Sojourn Times for Continuous-Parameter Markov Chains

1994 ◽  
pp. 69-105
Author(s):  
Attila Csenki
Keyword(s):  
Markov Chains ◽  
Sojourn Times ◽  
Continuous Parameter

Markov chains in small time intervals

1981 ◽  
Vol 18 (3) ◽  
pp. 747-751
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probability ◽  
Small Time ◽  
Exact Order ◽  
Time Intervals ◽  
Continuous Parameter ◽  
Minimum Number

For a time-homogeneous continuous-parameter Markov chain we show that as t → 0 the transition probability pn,j (t) is at least of order where r(n, j) is the minimum number of jumps needed for the chain to pass from n to j. If the intensities of passage are bounded over the set of states which can be reached from n via fewer than r(n, j) jumps, this is the exact order.

scholarly journals On sojourn times at particular gene frequencies

1975 ◽  
Vol 25 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Edward Pollak ◽  
Barry C. Arnold
Keyword(s):  
Markov Chains ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Finite Population ◽  
Diffusion Method ◽  
Gene Frequencies ◽  
Sojourn Times ◽  
The Mean ◽  
Number Of Visits ◽  
Finite Markov Chains

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

scholarly journals A note on repeated sequences in Markov chains

1987 ◽  
Vol 19 (03) ◽  
pp. 739-742 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Transition Matrix ◽  
Repeated Sequences ◽  
Markov Renewal Process ◽  
Conditional Mean ◽  
Sojourn Times

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

Some theorems on the transient covariance of Markov chains

1972 ◽  
Vol 9 (01) ◽  
pp. 214-218 ◽  
Author(s):  
John F. Reynolds
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Covariance Structure ◽  
Continuous Parameter ◽  
Transient Period

Several authors have considered the covariance structure of continuous parameter Markov chains. Most of this work has dealt with particular process ses, notably Morse (1955) who analysed the simple M/M/1 queue and Bene-(1961) who considered a telephone trunking model. Furthermore, the results obtained apply only when the process has attained its limiting (stationary) distribution. A recent paper by Reynolds (1968) gave some general results for finite chains, still assuming stationarity. This note generalises the results obtained therein, and considers the covariance structure during the transient period prior to attaining the stationary distribution where this exists. In the case where no such distribution exists, the results are valid throughout the whole lifetime of the process.

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.

Sojourn times for conditioned Markov chains in genetics

1976 ◽  
Vol 8 (4) ◽  
pp. 645-648 ◽  
Author(s):  
S. Tavaré
Keyword(s):  
Markov Chains ◽  
Sojourn Times

scholarly journals Seismic hazard assessment in the Northern Aegean Sea (Greece) through discrete Semi-Markov modeling.

2016 ◽  
Vol 47 (3) ◽  
pp. 1417
Author(s):  
C. E. Pertsinidou ◽  
G. Tsaklidis ◽  
E. Papadimitriou
Keyword(s):  
Markov Chains ◽  
Seismic Hazard ◽  
Seismic Hazard Assessment ◽  
Aegean Sea ◽  
Arbitrary Distribution ◽  
Markov Modeling ◽  
Sojourn Times ◽  
Markov Kernels ◽  
Northern Aegean Sea ◽  
State 1

Semi-Markov  chains  are  used  for  studying  the  evolution  of  seismicity  in  the Northern Aegean Sea (Greece). Their main difference from the Markov chains is that   they  allow the sojourn times (i.e. the time between successive earthquakes), to follow any arbitrary distribution. It is assumed that the time series of earthquakes that  occurred  in  Northern  Aegean  Sea  form  a  discrete  semi-Markov  chain. The probability law of the sojourn times, is considered to be the geometric distribution or   the   discrete  Weibull  distribution. Firstly,  the  data  are  classified  into  two categories that is, state 1: Magnitude 6.5  -7 and state 2 Magnitude>7, and secondly into three categories , that is    state 1: Magnitude 6.5-6.7, state 2: Magnitude 6.8-7.1 and state 3: Magnitude 7.2-7.4 . This methodology is followed in order to obtain more accurate results and find out whether there exists an impact of the different classification on the results. The parameters of the probability laws of the sojourn times are estimated and the semi-Markov kernels are  evaluated for all the above cases  .  The  semi-Markov  kernels  are  compared and  the   conclusions  are  drawn relatively to future seismic hazard in the area under study.

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