On the stability of the computation of the stationary probabilities of Markov chains using Perron complements

2003 ◽  
Vol 10 (7) ◽  
pp. 603-618 ◽  
Author(s):  
Michael Neumann ◽  
Jianhong Xu
Keyword(s):  
Markov Chains ◽  
The Stability ◽  
Stationary Probabilities

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

WINNER PLAYS COMPETITION MODELS

2019 ◽  
pp. 1-6
Author(s):  
Yang Cao ◽  
Sheldon M. Ross
Keyword(s):  
Markov Chains ◽  
Competition Models ◽  
Stationary Probabilities ◽  
Quantities Of Interest

Abstract Suppose there are n players in an ongoing competition, with player i having value v i , and suppose that a game between i and j is won by i with probability v i /(v i  + v j ). Consider the winner plays competition where in each stage two players play a game, and the winner keeps playing in the next game. We consider two models for choosing its opponent, analyze both models as Markov chains, and determine their stationary probabilities as well as other quantities of interest.

Conditions for the non-ergodicity of Markov chains with application to a communication system

1987 ◽  
Vol 24 (2) ◽  
pp. 339-346 ◽  
Author(s):  
Linn I. Sennott
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Communication System ◽  
System Performance ◽  
Multiple Access ◽  
Sufficient Condition ◽  
Performance Bounds ◽  
Communication System Performance ◽  
Null Recurrence ◽  
The Stability

We obtain a sufficient condition for the transience of a Markov chain, and a sufficient condition for its null recurrence. These are applied to characterize the stability of a multiple-access communication system. Performance bounds for the system are also obtained.

scholarly journals Maximal Coupling Procedure and Stability of Continuous-Time Markov Chains

2014 ◽  
Vol 10 ◽  
pp. 30-37
Author(s):  
Mokaedi V. Lekgari
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Continuous Time ◽  
Stopping Times ◽  
Continuous Time Markov Chains ◽  
Maximal Coupling ◽  
Countable State ◽  
Coupling Procedure ◽  
The Stability

In this study we first investigate the stability of subsampled discrete Markov chains through the use of the maximal coupling procedure. This is an extension of the available results on Markov chains and is realized through the analysis of the subsampled chain ΦΤn, where {Τn, nєZ+}is an increasing sequence of random stopping times. Then the similar results are realized for the stability of countable-state Continuous-time Markov processes by employing the skeleton-chain method.

Observing Customer Segment Stability Using Soft Computing Techniques and Markov Chains Within Data Mining Framework

2018 ◽  
pp. 1440-1457
Author(s):  
Abdulkadir Hiziroglu
Keyword(s):  
Data Mining ◽  
Markov Chains ◽  
Soft Computing ◽  
Internal Validity ◽  
Real World Data ◽  
Soft Computing Techniques ◽  
Customer Segment ◽  
Managerial Implications ◽  
Classification Prediction ◽  
The Stability

This study proposes a model that utilizes soft computing and Markov Chains within a data mining framework to observe the stability of customer segments. The segmentation process in this study includes clustering of existing consumers and classification-prediction of segments for existing and new customers. Both a combination and an integration of soft computing techniques were used in the proposed model. Segmenting customers was done according to the purchasing behaviours of customers based on RFM (Recency, Frequency, Monetary) values. The model was applied to real-world data that were procured from a UK retail chain covering four periods of shopping transactions of around 300,000 customers. Internal validity was measured by two different clustering validity indices and a classification accuracy test. Some meaningful information associated with segment stability was extracted to provide practitioners a better understanding of segment stability over time and useful managerial implications.

Insensitive bounds for the stationary distribution of non-reversible Markov chains

1988 ◽  
Vol 25 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Arie Hordijk ◽  
AD Ridder
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Service Time ◽  
Form Solution ◽  
Tree Form ◽  
Standard Product ◽  
Reversible Markov Chains ◽  
Stationary Probabilities ◽  
Networks Of Queues ◽  
General Method

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

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