Findings about the two-state BMMPP for modeling point processes in reliability and queueing systems

The Batch Markov Modulated Poisson Process (BMMPP) is a subclass of the versatile Batch Markovian Arrival process (BMAP) which have been widely used for the modeling of dependent and correlated simultaneous events (as arrivals, failures or risk events, real-time multimedia communications). Both theoretical and applied aspects are examined in this paper. On one hand, the identifiability of the stationary BMMPP2(K) is proven, where K is the maximum batch size. This is a powerful result when inferential tasks related to real data sets are carried out. On the other hand, some findings concerning the correlation and autocorrelation structures are provided.The first and second authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, research project ECO2015-66593-P. The Third author acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, research project MTM2015-65915-R; and also from Junta de Andalucía, and BBVA Fundation, research project P11- FQM-7603 and FQM-32

Mathematical Analysis of Queue with Phase Service: An Overview

We discuss various aspects of phase service queueing models. A large number of models have been developed in the area of queueing theory incorporating the concept of phase service. These phase service queueing models have been investigated for resolving the congestion problems of many day-to-day as well as industrial scenarios. In this survey paper, an attempt has been made to review the work done by the prominent researchers on the phase service queues and their applications in several realistic queueing situations. The methodology used by several researchers for solving various phase service queueing models has also been described. We have classified the related literature based on modeling and methodological concepts. The main objective of present paper is to provide relevant information to the system analysts, managers, and industry people who are interested in using queueing theory to model congestion problems wherein the phase type services are prevalent

Analysis of queueing models with batch service

This dissertation is the result of my research work at the SMACS research group (Department of Telecommunications and Information Processing, Ghent University) and it concerns the analysis of queueing models with batch service. A queueing model basically is a mathematical abstraction of a situation where customers arrive and queue up until they receive some kind of service. These phenomena are omnipresent in real life: people waiting at a counter of a post office or bank, people in the waiting room of a doctor, airplanes waiting to take off, people waiting until they get connected with the call center, data packets which are temporarily stored into a buffer until the transmisssion channel is available, et cetera. The analysis of queueing models constitutes the subject of the applied mathematical discipline called queueing theory and amounts to answering questions such as “How many customers are waiting on average?”, “How long do customers have to wait?”, “Is there a large variation on the waiting time?”, “What is the probability that data packets are lost due to a full buffer?”, “What is the probability that a customer suffers a lengthy delay?”, et cetera. In queueing theory, the number of customers and their waiting time are often denominated by respectively buffer content and customer delay. In addition, the probability that a quantity, such as the buffer content or customer delay, is very large or lengthy, is generally called a tail probability. The models we investigate throughout this dissertation have in common that customers can be served in batches, meaning that several customers can be served simultaneously. An elevator can be viewed as a classic example, as several people can be transported simultaneously to another floor. Also, in a variety of production or transport processes several goods can be processed together. Furthermore, in quality control, classification of items as good or bad can often be achieved more economically by examining the items in groups rather than individually. If the result of a group test is good, all items within it can then be classified as good, whereas one or more items are bad in the opposite case, where the items can then be retested by considering smaller groups. Group testing is especially of importance when the percentage of bad items is small. In addition, in telecommunications networks, packets with the same destination and quality of service (QoS) requirements are often aggregated into so-called bursts and these bursts are transmitted over the network. This is mainly done for efficiency reasons, since only one header per aggregated burst has to be constructed, instead of one header per single information unit, thus leading to an increased throughput. Technologies using packet aggregation include for instance Optical burst switched (OBS) networks and IEEE 802.11n wireless local area networks (WLANs). An inherent aspect of batch service is that newly arriving customers cannot join the ongoing service, even if there is free capacity (we denominate the maximum number of customers that can be served simultaneously by server capacity). For instance, an arriving person cannot enter an elevator that has just left, even if space is available. This person has to wait until the elevator has transported its occupants to their requested floors and has returned, which might take a long time in high buildings. In view of this, it is of importance to take a well-considered decision when the server becomes available and finds less customers than it can serve in theory. This decision is called the service policy. A whole spectrum of service policies exist. The server could, for instance, start serving the already present customers immediately. Although the present customers benefit from this approach, capacity is wasted: customers that arrive later cannot join the ongoing service. An alternative for this so-called immediate-batch service policy is the full-batch service policy. In this case, the available server postpones service until the number of present customers reaches or exceeds the server capacity, which, in turn, has a negative effect on the delay of the customers waiting to form a full batch (postponing delay). The threshold-based policy is a kind of compromise between immediate-batch service policy and full-batch service policy. When the number of present customers is below some service threshold, service is postponed, whereas service is initiated when the number of present customers reaches or exceeds this threshold. It is important to realize that even with this compromise, long postponing delays are possible. Therefore, in this dissertation, we combine a thresholdbased policy with a timer mechanism that avoids excessive postponing delays. The purpose of this dissertation is to calculate a large spectrum of performance measures, which enable to evaluate a broad set of situations with batch service and aid in selecting an efficient service policy. The studied performance measures are moments, such as the mean value and variance, and tail probabilities of the buffer content and the customer delay. This dissertation is structured as follows. In chapter 1, we motivate our work and we introduce crucial concepts such as probability generating functions (PGFs), whose useful properties are frequently relied upon throughout the analysis. Then we deduce moments and tail probabilities of the buffer content in chapter 2. The resulting formulas still contain unknown probabilities that have to be calculated numerically. As this might become unfeasible in some cases, we compute in chapter 3 approximations for the buffer content. Next, moments and tail probabilities of the customer delay are covered in respectively chapters 4 and 5. In order to analyze the moments, we conceive the customer delay as the sum of two non-overlapping parts, whereas for the tail probabilities, it turns out to be more convenient to interpret the delay as the maximum of two time periods. Further, in real life the customer arrival process often exhibits some kind of dependency. For instance, if a large amount of customers have recently arrived, it is likely that many customers arrive in the near future, as it might be an indication of a peak moment. Therefore, we investigate in chapter 6 the influence of dependency in the arrival process on the behaviour of batch-service phenomena and on the selection of an efficient service policy. Finally, the main contributions are summarized in chapter 7

The power-series algorithm:A numerical approach to Markov processes

Abstract: The development of computer and communication networks and flexible manufacturing systems has led to new and interesting multidimensional queueing models. The Power-Series Algorithm is a numerical method to analyze and optimize the performance of such models. In this thesis, the applicability of the algorithm is extended. This is illustrated by introducing and analyzing a wide class of queueing networks with very general dependencies between the different queues. The theoretical basis of the algorithm is strengthened by proving analyticity of the steady-state distribution in light traffic and finding remedies for previous imperfections of the method. Applying similar ideas to the transient distribution renders new analyticity results. Various aspects of Markov processes, analytic functions and extrapolation methods are reviewed, necessary for a thorough understanding and efficient implementation of the Power-Series Algorithm.

Modelling activities in a Critical Care Unit

The Critical Care Unit (CCU) is the sector of the hospital where, as the name suggests, critically ill patients receive treatment. The main aim of this research is to identify and apply suitable Operational Research techniques to model patient flow in the CCU at the University Hospital of Wales, Cardiff. The Operational Research techniques employed in this thesis include queueing theory and simulation. These methods have been utilised previously in the field of healthcare with much success. The thesis begins by considering two aspects of queueing theory, namely batch service queueing theory and batch arrival queueing theory. The latter of these is utilised to model patient flow within the CCU. Although queueing theory may be used as a good approximation to activities in the Unit, it does not incorporate all aspects of real-life. Thus discrete-event simulation is suggested as an alternative approach. Two types of statistical analysis, CART and Regression, are applied to both length of stay and mortality variables. The results from these statistical tests are compiled and investigated in more depth. Finally, a discrete event simulation model is built in Visual Basic for Applications, for Microsoft Excel. This simulation model incorporates many of the complexities of a CCU, such as patient priority and cancellation of scheduled patients if all beds on the Unit are occupied. The model is then used to test various "what-if type" scenarios, including the possibility of funding additional beds, the concept of ring-fencing of beds for different levels of care, and the likely effect of reducing the impact of bed-blocking

A study of current and possible future industrial engineering methodologies used to increase energy effieciency

Thesis (MScEng)--Stellenbosch University, 2012.ENGLISH ABSTRACT: Energy-related costs are increasing steadily. This is especially true in South Africa where we have been dealing with an energy crisis during the past couple of years. The increase in energy-related costs puts energy dependent industries under financial pressure. It is therefore imperative to find ways to improve the efficiency with which energy is being consumed in order to decrease the amount of money that has to be allocated to energy costs. The efficient consumption of energy at a facility is crucial and to increase that efficiency, Energy Management Programs (EMPs) should be implemented. An important component of EMPs is ascertaining the current energy consumption of a facility in order to identify areas where possible improvements can be made. This is done by completing an energy audit at the facility. After the energy audit has been completed and Energy Conservation Methods (ECMs) have been identified, the implementation of these methods should commence. The aim of this study is to determine how Industrial Engineering (IE) methods can play a more integral role in making South Africa more energy efficient. As part of this study, research was done to identify current EMCs being implemented in different areas and for different equipment in facilities. This information was compared to IE methods to identify the possible relationship between the ECMs and IE methods. Content analyses were completed on both IE and energy efficiency corpora using the Content Analysis Toolkit (CAT) program. These analyses identified important topics in these corpora and correlations between these topics in order to show correlations between the IE and energy efficiency fields. The most significant correlations identified, were between statistical methods and various energy efficiency topics. A case study was completed at a company in the Western Cape that manufactures electronic and integrated circuit products to implement the relevant ECMs. As part of the case study, an energy audit was completed at the facility. The implementation of a number of the ECMs has shown reductions in the daily kilowatt hours (kWh) consumptions. These reductions were obtained through the implementation of a Shut Down Management program, which highlights the importance of management in an energy conservation project. The application of optimisation algorithms for energy efficient design was examined through the optimisation of lighting design, using a Genetic Algorithm. It was found that a Genetic Algorithm is applicable to lighting design but requires further refinement in order to generate the most optimal design solutions.AFRIKAANSE OPSOMMING: Kostes verbonde aan energieverbruik is voortdurend besig om toe te neem.Dit is veral relevant in Suid-Afrika waar ons tans ’n energiekrisis beleef. Hierdie toename in energieverwante kostes plaas energie-afhanklike industrieë onder groot finansiële druk. Dit is daarom belangrik om maniere te vind om energieverbruik meer effektief te maak sodat die bedrag geld wat aan energieverwante kostes toegestaan word, verminder kan word. Effektiewe energieverbruik by ’n fasiliteit is kritiek en om hierdie effektiwiteit te verbeter behoort ’n energiebestuursprogram by die fasiliteit geïmplimenteer te word. ’n Belangrike komponent van energiebestuursprogramme is die bepaling van die huidige energieverbruik en dit word gebruik om die areas te identifiseer waar moontlike verbeteringe aangebring kan word. Die energieverbruik word bepaal deur ’n energie-oudit. Nadat die energie-oudit voltooi en die energiebesparingsmetodes bepaal is, moet hierdie metodes by die fasiliteit geïmplementeer word. Hierdie studie probeer vasstel hoe bedryfsingenieurswesemetodes ‘n groter rol kan speel in die proses om Suid-Afrika meer energie-effektief te maak. Navorsing is gedoen oor energiebesparingsmetodes wat in verskillende areas en vir verskillende toerusting in fasiliteite geïmplementeer word. Hierdie inligting is daarna vergelyk met bedryfsingenieurswesemetodes om juis die moontlike verhouding tussen hierdie twee tipe metodes te identifiseer. Analises was gedoen in bedryfsingenieurswese en energie-effektiwiteitskorpusse met die gebruik van die ‘Content Analysis Toolkit’ program. Belangrike onderwerpe en verwantskappe tussen hierdie onderwerpe in die korpusse is identifiseer om sodoende korrelasies tussen die bedryfsingenieurswese- en energie-effektiwiteitsveld uit te lig. Die mees betekenisvolle korrelasies was tussen statistiese metodes en verskeie energieeffektiwiteitsonderwerpe identifiseer. ’n Gevallestudie is by ’n maatskappy in die Wes-Kaap wat geïntegreerde elektroniese stroombane vervaardig gedoen, om die relevante energiebesparingsmetodes te implementeer. ’n Energie-oudit is as deel van die gevallestudie by die fasiliteit gedoen. Die aantal energiebesparingsmetodes wat wel geïmplementeer is, het ’n verlaging in die kilowatture (kWh) teweeggebring. Hierdie verlagings is verkry deur die implementering van ’n afskakelingsbestuursprogram wat die belangrikheid van bestuur in ’n energiebesparingsprogram uitlig. Die toepaslikheid van optimiseringsalgoritmes vir energie-effektiewe ontwerp is getoets deur die optimisering van ’n liguitlegontwerp met behulp van ’n genetiese algoritme. Daar is gevind dat ’n genetiese algoritme wel toegepas kan word, maar dat dit verbeteringe benodig

Modelling critical care unit activities through queueing theory

Critical Care Units (CCUs) are one of the most complex and expensive of all medical resources and hospital managers are challenged to meet the demand for critical care services with adequate capacity. The pressure on critical care beds is continuously increasing as new medical equipment provides the opportunity to save more patients lives. It is therefore crucial that beds are managed well and used efficiently. This thesis describes two major projects, the first undertaken in conjunction with the CCU at the University Hospital of Wales in Cardiff (UHW); and the second with two CCUs from the Aneurin Bevan Health Board. In the first project data has been analysed to determine the flow of patients through the Unit. Admissions to CCUs were categorised under two headings: emergency, and elective. The length of stay in the CCU is heavily dependent on the admission category. In this thesis, both computer simulation and theoretical queueing models have been considered, which show how improvements in bed management may be achieved by considering these two categories of patients separately. The vast majority of previous literature in this field is concerned only with steady-state conditions, whereas in reality the processes are time-dependent. This thesis goes some way to addressing this deficiency. The second project relates to work undertaken with managers from the Royal Gwent Hospital in Newport and at the Nevill Hall Hospital in Abergavenny. Data from both hospitals have been analysed to define arrival and service processes. A state-dependent theoretical queueing model has been considered which has been used to investigate the significance of combining the two units. The model has been also utilised to advise on the number of beds the new combined unit should have in order to satisfy targets quoted by the hospital managers. In the final part of the thesis, consideration has been given to the impact of collaboration, or lack thereof, between hospitals using a game theoretical approach. The effect of patient diversion has been studied. To formally investigate the impact of patients transfers, a Markov chain model of the two CCUs has been set-up, each admitting two arrival streams: namely, their own patients and transfers from other hospital. Four different models were considered and for each model the effect of targets, demand and capacity were studied. The efficiency of a system which degrades due to selfish behaviour of its agents has been measured in terms of Price of Anarchy