on the amount of therapy received over time. This directly relates to the amount of treatment allocated in the weekly timetables. For the physiotherapy department, these take about eight hours to produce each week by hand. In order to ask the valuable what-if questions that relate to treatment intensity, it is therefore necessary to produce an automated scheduling program that replicates the manual assignment of therapy. The quality of timetables produced using this program was, in fact, considerably better than its alternative and so replaced the by-hand approach. Other benefits are more clinical time (since less employee input is required)and a convenient output of data and performance measures that are required for audit purposes. Once the model is constructed a number of relevant hypothetical scenarios are considered. Such as, what if delays to discharge are reduced by 50%? Also, through the scheduling program, the effect of changes to the composition of staff or therapy sessions can be evaluated, for example, what if the number of therapists is increased by one third? The effects of such measures are analysed by studying performance measures (such as throughput and occupancy) and the associated costs