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.

A New Class of Processes for Formalizing and Generalizing Individual-Based Models: The Semi-Semi-Markov Processes

2000 Mathematics Subject Classification: 60K15, 60K20, 60G20,60J75, 60J80, 60J85, 60-08, 90B15.

Adaptive estimation of hidden semi-Markov chains with parameterised transition probabilities and exponential decaying states

This paper develops maximum likelihood (ML) estimation schemes for finite-state semi-Markov chains in white Gaussian noise. We assume that the semi-Markov chain is characterised by transition probabilities of known parametric from with unknown parameters. We reformulate this hidden semi-Markov model (HSM) problem in the scalar case as a two-vector homogeneous hidden Markov model (HMM) problem in which the state consist of the signal augmented by the time to last transition. With this reformulation we apply the expectation Maximumisation (EM ) algorithm to obtain ML estimates of the transition probabilities parameters, Markov state levels and noise variance. To demonstrate our proposed schemes, motivated by neuro-biological applications, we use a damped sinusoidal parameterised function for the transition probabilities.

Optimal maintenance and replacement via a semi-Markov decision model

Optimal bang-coast maintenance policies for a machine, subject to failure, are considered. The approach utilizes a semi-Markov model for the system. A simplified model for modifying the probability of machine failure with maintenance is employed. A numerical example is presented to illustrate the procedure and results.

Activity recognition and abnormality detection with the switching hidden semi-Markov model

This paper addresses the problem of learning and recognizing human activities of daily living (ADL), which is an important research issue in building a pervasive and smart environment. In dealing with ADL, we argue that it is beneficial to exploit both the inherent hierarchical organization of the activities and their typical duration. To this end, we introduce the Switching Hidden Semi-Markov Model (S-HSMM), a two-layered extension of the hidden semi-Markov model (HSMM) for the modeling task. Activities are modeled in the S-HSMM in two ways: the bottom layer represents atomic activities and their duration using HSMMs; the top layer represents a sequence of high-level activities where each high-level activity is made of a sequence of atomic activities. We consider two methods for modeling duration: the classic explicit duration model using multinomial distribution, and the novel use of the discrete Coxian distribution. In addition, we propose an effective scheme to detect abnormality without the need for training on abnormal data. Experimental results show that the S-HSMM performs better than existing models including the flat HSMM and the hierarchical hidden Markov model in both classification and abnormality detection tasks, alleviating the need for presegmented training data. Furthermore, our discrete Coxian duration model yields better computation time and generalization error than the classic explicit duration model.

Topic transition detection using hierarchical hidden Markov and semi-Markov models

In this paper we introduce a probabilistic framework to exploit hierarchy, structure sharing and duration information for topic transition detection in videos. Our probabilistic detection framework is a combination of a shot classification step and a detection phase using hierarchical probabilistic models. We consider two models in this paper: the extended Hierarchical Hidden Markov Model (HHMM) and the Coxian Switching Hidden semi-Markov Model (S-HSMM) because they allow the natural decomposition of semantics in videos, including shared structures, to be modeled directly, and thus enabling efficient inference and reducing the sample complexity in learning. Additionally, the S-HSMM allows the duration information to be incorporated, consequently the modeling of long-term dependencies in videos is enriched through both hierarchical and duration modeling. Furthermore, the use of the Coxian distribution in the S-HSMM makes it tractable to deal with long sequences in video. Our experimentation of the proposed framework on twelve educational and training videos shows that both models outperform the baseline cases (flat HMM and HSMM) and performances reported in earlier work in topic detection. The superior performance of the S-HSMM over the HHMM verifies our belief that duration information is an important factor in video content modeling.

Propagation model of smartphone worms based on semi-Markov process and social relationship graph

Smartphone applications are getting more and more popular and pervasive in our daily life, and are also attractive to malware writers due to their limited computing source and vulnerabilities. At the same time, we possess limited understanding of our opponents in cyberspace. In this paper, we investigate the propagation model of SMS/MMS-based worms through integrating semi-Markov process and social relationship graph. In our modeling, we use semi-Markov process to characterize state transition among mobile nodes, and hire social network theory, a missing element in many previous works, to enhance the proposed mobile malware propagation model. In order to evaluate the proposed models, we have developed a specific software, and collected a large scale real-world data for this purpose. The extensive experiments indicate that the proposed models and algorithms are effective and practical. © 2014 Elsevier Ltd. All rights reserved.

Maintenance strategy optimization using a continuous-state partially observable semi-Markov decision process

Due to the limitation of current condition monitoring technologies, the estimates of asset health states may contain some uncertainties. A maintenance strategy ignoring this uncertainty of asset health state can cause additional costs or downtime. The partially observable Markov decision process (POMDP) is a commonly used approach to derive optimal maintenance strategies when asset health inspections are imperfect. However, existing applications of the POMDP to maintenance decision-making largely adopt the discrete time and state assumptions. The discrete-time assumption requires the health state transitions and maintenance activities only happen at discrete epochs, which cannot model the failure time accurately and is not cost-effective. The discrete health state assumption, on the other hand, may not be elaborate enough to improve the effectiveness of maintenance. To address these limitations, this paper proposes a continuous state partially observable semi-Markov decision process (POSMDP). An algorithm that combines the Monte Carlo-based density projection method and the policy iteration is developed to solve the POSMDP. Different types of maintenance activities (i.e., inspections, replacement, and imperfect maintenance) are considered in this paper. The next maintenance action and the corresponding waiting durations are optimized jointly to minimize the long-run expected cost per unit time and availability. The result of simulation studies shows that the proposed maintenance optimization approach is more cost-effective than maintenance strategies derived by another two approximate methods, when regular inspection intervals are adopted. The simulation study also shows that the maintenance cost can be further reduced by developing maintenance strategies with state-dependent maintenance intervals using the POSMDP. In addition, during the simulation studies the proposed POSMDP shows the ability to adopt a cost-effective strategy structure when multiple types of maintenance activities are involved.

Risk Minimizing Option Pricing in a Semi-Markov Modulated Market

We address risk minimizing option pricing in a semi-Markov modulated market where the floating interest rate depends on a finite state semi-Markov process. The growth rate and the volatility of the stock also depend on the semi-Markov process. Using the Föllmer–Schweizer decomposition we find the locally risk minimizing price for European options and the corresponding hedging strategy. We develop suitable numerical methods for computing option prices.

Portfolio Optimization in a Semi-Markov Modulated Market

We address a portfolio optimization problem in a semi-Markov modulated market. We study both the terminal expected utility optimization on finite time horizon and the risk-sensitive portfolio optimization on finite and infinite time horizon. We obtain optimal portfolios in relevant cases. A numerical procedure is also developed to compute the optimal expected terminal utility for finite horizon problem.

Non-Stationary Semi-Markov Decision Processes on a Finite Horizon

We introduce and study a class of non-stationary semi-Markov decision processes on a finite horizon. By constructing an equivalent Markov decision process, we establish the existence of a piecewise open loop relaxed control which is optimal for the finite horizon problem.