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.

Analysis of Stochastic Volatility Sequences Generated by Product Autoregressive Models

The classical methods of analysing time series by Box-Jenkins approach assume that the observed series uctuates around changing levels with constant variance. That is, the time series is assumed to be of homoscedastic nature. However, the nancial time series exhibits the presence of heteroscedasticity in the sense that, it possesses non-constant conditional variance given the past observations. So, the analysis of nancial time series, requires the modelling of such variances, which may depend on some time dependent factors or its own past values. This lead to introduction of several classes of models to study the behaviour of nancial time series. See Taylor (1986), Tsay (2005), Rachev et al. (2007). The class of models, used to describe the evolution of conditional variances is referred to as stochastic volatility modelsThe stochastic models available to analyse the conditional variances, are based on either normal or log-normal distributions. One of the objectives of the present study is to explore the possibility of employing some non-Gaussian distributions to model the volatility sequences and then study the behaviour of the resulting return series. This lead us to work on the related problem of statistical inference, which is the main contribution of the thesis

Analysis of Some Single and Two Commodity Inventory Problems

This thesis is devoted to the study of some stochastic models in inventories. An inventory system is a facility at which items of materials are stocked. In order to promote smooth and efficient running of business, and to provide adequate service to the customers, an inventory materials is essential for any enterprise. When uncertainty is present, inventories are used as a protection against risk of stock out. It is advantageous to procure the item before it is needed at a lower marginal cost. Again, by bulk purchasing, the advantage of price discounts can be availed. All these contribute to the formation of inventory. Maintaining inventories is a major expenditure for any organization. For each inventory, the fundamental question is how much new stock should be ordered and when should the orders are replaced. In the present study, considered several models for single and two commodity stochastic inventory problems. The thesis discusses two models. In the first model, examined the case in which the time elapsed between two consecutive demand points are independent and identically distributed with common distribution function F(.) with mean  (assumed finite) and in which demand magnitude depends only on the time elapsed since the previous demand epoch. The time between disasters has an exponential distribution with parameter . In Model II, the inter arrival time of disasters have general distribution (F.) with mean  ( ) and the quantity destructed depends on the time elapsed between disasters. Demands form compound poison processes with inter arrival times of demands having mean 1/. It deals with linearly correlated bulk demand two Commodity inventory problem, where each arrival demands a random number of items of each commodity C1 and C2, the maximum quantity demanded being a (< S1) and b(

Studies on the Effect of Randomness on the Synchronization of Coupled Systems and on the Dynamics of Intermittently driven systems

Nonlinear dynamics has emerged into a prominent area of research in the past few Decades.Turbulence, Pattern formation,Multistability etc are some of the important areas of research in nonlinear dynamics apart from the study of chaos.Chaos refers to the complex evolution of a deterministic system, which is highly sensitive to initial conditions. The study of chaos theory started in the modern sense with the investigations of Edward Lorentz in mid 60's. Later developments in this subject provided systematic development of chaos theory as a science of deterministic but complex and unpredictable dynamical systems. This thesis deals with the effect of random fluctuations with its associated characteristic timescales on chaos and synchronization. Here we introduce the concept of noise, and two familiar types of noise are discussed. The classifications and representation of white and colored noise are introduced. Based on this we introduce the concept of randomness that we deal with as a variant of the familiar concept of noise. The dynamical systems introduced are the Rossler system, directly modulated semiconductor lasers and the Harmonic oscillator. The directly modulated semiconductor laser being not a much familiar dynamical system, we have included a detailed introduction to its relevance in Chaotic encryption based cryptography in communication. We show that the effect of a fluctuating parameter mismatch on synchronization is to destroy the synchronization. Further we show that the relation between synchronization error and timescales can be found empirically but there are also cases where this is not possible. Studies show that under the variation of the parameters, the system becomes chaotic, which appears to be the period doubling route to chaos.

Detection of Flock Movement in Spatio-Temporal Database using Clustering Techniques - an Experience

In this paper, moving flock patterns are mined from spatio- temporal datasets by incorporating a clustering algorithm. A flock is defined as the set of data that move together for a certain continuous amount of time. Finding out moving flock patterns using clustering algorithms is a potential method to find out frequent patterns of movement in large trajectory datasets. In this approach, SPatial clusteRing algoRithm thrOugh sWarm intelligence (SPARROW) is the clustering algorithm used. The advantage of using SPARROW algorithm is that it can effectively discover clusters of widely varying sizes and shapes from large databases. Variations of the proposed method are addressed and also the experimental results show that the problem of scalability and duplicate pattern formation is addressed. This method also reduces the number of patterns produced

Analysis of Some Stochastic Inventory Models with Pooling Retrial of Customers

The thesis deals with analysis of some Stochastic Inventory Models with Pooling/Retrial of Customers.. In the first model we analyze an (s,S) production Inventory system with retrial of customers. Arrival of customers from outside the system form a Poisson process. The inter production times are exponentially distributed with parameter µ. When inventory level reaches zero further arriving demands are sent to the orbit which has capacity M(<∞). Customers, who find the orbit full and inventory level at zero are lost to the system. Demands arising from the orbital customers are exponentially distributed with parameter γ. In the model-II we extend these results to perishable inventory system assuming that the life-time of each item follows exponential with parameter θ. The study deals with an (s,S) production inventory with service times and retrial of unsatisfied customers. Primary demands occur according to a Markovian Arrival Process(MAP). Consider an (s,S)-retrial inventory with service time in which primary demands occur according to a Batch Markovian Arrival Process (BMAP). The inventory is controlled by the (s,S) policy and (s,S) inventory system with service time. Primary demands occur according to Poissson process with parameter λ. The study concentrates two models. In the first model we analyze an (s,S) Inventory system with postponed demands where arrivals of demands form a Poisson process. In the second model, we extend our results to perishable inventory system assuming that the life-time of each item follows exponential distribution with parameter θ. Also it is assumed that when inventory level is zero the arriving demands choose to enter the pool with probability β and with complementary probability (1- β) it is lost for ever. Finally it analyze an (s,S) production inventory system with switching time. A lot of work is reported under the assumption that the switching time is negligible but this is not the case for several real life situation.

Gamma stochastic volatility models

This paper presents gamma stochastic volatility models and investigates its distributional and time series properties. The parameter estimators obtained by the method of moments are shown analytically to be consistent and asymptotically normal. The simulation results indicate that the estimators behave well. The insample analysis shows that return models with gamma autoregressive stochastic volatility processes capture the leptokurtic nature of return distributions and the slowly decaying autocorrelation functions of squared stock index returns for the USA and UK. In comparison with GARCH and EGARCH models, the gamma autoregressive model picks up the persistence in volatility for the US and UK index returns but not the volatility persistence for the Canadian and Japanese index returns. The out-of-sample analysis indicates that the gamma autoregressive model has a superior volatility forecasting performance compared to GARCH and EGARCH models.

Stochastic processes and their applications time dependent solutions for some queueing and inventory models

The objective of this thesis is to study the time dependent behaviour of some complex queueing and inventory models. It contains a detailed analysis of the basic stochastic processes underlying these models. In the theory of queues, analysis of time dependent behaviour is an area.very little developed compared to steady state theory. Tine dependence seems certainly worth studying from an application point of view but unfortunately, the analytic difficulties are considerable. Glosod form solutions are complicated even for such simple models as M/M /1. Outside M/>M/1, time dependent solutions have been found only in special cases and involve most often double transforms which provide very little insight into the behaviour of the queueing systems themselves. In inventory theory also There is not much results available giving the time dependent solution of the system size probabilities. Our emphasis is on explicit results free from all types of transforms and the method used may be of special interest to a wide variety of problems having regenerative structure.

Stochastic processes and their applications semi-markov analysis of some inventory and queuing problems

This thesis analyses certain problems in Inventories and Queues. There are many situations in real-life where we encounter models as described in this thesis. It analyses in depth various models which can be applied to production, storag¢, telephone traffic, road traffic, economics, business administration, serving of customers, operations of particle counters and others. Certain models described here is not a complete representation of the true situation in all its complexity, but a simplified version amenable to analysis. While discussing the models, we show how a dependence structure can be suitably introduced in some problems of Inventories and Queues. Continuous review, single commodity inventory systems with Markov dependence structure introduced in the demand quantities, replenishment quantities and reordering levels are considered separately. Lead time is assumed to be zero in these models. An inventory model involving random lead time is also considered (Chapter-4). Further finite capacity single server queueing systems with single/bulk arrival, single/bulk services are also discussed. In some models the server is assumed to go on vacation (Chapters 7 and 8). In chapters 5 and 6 a sort of dependence is introduced in the service pattern in some queuing models.

Queueing Theory And Other Applications Of Stochastic Processes Queueing And Inventory Models With Rest Periods

In this thesis we study the effect of rest periods in queueing systems without exhaustive service and inventory systems with rest to the server. Most of the works in the vacation models deal with exhaustive service. Recently some results have appeared for the systems without exhaustive service.

Probability Theory. Stochastic Processes And Their Applications Probabilistic Analysis Of Some Queueing And Inventory Models

In this thesis we attempt to make a probabilistic analysis of some physically realizable, though complex, storage and queueing models. It is essentially a mathematical study of the stochastic processes underlying these models. Our aim is to have an improved understanding of the behaviour of such models, that may widen their applicability. Different inventory systems with randon1 lead times, vacation to the server, bulk demands, varying ordering levels, etc. are considered. Also we study some finite and infinite capacity queueing systems with bulk service and vacation to the server and obtain the transient solution in certain cases. Each chapter in the thesis is provided with self introduction and some important references

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.

STUDIES ON HAEMOLYMPH CONSTITUENTS OF INDOPLANORBIS EXUSTUS (DESHAYES) AND LYMNAEA ACUMINATA (LAMARCK) F.RUFESCENS (GRAY) AND THE EFFECTS OF COPPER ON THE ACTIVITY PATTERN OF SELECTED TRANSAMINASES AND PHOSPHATASES

The problem investigated is on the haematological aspects of two freshwater pulmonate snails, Indoplanorbis exustus (Deshayes),and Lymnaea acuminata f.rufescens (Gray). An important aspect of the present investigation is to emphasize the utilization of freshwater organisms as models for research directed at understanding the basic biomedical problems that remain unresolved. Another aspect is to demonstrate how haemolymph can be treated as a tissue because of late, it has been shown that several parameters of blood can be taken as reliable indicators for diagnostic purposes, and also to monitor environmental pollution. The various haematological parameters studied are total haemocyte number,packed cell volume, haemoglobin, and inorganic as well organic constituents in three size groups of both the snail species. The effect of copper toxicity was measured in terms of total haemocyte count, and the activity pattern of selected phosphatases and transaminases.The study concluded that enzyme activity levels can be taken as reliable indicators to monitor pollution. Age is a factor that determines several of the physiological, biochemical and metabolic activities. This study also indicates that haemolymph can be taken as an organ system to study the various changes taking place at organ systems levels.

Stochastic Modelling Using Markov Chains

Department of Statistics, Cochin University of Science and Technology

Magnetic field-induced cluster formation and variation of magneto-optical signals in zinc-substituted ferrofluids

Fine magnetic particles (size≅100 Å) belonging to the series ZnxFe1−xFe2O4 were synthesized by cold co-precipitation methods and their structural properties were evaluated using X-ray diffraction. Magnetization studies have been carried out using vibrating sample magnetometry (VSM) showing near-zero loss loop characteristics. Ferrofluids were then prepared employing these fine magnetic powders using oleic acid as surfactant and kerosene as carrier liquid by modifying the usually reported synthesis technique in order to induce anisotropy and enhance the magneto-optical signals. Liquid thin films of these fluids were prepared and field-induced laser transmission through these films was studied. The transmitted light intensity decreases at the centre with applied magnetic field in a linear fashion when subjected to low magnetic fields and saturate at higher fields. This is in accordance with the saturation in cluster formation. The pattern exhibited by these films in the presence of different magnetic fields was observed with the help of a CCD camera and was recorded photographically.