Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable ‘matrix’ quantum mechanics, which is recently proposed by Odake and the author. The (q)Askey-scheme of hypergeometric orthogonal polynomials of a discrete...

We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund dual of birth and death chains are revisited in some detail, with emphasis on the non neutral Moran m...

This paper presents a more general class of MAP/MAP/1 exhaustive vacation queue, in which the Markov modulated arrival and service processes are dependent. This model class requires the evaluation of the busy period of quasi birth death process with arbitrary initial level, which is a new analysis element. The model is analyzed by applying matrix analytic methods for the underlying quasi birth ...

This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cutoff if and only if the product of spectral gap and mixing time tends to infinity. We est...

We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund dual of birth and death chains are revisited in some detail, with emphasis on the non neutral Moran m...

For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death proce...

We propose a Bayesian inference approach for a class of latent Markov models. These models are widely used for the analysis of longitudinal categorical data, when the interest is in studying the evolution of an individual unobservable characteristic. We consider, in particular, the basic latent Markov, which does not account for individual covariates, and its version that includes such covariat...

The classical models in risk theory consider a single type of claims. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous time Markov chain of pure birt...

This paper deals with the mathematical modeling and performance optimization for the Digesting system of a Paper Plant using Genetic Algorithm. The Digesting system of a Paper Plant has four main subsystems, arranged in series and parallel. Considering exponential distribution for the probable failures and repairs, the mathematical formulation of the problem is done using probabilistic approach...

This paper deals with the performance enhancement for crystallization unit of a sugar plant using genetic algorithm. The crystallization unit of a sugar industry has three main subsystems arranged in series. Considering exponential distribution for the probable failures and repairs, the mathematical formulation of the problem is done using probabilistic approach, and different...