Interval Transition Probabilities Research Articles

The use of Semi-Markov Model to determine the efficacy of CDC staging criterion in Nigeria (A case study of General Hospital Minna, Niger State)

Semi-Markov model in discrete state and time to study the trajection among the various defined HIV/AIDs stages has been presented in this paper. The model was used to determine the staging of HIV- infected clients reported at heart-to-heart clinic of General Hospital Minna, Niger State, Nigeria for a period of 10 years. The result shows that there is no transition from asymptomatic stage of HIV to late/advance AIDs such that φ14 (n) = 0; which led to gradual increment in the graphs of interval transition probabilities for all n ∈ N.The study shows that there are some increments in the transition probabilities from stages 2, 3 and 4 to stage 1 from about 0.05745, 0.01055 and 0.00379 in the first month (n = 1) to about 0.37163, 0.17821 and 0.05862 in the forty four months (n = 44) and in the forty five months (n = 45) respectively. From the virtual transition probabilities, the graph is gradually decreasing after forty-five (45) months. The result show that φ11 (n), φ22 (n), φ33 (n) and φ44 (n)attained the values of about 0.56516, 0.50353, 0.53343 and 0.65623 respectively for the first few months or years. The percentage decrease is about 37.1%, 17.8% for stages 2, 3 in forty-four months and 5.8% for stage 4 in forty-five month duration. However, client’s transitions dropped slowly and attain stability of stage at infinity. In compliance to the medical view, all HIV-naïve patients transit to AIDs stage especially when therapeutic intervention is lacking. The model established in this study could assist the Medical Personnels’ (MP), Health Care Providers (HCP), Epidemiologists, Medical Statisticians and other funding organizations to plan for the treatment, tracking and intervention for the ever increasing scourge of HIV/AIDs.

Testing the Semi Markov Model Using Monte Carlo Simulation Method for Predicting the Network Traffic

Semi-Markov processes can be considered as a generalization of both Markov and renewal processes. One of the principal characteristics of these processes is that in opposition to Markov models, they represent systems whose evolution is dependent not only on their last visited state but on the elapsed time since this state. Semi-Markov processes are replacing the exponential distribution of time intervals with an optional distribution. In this paper, we give a statistical approach to test the semi-Markov hypothesis. Moreover, we describe a Monte Carlo algorithm able to simulate the trajectories of the semi-Markov chain. This simulation method is used to test the semi-Markov model by comparing and analyzing the results with empirical data. We introduce the database of Network traffic which is employed for applying the Monte Carlo algorithm. The statistical characteristics of real and synthetic data from the models are compared. The comparison between the semi-Markov and the Markov models is done by computing the Autocorrelation functions and the probability density functions of the Network traffic real and simulated data as well. All the comparisons admit that the Markovian hypothesis is rejected in favor of the more general semi Markov one. Finally, the interval transition probabilities which show the future predictions of the Network traffic are given.

An Application of a Semi Markov Model in Predicting the Fading Occurrences in Wireless Communication Channels

The most frustrating and troublesome issue in wireless communication is fading. Estimating the fading occurrence for the design of greatly reliable communication link is crucial. In this paper, we present a novel mathematical method for predicting the future signal fading on the basis of current and past data. The application of a Semi Markov Model as a generalization of the Markov Model is discussed for predicting the deep fading occurrence probabilities of the received envelope in wireless communications channels. This flexible model is given for assessing the system performance with the envelope correlation. 142 deep fading data whose amplitudes are lower than a mean of the amplitudes which occurred in a typical wireless system with the Jakes filtering are considered. The transition probability matrix and the holding time mass functions are calculated for the next 1 to 21 unit times. One unit time is regarded as the inverse of sampling frequency; moreover, the core matrix and the cumulative probability distribution of the waiting time are obtained. Calculating the interval transition probabilities for Amplitude to Amplitude transition for these deep fades demonstrates the forecasting occurrence probabilities in the future and the possibility of forecasting the fading occurrences in dimensions of time and amplitude.

ON THE EARTHQUAKE OCCURRENCES IN JAPAN AND THE SURROUNDING AREA VIA SEMI MARKOV MODELING

In the present paper, the earthquake occurrences in the area of Japan, are studied by a semi Markov model which is considered homogeneous in time. The data applied refer to earthquakes of large magnitude (Mw>6.0) during the period 1900-2012. We consider 9 seismic zones derived from the typical 11 zones for the area of Japan, due to the lack of data for 3 zones (9-th,10-th and 11-th). Also, we define 3 groups for the magnitudes, corresponding to 6-7,7.1-8 and M> 8.0. Thus, we consider for our semi Markov model a finite state space, S={ ( ,)j i ZR | i=1,...9, j=1,2,3}, where i Z defines the i-th seismic zone and j R states the j-th magnitude scale. We applied the data to describe the interval transition probabilities for the states and the model's limiting behaviour for which is sufficient an interval of time of seven years. The time unit of the model is considered to be one day. Some interesting results, concerning the interval transition probabilities and the limiting state vector, are derived.

Transient analysis of a finite source discrete-time queueing system using homogeneous Markov system with state size capacities (HMS/c)

In this article, a finite source discrete-time queueing system is modeled as a discrete-time homogeneous Markov system with finite state size capacities (HMS/c) and transition priorities. This Markov system is comprised of three states. The first state of the HMS/c corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the third state which represents the system's queue. In order to examine the variability of the state sizes recursive formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence the probability mass function of each state size can be evaluated. Also the expected time in queue is computed by means of the interval transition probabilities. The theoretical results are illustrated by a numerical example.

Survival probabilities for HIV infected patients through semi-Markov processes

SUMMARY In this paper we apply a parametric semi-Markov process to model the dynamic evolution of HIV-1 infected patients. The seriousness of the infection is rendered by the CD4+ T-lymphocyte counts. For this purpose we introduce the main features of nonhomogeneous semi-Markov models. After determining the transition probabilities and the waiting time distributions in each state of the disease, we solve the evolution equations of the process in order to estimate the interval transition probabilities. These quantities appear to be of fundamental importance for clinical predictions. We also estimate the survival probabilities for HIV infected patients and compare them with respect to certain categories, such as gender, age group or type of antiretroviral therapy. Finally we attach a reward structure to the aforementioned semi-Markov processes in order to estimate clinical costs. For this purpose we generate random trajectories from the semi-Markov processes through Monte Carlo simulation. The proposed model is then applied to a large database provided by ISS (Istituto Superiore di Sanità, Rome, Italy), and all the quantities of interest are computed.

Model of a conflict situation in information security management

The problem of information security managementemerged from the necessity to search for the ways toincrease the efficiency of the state information systemsfunctioning in modern conditions. To choose the adequatecounteractions to the information security threatsand attacks, it was recommended to apply the models ofconflict parties’ interaction and the models of conflictsituations in the conditions of vagueness. Conflict situationsin the information security management of the statewere modelled using strategically-probabilistic approach.The process of conflict was described as a variable processwith two states and known transition matrix. Formulaswere derived to calculate the interval transition probabilities between the states of attack and defence. Themodel of conflict situations in information security managementwas created. Due to formalization of the processof information security management system reaction toattacks, the automated forecasting was achieved for theintervals of safety and information security recovery durationafter attacks.

On the Distributions of the State Sizes of the Continuous Time Homogeneous Markov System with Finite State Capacities

In the present paper we study the evolution of a continuous time homogeneous Markov system whose states have finite capacities. We assume that the members of the system who overflow, due to the finite state capacities, leave the system. In order to investigate the variability of the state sizes we provide the evaluation of the intensity matrix for any time point and then we derive a formula concerning the derivate of the moments of the state sizes. As a consequence the distributions of the state sizes can be evaluated. Moreover, an alternative method of calculating the distributions of the state sizes by means of the interval transition probabilities is given. Finally we examine the distribution of the time needed for the leavers to leave the system. The theoretical results are illustrated by a numerical example.

On the use of uniformization to estimate transition probabilities in categorical condition data

Large groups of structures like bridges, pavements and sewer systems, are often inspected visually and their condition is quantified based on a discrete scale which results in categorical data. Markov chains have traditionally been used to model the uncertain rate at which these structures progress through such a condition scale. In order to determine optimal strategies for inspections and maintenance activities, these Markov chains must be fitted to the data obtained in the field. This paper presents details on how to implement the method of uniformization for the calculation of interval transition probabilities and their sensitivity towards model parameters. The primary benefit of uniformization is that it provides a robust and efficient method for use in maximum likelihood estimation of transition probabilities or intensities in Markov chains.

Numerical Approach for Assessing System Dynamic Availability Via Continuous Time Homogeneous Semi-Markov Processes

Continuous-time homogeneous semi-Markov processes (CTHSMP) are important stochastic tools to model reliability measures for systems whose future behavior is dependent on the current and next states occupied by the process as well as on sojourn times in these states. A method to solve the interval transition probabilities of CTHSMP consists of directly applying any general quadrature method to the N2 coupled integral equations which describe the future behavior of a CTHSMP, where N is the number of states. However, the major drawback of this approach is its considerable computational effort. In this work, it is proposed a new more efficient numerical approach for CTHSMPs described through either transition probabilities or transition rates. Rather than N2 coupled integral equations, the approach consists of solving only N coupled integral equations and N straightforward integrations. Two examples in the context of availability assessment are presented in order to validate the effectiveness of this method against the comparison with the results provided by the classical and Monte Carlo approaches. From these examples, it is shown that the proposed approach is significantly less time-consuming and has accuracy comparable to the method of N2 computational effort.

Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes

Non-homogeneous semi-Markov processes (NHSMP) are important stochastic tools for modeling reliability metrics over time for systems where the future behavior depends on the current and next states as well as on sojourn and process times. The classical method to solve the interval transition probabilities of NHSMPs consists of directly applying any general quadrature method to some non-convolution integral equations. However, this approach has a considerable computational effort. Namely, N 2-coupled integral equations with two variables must be solved, where N is the number of states. Therefore, this article proposes a more efficient mathematical formulation and numerical treatment, which are based on transition frequency densities and general quadrature methods respectively, for NHSMPs. The approach consists of only solving N-coupled integral equations with one variable and N straightforward integrations. Two examples in the context of reliability are also presented. The first one addresses a case where a semi-analytical solution is available. Then an example of application concerning pressure–temperature optical monitoring systems for oil wells is discussed. In both cases, the proposed approach is validated via the comparison against the results obtained from the semi-analytical solution (for the first example) as well as from both the classic and the Monte Carlo methods.

Parametric and Non Homogeneous Semi-Markov Process for HIV Control

In AIDS control, physicians have a growing need to use pragmatically useful and interpretable tools in their daily medical taking care of patients. Semi-Markov process seems to be well adapted to model the evolution of HIV-1 infected patients. In this study, we introduce and define a non homogeneous semi-Markov (NHSM) model in continuous time. Then the problem of finding the equations that describe the biological evolution of patient is studied and the interval transition probabilities are computed. A parametric approach is used and the maximum likelihood estimators of the process are given. A Monte Carlo algorithm is presented for realizing non homogeneous semi-Markov trajectories. As results, interval transition probabilities are computed for distinct times and follow-up has an impact on the evolution of patients.

The Effect of Parameters Change of a Semi‐Markov Model on Interval Transition Probabilities

AbstractThis paper examines the changes in parameters of a Semi‐Markov model and its impact on interval transition probabilities. Computational techniques are used to illustrate that the model is sensitive enough to depict changes in the interval transition probabilities if the paramters of the model are modified.

Use of Bayesian Analysis of Semi-Markov Process Models to Study Consumer Buying Behavior

SYNOPTIC ABSTRACTAnalytical techniques are developed for a combined treatment of the interpurchase-times and brand-choices of frequently purchased consumer goods, in the framework of a semi-Markov process. The inverse Gaussian distribution is used as a model for the interpurchase times. To account for heterogeneity of the consumer population, a natural conjugate prior for the model parameters is developed and the compounding distribution is fitted to panel data of toothpaste purchases. Some important summary measures are derived, including: the market share of a brand as a function of time; the long-run behavior of the interval transition probabilities and the market share; and, the distribution of the number of purchases in a given time span.

System Analysis of Semi-Markov Processes

Flow graph analysis has proved a valuable approach to the study of probabilistic systems. This paper extends the analytic procedures to semi-Markov processes, Markov processes whose transition times can also be arbitrary random variables. First we present the basic theory of the semi-Markov process. We derive expressions for the interval transition probabilities, the probabilities that the system will be in each state after the passage of a time interval of length t, and find the limit of these probabilities when t is large. Then we develop the flow graph interpretation of these relations. We consider the special cases of counting transitions, transient processes, and first passage times. We investigate the effect on interval transition probabilities of starting the process in different ways, including choosing a time at random to begin observation of the process. We find that results are often most conveniently expressed by the matrix flow graph for the process.