Product Form Solution for Stochastic Automata Networks with synchronizations (original) (raw)

The Kronecker product and stochastic automata networks

Journal of Computational and Applied Mathematics, 2004

This paper can be thought of as a companion paper to Van Loan's The Ubiquitous Kronecker Product paper (J. Comput. Appl. Math. 123 (2000) 85). We collect and catalog the most useful properties of the Kronecker product and present them in one place. We prove several new properties that we discovered in our search for a stochastic automata network preconditioner. We conclude by describing one application of the Kronecker product, omitted from Van Loan's list of applications, namely stochastic automata networks.

Numerical Issues for Stochastic Automata Networks

1996

In this paper we consider some numerical issues in computing solutions to networks of stochastic automata (SAN). In particular our concern is with keeping the amount of computation per iteration to a minimum, since iterative methods appear to be the most e ective in determining numerical solutions. In a previous paper we presented complexity results concerning the vector-descriptor multiplication phase of the analysis. In this paper our concern is with implementation details. We experiment with the size and sparsity of individual automata; with the ordering of the automata; with the percentage and location of functional elements; with the occurrence of di erent types of synchronizing events and with the occurrence of cyclic dependencies within terms of the descriptor. We also consider the possible bene ts of grouping many small automata in a SAN with many small automata to create an equivalent SAN having a smaller number of larger automata.

Queueing networks and conditional product-forms

Proceedings of the 7th International Conference on Performance Evaluation Methodologies and Tools, 2014

Time reversibility plays an important role in the analysis of continuous and discrete time Markov chains (DTMCs). Specifically, the computation of the stationary distribution of a reversible Markov chain has been proved to be very efficient and does not require the solution of the system of global balance equations. A DTMC is reversible when the processes at forward and reversed time are probabilistically indistinguishable. In this paper we introduce the concept of ρ-reversibility, i.e., a notion of reversibility modulo a renaming of the states, and we contrast it with the previous definition of dynamic reversibility especially with respect to the assumptions on the state renaming function. We also discuss the applications of discrete time reversibility in the embedded and uniformized chains of continuous time processes.

Analysis of stochastic Petri nets with signals

Performance Evaluation, 2012

Product-form models facilitate the efficient analysis of large stochastic models and have been sought after for some three decades. Apart from the dominating work on queueing networks, some product-forms were found for stochastic Petri nets (SPNs) that allow fork-join constructs and for queueing networks extended to include special customers called signals, viz. G-networks. We appeal to the Reversed Compound Agent Theorem (RCAT) to prove new product-form solutions for SPNs in which there are special transitions, the firings of which act in a similar way to signals in G-networks, but which may be generated by synchronised firings (or service completions) and may affect several places simultaneously. We show that SPNs with signals are strict generalisations of G-networks with negative customers, triggers and catastrophes, and illustrate with copious examples.

Aggregation of stochastic automata networks with replicas

2004

We present techniques for computing the solution of large Markov chain models whose generators can be represented in the form of a generalized tensor algebra, such as Stochastic Automata Networks (SAN). Many large systems include a number of replications of identical components. This paper exploits replication by aggregating similar components. This leads to a state space reduction, based on lumpability. We define SAN with replicas, and we show how such SAN models can be strongly aggregated, taking functional rates into account. A tensor representation of the matrix of the aggregated Markov chain is proposed, allowing to store this chain in a compact manner and to handle larger models with replicas more efficiently. Examples and numerical results are presented to illustrate the reduction in state space and, consequently, the memory and processing time gains.

Parametric Analysis of Stochastic AutomataNetworks

1993

We investigate some graph properties which may simplify the computation of the steady-state distribution of Markov chain. We consider the directed graph associated to a Markov chain derived from Stochastic Automata Network and we give an algorithm to solve the Kolmogorov equations for the steady-state distribution. Then, we present an extension of this algorithm which allows the parametric analysis of Stochas-tic Automata Networks.

A framework for modeling stochastic flow and synchronization networks

Proceedings of the 2011 Winter Simulation Conference (WSC), 2011

Motivated mainly by infrastructure-network management problems, our group has been pursuing analysis and design of various models for network dynamics, which vary in their specifics but broadly can be viewed as either stochastic flow or synchronization processes defined on a graph. So as to obtain a common framework for these models, here we introduce broad and complementary models for linear stochastic flow and synchronization dynamics in networks, that are structured only in that the network's state evolution is Markov and conditionally linear. We first provide mathematical and graphical formulations for each model, and then verify that the models are broad enough to capture several common synchronization/flow networks. As a first analysis, graph-theoretic characterizations of these models' asymptotics are given; these results generalize and enhance known graphical characterizations of existing synchronization/flow models. A comparison of the stochasticity of different flow network models within the framework is also included.

A survey of product form queueing networks with blocking and their equivalences

Annals of Operations Research, 1994

Queueing network models have been extensively used to represent and analyze resource sharing systems, such as production, communication and information systems. Queueing networks with blocking are used to represent systems with finite capacity resources and with resource constraints. Different blocking mechanisms have been defined and analyzed in the literature to represent distinct behaviors of real systems with limited resources. Exact product form solutions of queueing networks with blocking have been derived, under special constraints, for different blocking mechanisms. In this paper we present a survey of product form solutions of queueing networks with blocking and equivalence properties among different blocking network models. By using such equivalences we can extend product form solution to queueing network models with different blocking mechanisms. The equivalence properties include relationships between open and closed product form queueing networks with different blocking mechanisms.

Lumping and reversed processes in cooperating automata

Annals of Operations Research, 2014

Performance evaluation of computer software or hardware architectures may rely on the analysis of a complex stochastic model whose specification is usually given in terms of a high level formalism such as queueing networks, stochastic Petri nets, stochastic automata or Markovian process algebras. Compositionality is a key-feature of many of these formalisms and allows the modeller to combine several (simple) components to form a complex architecture. However, although these formalisms lead to relative compact specifications of possibly complex models, the derivation of the performance indices may be computationally very time and space consuming since the set of possible states of the model tends to grow exponentially (or even faster) with the number of components. In this paper we focus on models with underlying continuous time Markov chains (CTMCs) and we introduce a notion of typed lumpability, which gives sufficient conditions under which a lumping of the process can be derived, allowing the exact computation of marginal stationary probabilities of the cooperating components. The peculiarity of our method relies on the fact that lumping is applied at the component-level rather than to the CTMC underlying the joint process, thus reducing both the memory requirements and the computational cost of the subsequent solution of the model. Moreover, we investigate the properties of the lumping of reversed automata and we prove that, if these are reversible, a conditional product-form solution of their cooperation with other non-blocking automata can be derived. Although conditional product-forms have been previously investigated by other authors with the aim of approximating non-product-form models, the contribution of this paper consists in giving sufficient conditions for this approach to yield exact results and providing examples to support the modeller's intuition.