Asymptotic expansion of the absorption-time distribution for a markov chain

Cybernetics ◽  
1975 ◽  
Vol 9 (4) ◽  
pp. 694-697
Author(s):  
V. S. Korolyuk ◽  
I. P. Penev ◽  
A. F. Turbin
Keyword(s):  
Asymptotic Expansion ◽  
Markov Chain ◽  
Time Distribution ◽  
Absorption Time

On the time a Markov chain spends in a lumped state

1997 ◽  
Vol 34 (02) ◽  
pp. 340-345
Author(s):  
Tommy Norberg
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Continuous Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Sojourn Time Distribution

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

scholarly journals Antiduality and Möbius monotonicity: generalized coupon collector problem

2019 ◽  
Vol 23 ◽  
pp. 739-769
Author(s):  
Paweł Lorek
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Partial Ordering ◽  
Absorption Time ◽  
Absorbing Markov Chain ◽  
Coupon Collector ◽  
Strong Stationary Duality ◽  
Finite State ◽  
A Chain ◽  
Finite State Space

For a given absorbing Markov chain X* on a finite state space, a chain X is a sharp antidual of X* if the fastest strong stationary time (FSST) of X is equal, in distribution, to the absorption time of X*. In this paper, we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence – utilizing known results on the limiting distribution of the absorption time – we indicate separation cutoffs (with their window sizes) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its FSST is distributed as a prescribed mixture of sums of geometric random variables.

A BMAP/PH/N SYSTEM WITH IMPATIENT REPEATED CALLS

2007 ◽  
Vol 24 (03) ◽  
pp. 293-312 ◽  
Author(s):  
VALENTINA I. KLIMENOK ◽  
DMITRY S. ORLOVSKY ◽  
ALEXANDER N. DUDIN
Keyword(s):  
Markov Chain ◽  
Time Distribution ◽  
Arrival Process ◽  
Service Time Distribution ◽  
State Distribution ◽  
Unsuccessful Attempt ◽  
Batch Markovian Arrival Process ◽  
Repeated Calls ◽  
Phase Type ◽  
Multi Server

A multi-server queueing model with a Batch Markovian Arrival Process, phase-type service time distribution and impatient repeated customers is analyzed. After any unsuccessful attempt, the repeated customer leaves the system with the fixed probability. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Stability condition and an algorithm for calculating the stationary state distribution of this Markov chain are obtained. Main performance measures of the system are calculated. Numerical results are presented.

scholarly journals Hurst Index of Functions of Long-Range-Dependent Markov Chains

2012 ◽  
Vol 49 (02) ◽  
pp. 451-471
Author(s):  
Barlas Oğuz ◽  
Venkat Anantharam
Keyword(s):  
Markov Chain ◽  
Long Range ◽  
Queueing Networks ◽  
Time Distribution ◽  
Return Time ◽  
Indicator Function ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Hurst Index ◽  
Positive Recurrent

A positive recurrent, aperiodic Markov chain is said to be long-range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long-range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queueing networks, source compression, and finance.

scholarly journals Hurst Index of Functions of Long-Range-Dependent Markov Chains

2012 ◽  
Vol 49 (2) ◽  
pp. 451-471 ◽  
Author(s):  
Barlas Oğuz ◽  
Venkat Anantharam
Keyword(s):  
Markov Chain ◽  
Long Range ◽  
Queueing Networks ◽  
Time Distribution ◽  
Return Time ◽  
Indicator Function ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Hurst Index ◽  
Positive Recurrent

A positive recurrent, aperiodic Markov chain is said to be long-range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long-range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queueing networks, source compression, and finance.

On the time a Markov chain spends in a lumped state

1997 ◽  
Vol 34 (2) ◽  
pp. 340-345 ◽  
Author(s):  
Tommy Norberg
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Continuous Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Sojourn Time Distribution

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

scholarly journals Estimation of Urban Link Travel Time Distribution Using Markov Chains and Bayesian Approaches

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Wenwen Qin ◽  
Meiping Yun
Keyword(s):  
Markov Chain ◽  
Travel Time ◽  
Time Distribution ◽  
State Transition ◽  
Transition Function ◽  
Travel Times ◽  
Travel Time Distribution ◽  
Multimodal Distributions ◽  
State Transition Function ◽  
Link Travel Time

Despite the wide application of Floating Car Data (FCD) in urban link travel time and congestion estimation, the sparsity of observations from a low penetration rate of GPS-equipped floating cars make it difficult to estimate travel time distribution (TTD), especially when the travel times may have multimodal distributions that are associated with the underlying traffic states. In this case, the study develops a Bayesian approach based on particle filter framework for link TTD estimation using real-time and historical travel time observations from FCD. First, link travel times are classified by different traffic states according to the levels of vehicle delays. Then, a state-transition function is represented as a Transition Probability Matrix of the Markov chain between upstream and current links with historical observations. Using the state-transition function, an importance distribution is constructed as the summation of historical link TTDs conditional on states weighted by the current link state probabilities. Further, a sampling strategy is developed to address the sparsity problem of observations by selecting the particles with larger weights in terms of the importance distribution and a Gaussian likelihood function. Finally, the current link TTD can be reconstructed by a generic Markov Chain Monte Carlo algorithm incorporating high weighted particles. The proposed approach is evaluated using real-world FCD. The results indicate that the proposed approach provides good accurate estimations, which are very close to the empirical distributions. In addition, the approach with different percentage of floating cars is tested. The results are encouraging, even when multimodal distributions and very few or no observations exist.

Asymptotic behavior of the distribution of the absorption time of a Markov chain

Cybernetics ◽  
1974 ◽  
Vol 8 (2) ◽  
pp. 193-196
Author(s):  
V. S. Korolyuk ◽  
I. P. Penev ◽  
A. F. Turbin
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Absorption Time
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