Exact Monte Carlo simulation for fork-join networks

In a fork-join network each incoming job is split into K tasks and the K tasks are simultaneously assigned to K parallel service stations for processing. For the distributions of response times and queue lengths of fork-join networks, no explicit formulae are available. Existing methods provide only analytic approximations for the response time and the queue length distributions. The accuracy of such approximations may be difficult to justify for some complicated fork-join networks. In this paper we propose a perfect simulation method based on coupling from the past to generate exact realisations from the equilibrium of fork-join networks. Using the simulated realisations, Monte Carlo estimates for the distributions of response times and queue lengths of fork-join networks are obtained. Comparisons of Monte Carlo estimates and theoretical approximations are also provided. The efficiency of the sampling algorithm is shown theoretically and via simulation.

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Attractive regular stochastic chains: perfect simulation and phase transition

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.7 ◽

2013 ◽

Vol 34 (5) ◽

pp. 1567-1586 ◽

Cited By ~ 6

Author(s):

SANDRO GALLO ◽

DANIEL Y. TAKAHASHI

Keyword(s):

Phase Transition ◽

Infinite Order ◽

Concentration Of Measure ◽

Finite Alphabet ◽

Perfect Simulation ◽

Exponential Rate ◽

The Past ◽

Coupling From The Past ◽

Probability Kernel ◽

Independent And Identically Distributed

AbstractWe prove that uniqueness of the stationary chain, or equivalently, of the$g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an independent and identically distributed (i.i.d.) process with countable alphabet; (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson–Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.

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

Monte Carlo Methods and Applications ◽

10.1515/mcma-2019-2050 ◽

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.

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Perfect simulation using dominated coupling from the past with application to area-interaction point processes and wavelet thresholding

Probability and Mathematical Genetics ◽

10.1017/cbo9781139107174.005 ◽

2011 ◽

pp. 64-90 ◽

Cited By ~ 1

Author(s):

G. K. Ambler ◽

B. W. Silverman

Keyword(s):

Point Processes ◽

Wavelet Thresholding ◽

Perfect Simulation ◽

Interaction Point ◽

The Past ◽

Coupling From The Past

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Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

Advances in Applied Probability ◽

10.1239/aap/1013540247 ◽

2000 ◽

Vol 32 (3) ◽

pp. 844-865 ◽

Cited By ~ 84

Author(s):

Wilfrid S. Kendall ◽

Jesper Møller

Keyword(s):

Point Process ◽

Discrete Time ◽

Point Processes ◽

Random Point ◽

Perfect Simulation ◽

Stable Point ◽

Birth And Death Processes ◽

The Past ◽

Coupling From The Past ◽

Equilibrium Distributions

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

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Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

Advances in Applied Probability ◽

10.1017/s0001867800010284 ◽

2000 ◽

Vol 32 (03) ◽

pp. 844-865 ◽

Cited By ~ 18

Author(s):

Wilfrid S. Kendall ◽

Jesper Møller

Keyword(s):

Point Process ◽

Discrete Time ◽

Point Processes ◽

Random Point ◽

Perfect Simulation ◽

Stable Point ◽

Birth And Death Processes ◽

The Past ◽

Coupling From The Past ◽

Equilibrium Distributions

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

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Perfect posterior simulation for mixture and hidden Markov models

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157007000022 ◽

2010 ◽

Vol 13 ◽

pp. 246-259

Author(s):

Kasper K. Berthelsen ◽

Laird A. Breyer ◽

Gareth O. Roberts

Keyword(s):

Bayesian Inference ◽

Statistical Models ◽

Markov Models ◽

Hidden Markov ◽

Perfect Simulation ◽

Unknown Parameters ◽

The Past ◽

Coupling From The Past ◽

Posterior Simulation ◽

Unknown Mixture

AbstractIn this paper we present an application of the read-once coupling from the past algorithm to problems in Bayesian inference for latent statistical models. We describe a method for perfect simulation from the posterior distribution of the unknown mixture weights in a mixture model. Our method is extended to a more general mixture problem, where unknown parameters exist for the mixture components, and to a hidden Markov model.

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Coupling from the past with randomized quasi-Monte Carlo

Mathematics and Computers in Simulation ◽

10.1016/j.matcom.2009.09.003 ◽

2010 ◽

Vol 81 (3) ◽

pp. 476-489 ◽

Cited By ~ 3

Author(s):

P. L’Ecuyer ◽

C. Sanvido

Keyword(s):

Monte Carlo ◽

Quasi Monte Carlo ◽

The Past ◽

Coupling From The Past

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Generating a Random Sink-free Orientation in Quadratic Time

The Electronic Journal of Combinatorics ◽

10.37236/1627 ◽

2002 ◽

Vol 9 (1) ◽

Cited By ~ 2

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.

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Widening and clustering techniques allowing the use of monotone CFTP algorithm

Monte Carlo Methods and Applications ◽

10.1515/mcma-2015-0111 ◽

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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