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.

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.

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.

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

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.

Phase Transitions of Trace Metals in the Aquatic Environment of Kuttanad, Kerala

This thesis is a modest attempt in assessing the trace metal levels and their behavior in the aquatic environment of Kuttanad, an aquatic system that is severely affected by man’s intervention on natural processes, by study seriously evaluating the levels of trace metals in dissolved and particulate phases and also in the different chemical fractions of the sediments. Understanding of the distributions, variations and transfer processes of trace metals in different environmental phases in the backwaters of Kuttanad is vital for the assessment of the water pollution problems and study the ecology of the area which contributes 20% of the rice production in the State of Kerala. Kuttand is a low-lying, shallow bay formed as a result of geological uplift. The major economic activity is agriculture involving 40% of the population. About 1.5% of the people are engaged in aquaculture. The trace metal distribution in the Kuttand backwaters is considerably influenced by the tropical features of the location and by human activities including agricultural activities and construction of salinity barrier. In this study an attempt is made to differentiate the metals in the sediment into exchangeable, reducible and resistant fractions in the sediments.

On Szegö’s Type Theorems

This study is to look the effect of change in the ordering of the Fourier system on Szegö’s classical observations of asymptotic distribution of eigenvalues of finite Toeplitz forms.This is done by checking proofs and Szegö’s properties in the new set up.The Fourier system is unconditional [19], any arbitrary ordering of the Fourier system forms a basis for the Hilbert space L2 [-Π, Π].Here study about the classical Szegö’s theorem.Szegö’s type theorem for operators in L2(R+) and check its validity for certain multiplication operators.Since the trigonometric basis is not available in L2(R+) or in L2(R) .This study discussed about the classes of orderings of Haar System in L2 (R+) and in L2(R) in which Szegö’s Type TheoreT Am is valid for certain multiplication operators.It is divided into two sections. In the first section there is an ordering to Haar system in L2(R+) and prove that with respect to this ordering, Szegö’s Type theorem holds for general class of multiplication operators Tƒ with multiplier ƒ ε L2(R+), subject to some conditions on ƒ.Finally in second section more general classes of ordering of Haar system in L2(R+) and in L2(R) are identified in such a way that for certain classes of multiplication operators the asymptotic distribution of eigenvalues exists.

STUDIES ON TAXONOMY, DISTRIBUTION, ECOLOGY AND REPRODUCTIVE POTENTIAL OF ROTIFERS FROM SELECTED CENTRES IN COCHIN BACKWATER SYSTEM, KERALA

The studies were conducted in nine stations with varying ecological characteristics along Cochin backwaters and adjoining canals. Many workers opined that the distribution of rotifers is cosmopolitan. The significance of rotifers as first food for early larvae was indicated by Fujita. Aquaculture is a fast growing field in fisheries sector and it is gaining more importance as the fish landings and supply are getting irregular. A consistent supply of fish/shellfish can only be achieved through aquaculture. The success of any culture activity depends on the timely production of seeds of finfishes/shellfishes. The availability of wild seed is seasonal and erratic. So, a dependable source of seed of fishes and shellfishes is possible only through large scale production in hatchery. A successful seed production activity depends on the availability of a variety of suitable live feed organisms in sufficient quantities at the proper time for use in the larval stages. As the live feeds promote high growth rates, easy digestion, assimilation and the quality of not contaminating the culture water when compared to other artificial feeds, make the culture of live feed organisms the principal means of providing food for the larvae of finfishes and shellfishes. Rotifers are considered to be an excellent and indispensable food for larvae of many finfishes and crustaceans. It (1960) was the first to culture Brachionus plicatilis for feeding marine fish larvae, and now it is being extensively used as live feed in hatcheries all over the world. They are a group of microscopic organisms coming under the Phylum Rotifera which comprises of about 2000 species. Their slow swimming habits, ability to tolerate a wide range of salinities, parthenogenetic mode of reproduction and ability to get enriched easily, make rotifers an ideal live feed organism. The major factors such as temperature, salinity and food that influence the reproductive potential and thereby the population size of rotifer, Salinity is one of the most important aspect influencing the reproductive rate of rotifers. The feed type and feed concentration play a vital role in influencing the reproductive rate of rotifers. For culture of rotifers, the commonly used micro algae belong to Chlorella, Nannochloropsis, Isochrysis and Tetraselmis. While some studies have suggested that, algal diet has little effect on reproductive rates in 1979 while using the rotifer, Brachionus plicatilis as feed for the larvae of red sea bream, Pagrus major. It is generally accepted that rotifers play a pivotal role in the successful rearing of marine fish larvae.

Liquid-phase Friedel-Crafts acylation using NixMn(l-x)Fe2O4 spinel catalysts

The liquid-phase Friedel-Crafts acylation of toluene using benzoyl chloride as benzoylating agent has heen carried out over Nix, Mn(l-x)Fe2 O4 (x=O, 0.2, 0.4, 0.6, 0.8 and 1.0) type systems under different reaction conditions. It is observed that the systems with high 'x' values are effective for the conversion of BOC and the selective formation of 4-MBP. Selectivity for 4-MBP over MnFe2O4 is more than 90% under the optimized reaction conditions. Sites of moderate acidity is effective in calalyzing the benzoylation reaction.

Selective N-monomethylation of aniline over Zn1-xNixFe2O4 (x=0, 0.2, 0.5, 0.8 and 1) type systems

The reaction of aniline with methanol was carried out over Zn1-xNixFe2O4 (x= 0, 0.2, 0.5, 0.8 and 1) type systems in a fixed-bed down-flow reactor. It was observed that systems possessing low ``x'' values are highly selective and active for mono N-alkylation of aniline leading to N-methyl aniline. Selectivity for N-methyl aniline over ZnFe2O4 was more than 99% under the optimized reaction conditions. Even at methanol to aniline molar ratio of 2, the yield of N-methyl aniline was nearly 55.5%, whereas its yield exceeded 67% at the molar ratio of 7. The Lewis acid sites of the catalysts are mainly responsible for the good catalytic performance. Cation distribution in the spinel lattice influences their acido-basic properties, and hence, these factors have been considered as helpful to evaluate the activity and stability of the systems.

Liquid phase benzylation of o-xylene over pillared clays

Iron, aluminium and mixed iron aluminium pillared clays have been prepared by partial hydrolysis method and doped with IO% Mo, V and Cr. The samples have been characterised by XRD, FTIR and surface area and pore Volume measurements. The surface acid site distribution has been determined by temperature programmed desorption of ammonia. Vanadia incorporated systems show maximum acidity. Benzylation of o-xylene has been done as probe reaction to test catalytic activity. Benzyl chloride is a superior benzylating agent compared to benzyl alcohol in activity and selectivity. Cent percent selectivity towards monobenzylated product is obtained in all the cases. Fe pillared systems exhibit maximum activity. The catalytic activities of the systems can be correlated with the amount of strong add sites. The effects of various reaction variables on the reaction have been studied. Presence of moisture has a diminishing effect on the reaction rate.

Selective N-monomethylation of aniline using Zn1-x CoxFe2O4( x=0, 0.2, 0.5, 0.8 and 1.0)type systems

A series of ferrites having the general formula Zn1-xCoxFe2O4 (x=0, 0.2, 0.5, 0.8 and 1.0)were prepared by soft chemical route. The materials were characterized by adopting various physico-chemical methods. The reaction of aniline with methanol was studied in a fixed-bed reactor system as a potential source for the production of various methyl anilines. It was observed that systems possessing low ‘ x’ values are highly selective and active for N-monoalkylation of aniline leading to N-methylaniline. Reaction parameters were properly varied to optimize the reaction conditions for obtaining N-methylaniline selectively and in better yield. Among the systems Zn0.8Co0.2Fe2O4 is remarkable due to its very high activity and excellent stability. Under the optimized conditions N-methylaniline selectivity exceeded 98%. Even at a methanol to aniline molar ratio of 2, the yield of N-methylaniline was nearly 50%, whereas its yield exceeded 71% at the molar ratio of 5. ZnFe2O4, though executed better conversion than Zn0.8Co0.2Fe2O4 in the initial period of the run, deactivates quickly as the reaction proceeds. The Lewis acidity of the catalysts is mainly responsible for the good performance. Cation distribution in the spinel lattice influences their acido-basic properties and, hence, these factors have been considered as helpful parameters to evaluate the activity of the systems.

Catalytic activity of some of the perovskite-type mixed oxides (ABO3) consisting of rare earth and 3d transition metals

The catalytic activity of some of the ABO3 (A = La, Pr and Sm, B= Cr, Mn, Fe, Co and Ni) perovskite-type oxides for the liquid phase reduction of ketone and oxidation of alcohol in 2-propanol medium has been studied. The data have been correlated with the surface electron donor properties of these oxides. The surface electron donor properties have been determined from the adsorption of electron acceptors of varying electron affinities on the oxide surface.

Photoacoustic studies on n-type InP

We discuss an open photoacoustic cell study on sulfer-doped n-type InP wafer. The thermal diffusivity of the sample is evaluated from the phase data associated with the photoacoustic signal as a function of the modulation frequency under heat transmission configuration. Analysis is made on the basis of the Rosencwaig-Gersho theory and the results are compared with those from earlier reported photoacoustic studies of semiconductors. Our investigation clearly indicates that the instantaneous thermalization process is the major heat diffusion mechanism responsible for the photoacoustic signal generation in an InP sample.