On Queues with Interruptions and Repeat or Resumption of Service

This thesis entitled' On Queues with Interruptions and Repeat or Resumption of Service' introduces several new concepts into queues with service interruption. It is divided into Seven chapters including an introductory chapter. The following are keywords that we use in this thesis: Phase type (PH) distribution, Markovian Arrival Process (MAP), Geometric Distribution, Service Interruption, First in First out (FIFO), threshold random variable and Super threshold random variable. In the second chapter we introduce a new concept called the 'threshold random variable' which competes with interruption time to decide whether to repeat or resume the interrupted service after removal of interruptions. This notion generalizes the work reported so far in queues with service interruptions. In chapter 3 we introduce the concept of what is called 'Super threshold clock' (a random variable) which keeps track of the total interruption time of a customer during his service except when it is realized before completion of interruption in some cases to be discussed in this thesis and in other cases it exactly measures the duration of all interruptions put together. The Super threshold clock is OIl whenever the service is interrupted and is deactivated when service is rendered. Throughout this thesis the first in first out service discipline is followed except for priority queues.

Stochastic Modelling Using Markov Chains

Department of Statistics, Cochin University of Science and Technology

Queues with Customer Induced Interruption

In this thesis we have introduced and studied the notion of self interruption of service by customers. Service interruption in queueing systems have been extensively discussed in literature (see, Krishnamoorthy, Pramod and Chakravarthy [38]) for the most recent survey. So far all work reported deal with cases in which service interruptions are generated by sources other than customers. However, there are situations where interruptions are due to the customers rather than the system. Such situations are especially arise at doctors clinic, banks, reservation counter etc. Our attempt is to quantify a few of such problems. Systematically we have proceed from single server queue (in Chapter 2) to multi-server queues (Chapter 3). In Chapte 4, we have studied a very general multiserver queueing model with service interruption and protection of service phases. We also introduced customer interruption in a retrial setup (in Chapter 5). All models (from Chapter 2 to Chapter 4) that were analyzed involve 'non-preemptive priority' for interrupted customers where as in the model discussed in Chapter 5 interruption of service by customers is not encouraged. So the interrupted customers cannot access the server as long as there are primary customers in the system. In Chapter 5 we have obtained an explicit expression for the stability condition of the system. In all models analyzed in this thesis, we have assumed that no more than one interruption is allowed for a customer while in service. Since the models are not analytically tractable, a large number of numerical illustrations were given in each chapter it illustrate the working of the systems. We can extend the models discussed in this thesis to several directions. For example some of the models can be analyzed with both server induced and customer induced interruptions the results for which are not available till date. Another possible extension of work is to the case where there is no bound on the number of interruptions a customer is permitted to have before service completion. More complex is the case where a customer is permitted to have a nite number (K ≥ 2) of We can extend the models discussed in this thesis to several directions.

Queueing Models with Vacations and Working Vacations

The thesis entitled “Queueing Models with Vacations and Working Vacations" consists of seven chapters including the introductory chapter. In chapters 2 to 7 we analyze different queueing models highlighting the role played by vacations and working vacations. The duration of vacation is exponentially distributed in all these models and multiple vacation policy is followed.In chapter 2 we discuss an M/M/2 queueing system with heterogeneous servers, one of which is always available while the other goes on vacation in the absence of customers waiting for service. Conditional stochastic decomposition of queue length is derived. An illustrative example is provided to study the effect of the input parameters on the system performance measures. Chapter 3 considers a similar setup as chapter 2. The model is analyzed in essentially the same way as in chapter 2 and a numerical example is provided to bring out the qualitative nature of the model. The MAP is a tractable class of point process which is in general nonrenewal. In spite of its versatility it is highly tractable as well. Phase type distributions are ideally suited for applying matrix analytic methods. In all the remaining chapters we assume the arrival process to be MAP and service process to be phase type. In chapter 4 we consider a MAP/PH/1 queue with working vacations. At a departure epoch, the server finding the system empty, takes a vacation. A customer arriving during a vacation will be served but at a lower rate.Chapter 5 discusses a MAP/PH/1 retrial queueing system with working vacations.In chapter 6 the setup of the model is similar to that of chapter 5. The signicant dierence in this model is that there is a nite buer for arrivals.Chapter 7 considers an MMAP(2)/PH/1 queueing model with a nite retrial group

Analysis of some Stochastic models in inventories and Queues

The thesis entitled Analysis of Some Stochastic Models in Inventories and Queues. This thesis is devoted to the study of some stochastic models in Inventories and Queues which are physically realizable, though complex. It contains a detailed analysis of the basic stochastic processes underlying these models. In this thesis, (s,S) inventory systems with nonidentically distributed interarrival demand times and random lead times, state dependent demands, varying ordering levels and perishable commodities with exponential life times have been studied. The queueing system of the type Ek/Ga,b/l with server vacations, service systems with single and batch services, queueing system with phase type arrival and service processes and finite capacity M/G/l queue when server going for vacation after serving a random number of customers are also analysed. The analogy between the queueing systems and inventory systems could be exploited in solving certain models. In vacation models, one important result is the stochastic decomposition property of the system size or waiting time. One can think of extending this to the transient case. In inventory theory, one can extend the present study to the case of multi-item, multi-echelon problems. The study of perishable inventory problem when the commodities have a general life time distribution would be a quite interesting problem. The analogy between the queueing systems and inventory systems could be exploited in solving certain models.

Development and Analysis of a Compact Dual-Band Coplanar Antenna

The coplanar wave guide is an attractive device in microwave integrated circuits due to its uniplanar nature, ease of fabrication and low production cost. Several attempts are already done to explore the radiating modes in coplanar wave guide transmission lines. Usually coplanar wave guides are excited by an SMA connector with its centre conductor connected to the exact middle of the centre strip and the outer ground conductor to the two ground strips. The mode excited on it is purely a bound mode. The E-field distribution in the two slots are out of phase and there for cancels at the far field. This thesis addresses an attempt to excite an in phase E-field distribution in the two slots of the co planar wave guide by employing a feed asymmetry, in order to get radiation from the two large slot discontinuities of the coplanar waveguide. The omni directional distribution of the radiating energy can be achieved by widening the centre strip.The first part of the thesis deals with the investigations on the resonance phenomena of conventional coplanar waveguides at higher frequency bands. Then an offset fed open circuited coplanar waveguide supporting resonance/radiation phenomena is analyzed. Finally, a novel compact co planar antenna geometry with dual band characteristics, suitable for mobile terminal applications is designed and characterized using the inferences from the above study.

Analysis of Some Stochastic Inventory Models with Pooling/ Retrial of Customers

In this thesis we have presented several inventory models of utility. Of these inventory with retrial of unsatisfied demands and inventory with postponed work are quite recently introduced concepts, the latt~~ being introduced for the first time. Inventory with service time is relatively new with a handful of research work reported. The di lficuity encoLlntered in inventory with service, unlike the queueing process, is that even the simplest case needs a 2-dimensional process for its description. Only in certain specific cases we can introduce generating function • to solve for the system state distribution. However numerical procedures can be developed for solving these problem.

On Multi- Server Queues With Consultation With Main Server

Queueing Theory is the mathematical study of queues or waiting lines. Queues abound in every day life - in computer networks, in tra c islands, in communication of electro-magnetic signals, in telephone exchange, in bank counters, in super market checkouts, in doctor's clinics, in petrol pumps, in o ces where paper works to be processed and many other places. Originated with the published work of A. K. Erlang in 1909 [16] on congestion in telephone tra c, Queueing Theory has grown tremendously in a century. Its wide range applications includes Operations Research, Computer Science, Telecommunications, Tra c Engineering, Reliability Theory, etc.