A kind of dual form for coupling from the past algorithm, to sample from Markov chain steady-state probability

2019 ◽  
Vol 25 (4) ◽  
pp. 317-327
Author(s):  
Abdelaziz Nasroallah ◽  
Mohamed Yasser Bounnite
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Steady State ◽  
Performance Comparison ◽  
State Probability ◽  
Steady State Probability ◽  
Dual Form ◽  
The Past ◽  
Coupling From The Past ◽  
History Of

Abstract The standard coupling from the past (CFTP) algorithm is an interesting tool to sample from exact Markov chain steady-state probability. The CFTP detects, with probability one, the end of the transient phase (called burn-in period) of the chain and consequently the beginning of its stationary phase. For large and/or stiff Markov chains, the burn-in period is expensive in time consumption. In this work, we propose a kind of dual form for CFTP called D-CFTP that, in many situations, reduces the Monte Carlo simulation time and does not need to store the history of the used random numbers from one iteration to another. A performance comparison of CFTP and D-CFTP will be discussed, and some numerical Monte Carlo simulations are carried out to show the smooth running of the proposed D-CFTP.

scholarly journals Estimation of extreme inter-day changes to peak electricity demand using Markov chain analysis: A comparative analysis with extreme value theory

2017 ◽  
Vol 28 (4) ◽  
Author(s):  
Caston Sigauke ◽  
Delson Chikobvu
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Return Period ◽  
Electricity Demand ◽  
State Probability ◽  
Markov Chain Analysis ◽  
State Problem ◽  
Steady State Probability ◽  
Chain Analysis ◽  
Steady State Probabilities

Uncertainty in electricity demand is caused by many factors. Large changes are usually attributed to extreme weather conditions and the general random usage of electricity by consumers. More understanding requires a detailed analysis using a stochastic process approach. This paper presents a Markov chain analysis to determine stationary distributions (steady state probabilities) of large daily changes in peak electricity demand. Such large changes pose challenges to system operators in the scheduling and dispatching of electrical energy to consumers. The analysis used on South African daily peak electricity demand data from 2000 to 2011 and on a simple two-state discrete-time Markov chain modelling framework was adopted to estimate steady-state probabilities of two states: positive inter-day changes (increases) and negative inter-day changes (decreases). This was extended to a three-state Markov chain by distinguishing small positive changes and extreme large positive changes. For the negative changes, a decrease state was defined. Empirical results showed that the steady state probability for an increase was 0.4022 for the two-state problem, giving a return period of 2.5 days. For the three state problem, the steady state probability of an extreme increase was 0.0234 with a return period of 43 days, giving approximately nine days in a year that experience extreme inter-day increases in electricity demand. Such an analysis was found to be important for planning, load shifting, load flow analysis and scheduling of electricity, particularly during peak periods.

A formula for singular perturbations of Markov chains

1994 ◽  
Vol 31 (03) ◽  
pp. 829-833 ◽  
Author(s):  
Jean B. Lasserre
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Probability Distribution ◽  
Markov Chains ◽  
Singular Perturbations ◽  
Singularly Perturbed ◽  
State Probability ◽  
Regular Perturbation ◽  
Steady State Probability ◽  
Fundamental Matrices

We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.

A formula for singular perturbations of Markov chains

1994 ◽  
Vol 31 (3) ◽  
pp. 829-833 ◽  
Author(s):  
Jean B. Lasserre
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Probability Distribution ◽  
Markov Chains ◽  
Singular Perturbations ◽  
Singularly Perturbed ◽  
State Probability ◽  
Regular Perturbation ◽  
Steady State Probability ◽  
Fundamental Matrices

We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.

Runway Utilization Analysis Based on the Stochastic Petri Net Model

2013 ◽  
Vol 411-414 ◽  
pp. 1750-1756
Author(s):  
Gao Yang Jiang ◽  
Jie Ning Wang ◽  
Chun Feng Zhang ◽  
Mei Dong
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Petri Net ◽  
Simulation Method ◽  
State Probability ◽  
Homogeneous Markov Chain ◽  
Steady State Probability ◽  
Stochastic Petri Net ◽  
Markov Chain Method ◽  
Key Indicators

Runway utilization is one of the key indicators of airport operational efficiency. Firstly, stochastic Petri net was introduced to built runway system operational model, and then we analyzed the reachability graph of this model, which not only prove the reachability and boundedness of this model, but also can be used to transform to homogeneous Markov chain. Secondly, The system steady-state probability expressions in various states were established based on the homogeneous Markov chain. Thirdly, runway utilization was calculated based on the steady-state probability expressions. During simulation, runway utilizations in various conditions were analyzed by changing some transitions fire rate. Both Markov chain method and petri net simulation method are useful for runway utilization improvement.

scholarly journals MATHEMATICAL ANALYSIS OF VEHICLE DELIVERY SCALE OF BIKE-SHARING RENTAL NODES

2018 ◽  
Vol XLII-3 ◽  
pp. 2229-2235
Author(s):  
Y. Zhai ◽  
J. Liu ◽  
L. Liu
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Transition Probability ◽  
Limit State ◽  
Linear Equations ◽  
Transition Probability Matrix ◽  
State Probability ◽  
Steady State Probability ◽  
Bike Sharing ◽  
State Scale

Aiming at the lack of scientific and reasonable judgment of vehicles delivery scale and insufficient optimization of scheduling decision, based on features of the bike-sharing usage, this paper analyses the applicability of the discrete time and state of the Markov chain, and proves its properties to be irreducible, aperiodic and positive recurrent. Based on above analysis, the paper has reached to the conclusion that limit state (steady state) probability of the bike-sharing Markov chain only exists and is independent of the initial probability distribution. Then this paper analyses the difficulty of the transition probability matrix parameter statistics and the linear equations group solution in the traditional solving algorithm of the bike-sharing Markov chain. In order to improve the feasibility, this paper proposes a "virtual two-node vehicle scale solution" algorithm which considered the all the nodes beside the node to be solved as a virtual node, offered the transition probability matrix, steady state linear equations group and the computational methods related to the steady state scale, steady state arrival time and scheduling decision of the node to be solved. Finally, the paper evaluates the rationality and accuracy of the steady state probability of the proposed algorithm by comparing with the traditional algorithm. By solving the steady state scale of the nodes one by one, the proposed algorithm is proved to have strong feasibility because it lowers the level of computational difficulty and reduces the number of statistic, which will help the bike-sharing companies to optimize the scale and scheduling of nodes.

scholarly journals Generating a Random Sink-free Orientation in Quadratic Time

10.37236/1627 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Henry Cohn ◽  
Robin Pemantle ◽  
James Propp
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Undirected Graph ◽  
Randomized Algorithm ◽  
Degree Of Approximation ◽  
The Past ◽  
Coupling From The Past ◽  
Mean Time ◽  
Random Sink

A sink-free orientation of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately from the uniform distribution on sink-free orientations in time $O(m^3 \log (1 / \varepsilon))$, where $m$ is the number of edges and $\varepsilon$ the degree of approximation. Huber (1998) uses coupling from the past to obtain an exact sample in time $O(m^4)$. We present a simple randomized algorithm inspired by Wilson's cycle popping method which obtains an exact sample in mean time at most $O(nm)$, where $n$ is the number of vertices.

scholarly journals The use of Markov Chain method to determine spare transformer number and location

2019 ◽  
Vol 9 (1) ◽  
pp. 1
Author(s):  
Musa Partahi Marbun ◽  
Ngapuli Irmea Sinisuka ◽  
Nanang Hariyanto
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
State Transition ◽  
State Probability ◽  
Transition Model ◽  
Steady State Probability ◽  
Actual Result ◽  
Markov Chain Method ◽  
State Transition Model

<span lang="EN-US">The purpose of this study is to develop a method to determine spare transformer number and location. Using Markov Chain method, state transition model and steady state probability was used on each 500-kV substation in order to analyze the effect of spare number and location variation with the reliability changes.  To give an actual result of the case study, calculation of spare transformer number and location on 500/150 kV transformers in Java Bali System was analyzed. The steady state probability results will vary depending on the number of spare transformer, these results can then be used to assess the spare transformer needed. The variation of spare transformer location can be used to analyze the best possible location of the spare in order to satisfy the reliability required. The methodology presented shows an integrated calculation for determining the spare transformer number and location.</span>

Widening and clustering techniques allowing the use of monotone CFTP algorithm

2015 ◽  
Vol 21 (4) ◽  
Author(s):  
Mohamed Yasser Bounnite ◽  
Abdelaziz Nasroallah
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Simulations ◽  
State Space ◽  
Stationary Distribution ◽  
Clustering Techniques ◽  
Clustering Technique ◽  
The Past ◽  
Coupling From The Past

AbstractThe standard Coupling From The Past (CFTP) algorithm is an interesting tool to sample from exact stationary distribution of a Markov chain. But it is very expensive in time consuming for large chains. There is a monotone version of CFTP, called MCFTP, that is less time consuming for monotone chains. In this work, we propose two techniques to get monotone chain allowing use of MCFTP: widening technique based on adding two fictitious states and clustering technique based on partitioning the state space in clusters. Usefulness and efficiency of our approaches are showed through a sample of Markov Chain Monte Carlo simulations.

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