Step Transition Probability Matrix Research Articles

MRSPN analysis of Semi-Markovian finite source retrial queues

In this paper, the analysis of Semi-Markovian single server retrial queues by means of Markov Regenerative Stochastic Petri Nets (MRSPN) is considered. We propose MRSPN models for the two retrial queues M/G/1/N/N and M/G/1/N/N with orbital search. By inspecting the reduced reachability graph of both MRSPN models, the qualitative analysis is obtained. The quantitative analysis is carried out after constructing their one step transition probability matrix and computing the steady state probability distribution of each tangible marking. As an example, the queue $$M/Hypo_{2}/1/2/2$$ is treated in order to illustrate the functionality of the MRSPN approach. The exact performance measures (mean number of customers in the system, mean response time, mean waiting time,...) are computed for different parameters of the two systems by an algorithm elaborated in Matlab environment.

The Reliability Analysis of Mechanical Leg (2UPS+UP) Based on a Discrete Time Repairable Markov Model

We have analyzed some reliability problems of the 2UPS+UP mechanism using continuous Markov repairable model in our previous work. According to the check and repair of the robot is periodic, the discrete time Markov repairable model should be more appropriate. Firstly we built up the discrete time repairable model and got the one step transition probability matrix. Secondly solved the steady state equations and got the steady state availability of the mechanical leg, by the solution of the difference equations the reliability and the mean time to first failure were obtained. In the end we compared the reliability indexes with the continuous model.

A continuous time version and a generalization of a Markov-recapture model for trapping experiments

Wileyto et al. [E.P. Wileyto, W.J. Ewens, M.A. Mullen, Markov-recapture population estimates: a tool for improving interpretation of trapping experiments, Ecology 75 (1994) 1109] propose a four-state discrete time Markov process, which describes the structure of a marking–capture experiment as a method of population estimation. They propose this method primarily for estimation of closed insect populations. Their method provides a mark–recapture estimate from a single trap observation by allowing subjects to mark themselves. The estimate of the unknown population size is based on the assumption of a closed population and a simple Markov model in which the rates of marking, capture, and recapture are assumed to be equal. Using the one step transition probability matrix of their model, we illustrate how to go from an embedded discrete time Markov process to a continuous time Markov process assuming exponentially distributed holding times. We also compute the transition probabilities after time t for the continuous time case and compare the limiting behavior of the continuous and discrete time processes. Finally, we generalize their model by relaxing the assumption of equal per capita rates for marking, capture, and recapture. Other questions about how their results change when using a continuous time Markov process are examined.

A stochastic model for space-division switches in computer networks

In computer networks, calls are routed through switches in order to reach at destinations. The simplest types of them are crosspoint switches which require one-to-one connection between inputs and outputs. If it assumed that all lines fully duplex and there are no self-connections, so many crosspoints are needed to provide the proper connectivity. To reduce this factor, space-division switches are designed to provide better connectivity under the condition of limited space. In this paper, we assume a Markov chain based stochastic model with the unit step transition probability matrix to examine the functioning and properties of space-division switches. The connectivity of system under this model is studied using a simulation procedure. Expressions for various connectivity probabilities are derived for the different stages of the space-division switching elements and simulation is performed for K-dependent and L-dependent Markov models.

Explicit inversion formulas for a model in diffuse tomography

Optical, or diffuse tomography, refers to the use of low energy probes to obtain images of highly scattering media. This represents an emerging area for the application of new mathematical tools to important problems in medical imaging. It is our contention that it poses important mathematical challenges dealing with the solution of large systems of nonlinear equations. In this paper we present a case for which a complete solution is possible; however the consideration of any larger system presents at the moment a number of unresolved difficulties. A variety of mathematical models of the underlying physics are reviewed in [1] including the ones considered in [4,7–10], where we have discussed both the “direct”, as well as the more interesting “inverse problem” for a discrete model. The time evolution of the system is governed by a Markov chain with discrete state space and discrete time. The one step transition probability matrix is denoted by P . Some of the states are “incoming”, some are “outgoing”, and the rest are “hidden” states. The first two types correspond to sources and detectors respectively, and the hidden ones represent the state of a photon while it moves in the interior of the object.

Molecular weight distribution in emulsion polymerization. II. The copolymer case

AbstractA model for evaluating instantaneous degree of polymerization distribution and the chain composition distribution of copolymers produced in emulsion is developed. The approach adopted is based on the mathematics of Markov processes and represents an extension of the one developed for homopolymers in Part I. As in the homopolymer case, the main aspect of the theoretical treatment is the definition of the proper one step transition probability matrix through the so called subprocess‐main process procedure. The model accounts for monomolecular and bimolecular termination (both by combination and disproportionation) and, in principle, it can be applied to any number of reacting monomer species as well as to any number of active chains per particle. However, only the 0–1–2 and 0–1–2–3 emulsion copolymerization systems are discussed in detail. In the case of the chain composition distribution, the model allows the calculation of its moments only, through the method of the Generating Function associated with the probability density function. The expression obtained for the instantaneous probability density functions, as well as for the corresponding cumulative distributions, are all in explicit form and involve only algebraic operations among matrices. Efficient numerical procedure for their application are reported in the Appendix. Illustrative calculations are reported for a 0–1–2–3 copolymerization system, simulating the copolymer styrene–methylmethacrylate. The effect of the various termination mechanisms on the distribution of degrees of polymerization and on the first two moments of the chain composition distribution is discussed in detail. Finally, the three dimensional overall distribution function of both chain length and composition is shown under the assumption of Gaussian type chain composition distribution.

A Statistical Study of Generalized Cell Mapping

In this paper a statistical error analysis of the generalized cell mapping method for both deterministic and stochastic dynamical systems is examined, based upon the statistical analogy of the generalized cell mapping method to the density estimation. The convergence of the mean square error of the one step transition probability matrix of generalized cell mapping for deterministic and stochastic systems is studied. For stochastic systems, a well-known trade-off feature of the density estimation exists in the mean square error of the one step transition probability matrix, which leads to an optimal design of generalized cell mapping for stochastic systems. The conclusions of the study are illustrated with some examples.