On the stochastic structure of parallelism and synchronization models for distributed algorithms

Plateau, Brigitte

doi:10.1145/317786.317819

Cited by 177 publications

(92 citation statements)

References 6 publications

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“…When the G-reset arrives at an empty queue, then its population jumps to n > 0 selected with probability π(n)/(1 − π (0) simplified model with an underlying Birth&Death process we resort again to the propagation of instantaneous transitions. First, we introduce the following notation: In the case that for a state s there are more than one outgoing passive transition with the same label, we can assign a probability of synchronization in an analogue way of what happens for SAN [62], PEPA [49,50]. Here, with an abuse of notation, we write as subscript of the symbol the probability that the synchronization occurs with that transition, e.g., from state s we may have two outgoing transitions with label a written as (a, p ) and (a, 1−p ).…”

Section: Gelenbeandfourneau Resets (G-resets)mentioning

confidence: 99%

Product-Form in G-Networks

Marin

2016

Prob. Eng. Inf. Sci.

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The introduction of the class of queueing networks called G-networks by Gelenbe has been a breakthrough in the field of stochastic modeling since it has largely expanded the class of models which are analytically or numerically tractable. From a theoretical point of view, the introduction of the G-networks has lead to very important considerations: first, a product-form queueing network may have non-linear traffic equations; secondly, we can have a product-form equilibrium distribution even if the customer routing is defined in such a way that more than two queues can change their states at the same time epoch. In this work, we review some of the classes of product-forms introduced for the analysis of the G-networks with special attention to these two aspects. We propose a methodology that, coherently with the product-form result, allows for a modular analysis of the G-queues to derive the equilibrium distribution of the network.

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Section: Gelenbeandfourneau Resets (G-resets)mentioning

confidence: 99%

Product-Form in G-Networks

Marin

2016

Prob. Eng. Inf. Sci.

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“…In terms of a rate matrix, the Kronecker product represents synchronization (Plateau, 1985). If we have two variables, X 1 and X 2 with rate 2 matrices R 1 and R 2 , R 1 ⊗ R 2 is a rate matrix over the state space X = X 1 × X 2 (joint assignments to X 1 and X 2 ).…”

Section: Kronecker Productmentioning

confidence: 99%

Tutorial on Structured Continuous-Time Markov Processes

Shelton¹,

Ciardo²

2014

jair

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A continuous-time Markov process (CTMP) is a collection of variables indexed by a continuous quantity, time. It obeys the Markov property that the distribution over a future variable is independent of past variables given the state at the present time. We introduce continuous-time Markov process representations and algorithms for filtering, smoothing, expected sufficient statistics calculations, and model estimation, assuming no prior knowledge of continuous-time processes but some basic knowledge of probability and statistics. We begin by describing "flat" or unstructured Markov processes and then move to structured Markov processes (those arising from state spaces consisting of assignments to variables) including Kronecker, decision-diagram, and continuous-time Bayesian network representations. We provide the first connection between decision-diagrams and continuoustime Bayesian networks. Tutorial GoalsThis tutorial is intended for readers interested in learning about continuous-time Markov processes, and in particular compact or structured representations of them. It is assumed that the reader is familiar with general probability and statistics and has some knowledge of discrete-time Markov chains and perhaps hidden Markov model algorithms.While this tutorial deals only with Markovian systems, we do not require that all variables be observed. Thus, hidden variables can be used to model long-range interactions among observations. In these models, at any given instant the assignment to all state variables is sufficient to describe the future evolution of the system. The variables themselves real-valued (continuous) times. We consider evidence or observations that can be regularly spaced, irregularly spaced, or continuous over intervals. These evidence patterns can change by model variable and time.We deal exclusively with discrete-state continuous-time systems. Real-valued variables are important in many situations, but to keep the scope manageable, we will not treat them here. We refer to the work of Särkkä (2006) for a machine-learning-oriented treatment of filtering and smoothing in such models. The literature on parameter estimation is more scattered. We will further constrain our discussion to systems with finite states, although many of the concepts can be extended to countably infinite state systems.We will be concerned with two main problems: inference and learning (parameter estimation). These were chosen as those most familiar to and applicable for researchers in artificial intelligence. At points we will also discuss the computation of steady-state properties, especially for model for which most research concentrates on this computation.

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“…This information regarding the automata and their types of transitions provides all the information needed to formally deÿne a SAN, as Atif and Plateau have done [2]. While this infrequent interaction (synchronizing transitions and functional transition rates) does complicate SANs, Plateau and her co-workers have shown that the SAN can still be represented in compact form as a sum of Kronecker products, known as the SAN descriptor [1,3,4]. Of course, too much interaction may complicate the SAN model to the point that all savings have been lost.…”

Section: Introductionmentioning

confidence: 99%

“…SANs were ÿrst proposed by Plateau [1] in 1985 and have been actively researched since. A SAN is a collection of individual stochastic automata that generally act independently of one another, requiring only infrequent interaction.…”

Section: Introductionmentioning

confidence: 99%

A Kronecker product approximate preconditioner for SANs

Langville

Stewart

2004

Numerical Linear Algebra App

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SUMMARYMany very large Markov chains can be modelled e ciently as stochastic automata networks (SANs). A SAN is composed of individual automata which, for the most part, act independently, requiring only infrequent interaction. SANs represent the generator matrix Q of the underlying Markov chain compactly as the sum of Kronecker products of smaller matrices. Thus, storage savings are immediate. The beneÿt of a SAN's compact representation, known as the descriptor, is often outweighed by its tendency to make analysis of the underlying Markov chain tough. While iterative or projections methods have been used to solve the system Q = 0, the time until these methods converge to the stationary solution is still unsatisfactory. SAN's compact representation has made the next logical research step of preconditioning thorny. Several preconditioners for SANs have been proposed and tested, yet each has enjoyed little or no success. Encouraged by the recent success of approximate inverses as preconditioners, we have explored their potential as SAN preconditioners. One particularly relevant ÿnding on approximate inverse preconditioning is the nearest Kronecker product approximation discovered by Pitsianis and Van Loan. In this paper, we extend the nearest Kronecker product technique to approximate the Q matrix for an SAN with a Kronecker product, A 1 ⊗ A 2 ⊗ · · · ⊗ A N . Then, we take M = A

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On the stochastic structure of parallelism and synchronization models for distributed algorithms

Cited by 177 publications

References 6 publications

Product-Form in G-Networks

Product-Form in G-Networks

Tutorial on Structured Continuous-Time Markov Processes

A Kronecker product approximate preconditioner for SANs

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