A MARKOVIAN TWO COMMODITY QUEUEING–INVENTORY SYSTEM WITH COMPLIMENT ITEM AND CLASSICAL RETRIAL FACILITY

This paper explores the two-commodity (TC) inventory system in which commodities are classified as major and complementary items. The system allows a customer who has purchased a free product to conduct Bernoulli trials at will. Under the Bernoulli schedule, any entering customer will quickly enter an orbit of infinite capability during the stock-out time of the major item. The arrival of a retrial customer in the system follows a classical retrial policy. These two products' re-ordering process occurs under the $(s, Q)$ and instantaneous ordering policies for the major and complimentary items, respectively. A comprehensive analysis of the retrial queue, including the system's stability and the steady-state distribution of the retrial queue with the stock levels of two commodities, is carried out. The various system operations are measured under the stability condition. Finally, numerical evidence has shown the benefits of the proposed model under different random situations

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

A study of some M[x]/G/1 type queues with random breakdowns and bernouilli schedule server vacations based on a single vacation policy

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Queueing systems arise in modelling of many practical applications related to computer sciences, telecommunication networks, manufacturing and production, human computer interaction, and so on. The classical queueing system, even vacation queues or queues subject to breakdown, might not be sufficiently realistic. The purpose of this research is to extend the work done on vacation queues and on unreliable queues by studying queueing systems which take into consideration both phenomena. We study the behavior of a batch arrival queueing system with a single server, where the system is subject to random breakdowns which require a repair process, and on the other hand, the server is allowed to take a vacation after finishing a service. The breakdowns are assumed to occur while serving a customer, and when the system breaks down, it enters a repair process immediately while the customer whose service is interrupted comes back to the head of the queue waiting for the service to resume. Server vacations are assumed to follow a Bernoulli schedule under single vacation policy. We consider the above assumptions for different queueing models: queues with generalized service time, queues with two-stages of heterogeneous service, queues with a second optional service, and queues with two types of service. For all the models mentioned above, it is assumed that the service times, vacation times, and repair times all have general arbitrary distributions. Applying the supplementary variable technique, we obtain probability generating functions of queue size at a random epoch for different states of the system, and some performance measures such as the mean queue length, mean waiting time in the queue, proportion of server's idle time, and the utilization factor. The results obtained in this research, show the effect of vacation and breakdown parameters upon main performance measures of interest. These effects are also illustrated using some numerical examples and graphs.This work is funded by the Ministry of Education, Kingdom of Bahrain

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

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

On the dynamic allocation of assets subject to failure and replenishment

Problems of the dynamic allocation of assets subject to both failure and replenishment are common. We consider a problem inspired by naval search, where unmanned aerial vehicles are required to search an area of ocean for targets. The vehicles will require refuelling or rearming; this is represented by the aspects of failure and replenishment. Similar models can arise from considering problems of search and rescue, environmental monitoring, or project management. We formulate several versions of the problem, initially using the framework of a Markov decision process, bearing in mind trade-offs between real-world fidelity and mathematical tractability. We first consider models where rewards are gained independently from different tasks, before moving on to consider a specific kind of dependence in the rewards. We use a variety of mathematical techniques, including restless bandits, to formulate near-optimal policies for a slew of models. We consider and investigate the various policies through comprehensive computational modelling. For the independent case, we find that a Whittle index policy is extremely close to optimal while being computationally efficient. For the dependent formulation, we create a class of policies guaranteed to contain the optimal, parameterise the space, then choose the best from a limited set of parameters, augmenting with a single step of policy improvement. We close with some thoughts about what we have learned, considerations about applying the results presented in this thesis, and a discussion of intensifications and extensions we did not have time to consider

Discrete Event Simulations

Considered by many authors as a technique for modelling stochastic, dynamic and discretely evolving systems, this technique has gained widespread acceptance among the practitioners who want to represent and improve complex systems. Since DES is a technique applied in incredibly different areas, this book reflects many different points of view about DES, thus, all authors describe how it is understood and applied within their context of work, providing an extensive understanding of what DES is. It can be said that the name of the book itself reflects the plurality that these points of view represent. The book embraces a number of topics covering theory, methods and applications to a wide range of sectors and problem areas that have been categorised into five groups. As well as the previously explained variety of points of view concerning DES, there is one additional thing to remark about this book: its richness when talking about actual data or actual data based analysis. When most academic areas are lacking application cases, roughly the half part of the chapters included in this book deal with actual problems or at least are based on actual data. Thus, the editor firmly believes that this book will be interesting for both beginners and practitioners in the area of DES