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.

Evaluation of Pareto/D/1/k Queue by Simulation

The finding that Pareto distributions are adequate to model Internet packet interarrival times has motivated the proposal of methods to evaluate steady-state performance measures of Pareto/D/1/k queues. Some limited analytical derivation for queue models has been proposed in the literature, but their solutions are often of a great mathematical challenge. To overcome such limitations, simulation tools that can deal with general queueing system must be developed. Despite certain limitations, simulation algorithms provide a mechanism to obtain insight and good numerical approximation to parameters of queues. In this work, we give an overview of some of these methods and compare them with our simulation approach, which are suited to solve queues with Generalized-Pareto interarrival time distributions. The paper discusses the properties and use of the Pareto distribution. We propose a real time trace simulation model for estimating the steady-state probability showing the tail-raising effect, loss probability, delay of the Pareto/D/1/k queue and make a comparison with M/D/1/k. The background on Internet traffic will help to do the evaluation correctly. This model can be used to study the long- tailed queueing systems. We close the paper with some general comments and offer thoughts about future work.

Generalization of the Modified Bessel Function and Its Generating Function

2000 Mathematics Subject Classification: 33C10, 33-02, 60K25

On Queueinginventory Models – product form solution; reservation, Cancellation and common life time

Queueing theory is the mathematical study of ‘queue’ or ‘waiting lines’ where an item from inventory is provided to the customer on completion of service. A typical queueing system consists of a queue and a server. Customers arrive in the system from outside and join the queue in a certain way. The server picks up customers and serves them according to certain service discipline. Customers leave the system immediately after their service is completed. For queueing systems, queue length, waiting time and busy period are of primary interest to applications. The theory permits the derivation and calculation of several performance measures including the average waiting time in the queue or the system, mean queue length, traffic intensity, the expected number waiting or receiving service, mean busy period, distribution of queue length, and the probability of encountering the system in certain states, such as empty, full, having an available server or having to wait a certain time to be served.

On the throughput-WIP trade-off in queueing systems, diminishing returns and the threshold property: A linear programming approach

We present a new unifying framework for investigating throughput-WIP(Work-in-Process) optimal control problems in queueing systems,based on reformulating them as linear programming (LP) problems withspecial structure: We show that if a throughput-WIP performance pairin a stochastic system satisfies the Threshold Property we introducein this paper, then we can reformulate the problem of optimizing alinear objective of throughput-WIP performance as a (semi-infinite)LP problem over a polygon with special structure (a thresholdpolygon). The strong structural properties of such polygones explainthe optimality of threshold policies for optimizing linearperformance objectives: their vertices correspond to the performancepairs of threshold policies. We analyze in this framework theversatile input-output queueing intensity control model introduced byChen and Yao (1990), obtaining a variety of new results, including (a)an exact reformulation of the control problem as an LP problem over athreshold polygon; (b) an analytical characterization of the Min WIPfunction (giving the minimum WIP level required to attain a targetthroughput level); (c) an LP Value Decomposition Theorem that relatesthe objective value under an arbitrary policy with that of a giventhreshold policy (thus revealing the LP interpretation of Chen andYao's optimality conditions); (d) diminishing returns and invarianceproperties of throughput-WIP performance, which underlie thresholdoptimality; (e) a unified treatment of the time-discounted andtime-average cases.

Optimization of multiclass queueing networks with changeover times via the achievable region approach: Part I, the single-station case

We address the performance optimization problem in a single-stationmulticlass queueing network with changeover times by means of theachievable region approach. This approach seeks to obtainperformance bounds and scheduling policies from the solution of amathematical program over a relaxation of the system's performanceregion. Relaxed formulations (including linear, convex, nonconvexand positive semidefinite constraints) of this region are developedby formulating equilibrium relations satisfied by the system, withthe help of Palm calculus. Our contributions include: (1) newconstraints formulating equilibrium relations on server dynamics;(2) a flow conservation interpretation of the constraintspreviously derived by the potential function method; (3) newpositive semidefinite constraints; (4) new work decomposition lawsfor single-station multiclass queueing networks, which yield newconvex constraints; (5) a unified buffer occupancy method ofperformance analysis obtained from the constraints; (6) heuristicscheduling policies from the solution of the relaxations.

Optimization of multiclass queueing networks with changeover times via the achievable region method: Part II, the multi-station case

We address the problem of scheduling a multi-station multiclassqueueing network (MQNET) with server changeover times to minimizesteady-state mean job holding costs. We present new lower boundson the best achievable cost that emerge as the values ofmathematical programming problems (linear, semidefinite, andconvex) over relaxed formulations of the system's achievableperformance region. The constraints on achievable performancedefining these formulations are obtained by formulatingsystem's equilibrium relations. Our contributions include: (1) aflow conservation interpretation and closed formulae for theconstraints previously derived by the potential function method;(2) new work decomposition laws for MQNETs; (3) new constraints(linear, convex, and semidefinite) on the performance region offirst and second moments of queue lengths for MQNETs; (4) a fastbound for a MQNET with N customer classes computed in N steps; (5)two heuristic scheduling policies: a priority-index policy, anda policy extracted from the solution of a linear programmingrelaxation.

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.

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

Queueing Models with Vacations and Working Vacations

The objective of the study of \Queueing models with vacations and working vacations" was two fold; to minimize the server idle time and improve the e ciency of the service system. Keeping this in mind we considered queueing models in di erent set up in this thesis. Chapter 1 introduced the concepts and techniques used in the thesis and also provided a summary of the work done. In chapter 2 we considered an M=M=2 queueing model, where one of the two heterogeneous servers takes multiple vacations. We studied the performance of the system with the help of busy period analysis and computation of mean waiting time of a customer in the stationary regime. Conditional stochastic decomposition of queue length was derived. To improve the e ciency of this system we came up with a modi ed model in chapter 3. In this model the vacationing server attends the customers, during vacation at a slower service rate. Chapter 4 analyzed a working vacation queueing model in a more general set up. The introduction of N policy makes this MAP=PH=1 model di erent from all working vacation models available in the literature. A detailed analysis of performance of the model was provided with the help of computation of measures such as mean waiting time of a customer who gets service in normal mode and vacation mode.

Study of Queueing Behaviour in IP Buffers

It is unquestioned that the importance of IP network will further increase and that it will serve as a platform for more and more services, requiring different types and degrees of service quality. Modern architectures and protocols are being standardized, which aims at guaranteeing the quality of service delivered to users. In this paper, we investigate the queueing behaviour found in IP output buffers. This queueing increases because multiple streams of packets with different length are being multiplexed together. We develop balance equations for the state of the system, from which we derive packet loss and delay results. To analyze these types of behaviour, we study the discrete-time version of the “classical” queue model M/M/1/k called Geo/Gx/1/k, where Gx denotes a different packet length distribution defined on a range between a minimum and maximum value.

Obesity And Inflammation And The Effect On The Hematopoietic System.

Bone marrow is organized in specialized microenvironments known as 'marrow niches'. These are important for the maintenance of stem cells and their hematopoietic progenitors whose homeostasis also depends on other cell types present in the tissue. Extrinsic factors, such as infection and inflammatory states, may affect this system by causing cytokine dysregulation (imbalance in cytokine production) and changes in cell proliferation and self-renewal rates, and may also induce changes in the metabolism and cell cycle. Known to relate to chronic inflammation, obesity is responsible for systemic changes that are best studied in the cardiovascular system. Little is known regarding the changes in the hematopoietic system induced by the inflammatory state carried by obesity or the cell and molecular mechanisms involved. The understanding of the biological behavior of hematopoietic stem cells under obesity-induced chronic inflammation could help elucidate the pathophysiological mechanisms involved in other inflammatory processes, such as neoplastic diseases and bone marrow failure syndromes.

The Value Of The 2005 International Society Of Urological Pathology (isup) Modified Gleason Grading System As A Predictor Of Biochemical Recurrence After Radical Prostatectomy.

To compare time and risk to biochemical recurrence (BR) after radical prostatectomy of two chronologically different groups of patients using the standard and the modified Gleason system (MGS). Cohort 1 comprised biopsies of 197 patients graded according to the standard Gleason system (SGS) in the period 1997/2004, and cohort 2, 176 biopsies graded according to the modified system in the period 2005/2011. Time to BR was analyzed with the Kaplan-Meier product-limit analysis and prediction of shorter time to recurrence using univariate and multivariate Cox proportional hazards model. Patients in cohort 2 reflected time-related changes: striking increase in clinical stage T1c, systematic use of extended biopsies, and lower percentage of total length of cancer in millimeter in all cores. The MGS used in cohort 2 showed fewer biopsies with Gleason score ≤ 6 and more biopsies of the intermediate Gleason score 7. Time to BR using the Kaplan-Meier curves showed statistical significance using the MGS in cohort 2, but not the SGS in cohort 1. Only the MGS predicted shorter time to BR on univariate analysis and on multivariate analysis was an independent predictor. The results favor that the 2005 International Society of Urological Pathology modified system is a refinement of the Gleason grading and valuable for contemporary clinical practice.

The Applicability Of Ordered Mesoporous Sba-15 And Its Hydrophobic Glutaraldehyde-bridge Derivative To Improve Ibuprofen-loading In Releasing System.

The mesoporous SBA-15 silica with uniform hexagonal pore, narrow pore size distribution and tuneable pore diameter was organofunctionalized with glutaraldehyde-bridged silylating agent. The precursor and its derivative silicas were ibuprofen-loaded for controlled delivery in simulated biological fluids. The synthesized silicas were characterized by elemental analysis, infrared spectroscopy, (13)C and (29)Si solid state NMR spectroscopy, nitrogen adsorption, X-ray diffractometry, thermogravimetry and scanning electron microscopy. Surface functionalization with amine containing bridged hydrophobic structure resulted in significantly decreased surface area from 802.4 to 63.0 m(2) g(-1) and pore diameter 8.0-6.0 nm, which ultimately increased the drug-loading capacity from 18.0% up to 28.3% and a very slow release rate of ibuprofen over the period of 72.5h. The in vitro drug release demonstrated that SBA-15 presented the fastest release from 25% to 27% and SBA-15GA gave near 10% of drug release in all fluids during 72.5 h. The Korsmeyer-Peppas model better fits the release data with the Fickian diffusion mechanism and zero order kinetics for synthesized mesoporous silicas. Both pore sizes and hydrophobicity influenced the rate of the release process, indicating that the chemically modified silica can be suggested to design formulation of slow and constant release over a defined period, to avoid repeated administration.

Electronic And Structural Study Of Pt-modified Au Vicinal Surfaces: A Model System For Pt-au Catalysts.

Two single crystalline surfaces of Au vicinal to the (111) plane were modified with Pt and studied using scanning tunneling microscopy (STM) and X-ray photoemission spectroscopy (XPS) in ultra-high vacuum environment. The vicinal surfaces studied are Au(332) and Au(887) and different Pt coverage (θPt) were deposited on each surface. From STM images we determine that Pt deposits on both surfaces as nanoislands with heights ranging from 1 ML to 3 ML depending on θPt. On both surfaces the early growth of Pt ad-islands occurs at the lower part of the step edge, with Pt ad-atoms being incorporated into the steps in some cases. XPS results indicate that partial alloying of Pt occurs at the interface at room temperature and at all coverage, as suggested by the negative chemical shift of Pt 4f core line, indicating an upward shift of the d-band center of the alloyed Pt. Also, the existence of a segregated Pt phase especially at higher coverage is detected by XPS. Sample annealing indicates that the temperature rise promotes a further incorporation of Pt atoms into the Au substrate as supported by STM and XPS results. Additionally, the catalytic activity of different PtAu systems reported in the literature for some electrochemical reactions is discussed considering our findings.