The joint distribution of sojourn times in finite Markov processes

1992 ◽  
Vol 24 (1) ◽  
pp. 141-160 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Chains ◽  
Joint Distribution ◽  
Probabilistic Reasoning ◽  
Fault Tolerant ◽  
Time Discretization ◽  
Cumulative Distribution ◽  
Multiprocessor System ◽  
Probability Mass Function ◽  
Sojourn Times ◽  
Absorbing Markov Chains

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

The joint distribution of sojourn times in finite Markov processes

1992 ◽  
Vol 24 (01) ◽  
pp. 141-160 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Chains ◽  
Joint Distribution ◽  
Probabilistic Reasoning ◽  
Fault Tolerant ◽  
Time Discretization ◽  
Cumulative Distribution ◽  
Multiprocessor System ◽  
Probability Mass Function ◽  
Sojourn Times ◽  
Absorbing Markov Chains

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

Absorbing Markov chains for analyzing COVID-19 infections

2021 ◽  
pp. 2250008
Author(s):  
Safaa K. Kadhem ◽  
Sadeq A. Kadhim
Keyword(s):  
Markov Chains ◽  
Markov Model ◽  
Dynamical Model ◽  
Health State ◽  
Random Movement ◽  
Expected Time ◽  
Absorbing Markov Chains ◽  
Death State ◽  
Key Aspects ◽  
Model Approach

Recently, there are many works that proposed modeling approaches to describe the random movement of individuals for COVID-19 infection. However, these models have not taken into account some key aspects for disease such the prediction of expected time of patients remaining at certain health state before entering an absorption state (e.g., exit out of the system for ever such as death state or recovery). Therefore, we propose a dynamical model approach called the absorbing Markov chains for analyzing COVID-19 infections. From this modeling approach, we seek to focus and predict two states of absorption: recovery and death, as these two conditions are considered as important indicators in assessment of the health level. Based on the absorbing Markov model, the study suggested that there is a gradually increase in the predicted death number, while a decrease in the number of recovered individuals.

scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

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