Iterative disaggregation for a class of lumpable discrete-time stochastic automata networks

Abstract

Cataloged from PDF version of article.Stochastic automata networks (SANs) have been developed and used in the last 15 years as a modeling formalism for\ud large systems that can be decomposed into loosely connected components. In this work, we concentrate on the not so much\ud emphasized discrete-time SANs. First, we remodel and extend an SAN that arises in wireless communications. Second, for an\ud SAN with functional transitions, we derive conditions for a special case of ordinary lumpability in which aggregation is done\ud automaton by automaton. Finally, for this class of lumpable discrete-time SANs we devise an efficient aggregation–iterative\ud disaggregation algorithm and demonstrate its performance on the SAN model of interest.\ud © 2002 Elsevier Science B.V. All rights reserved

Analyzing Markov Chains using Kronecker Products

“…For instance, the one in [123] is a model of synchronization via messages passing in a distributed system, that in [120] is a model of the mutual exclusion algorithm of Lamport, those in [73,74] are models of buffer admission mechanisms for asynchronous transfer mode (ATM) networks from telecommunications, that in [149] is a multiservice resource allocation policy for wireless ATM networks, and that in [69] is a model to compute the loss rate in a multistage ATM switch. The model in [88] is a larger, scalable, and extended version of that in [149]. It serves as a good example showing that the underlying MC of a discrete-time model based on Kronecker products can be relatively dense and numerically difficult to analyze.…”

Lumpable continuous-time stochastic automata networks

“…See also [22] and the references therein for examples of lumpable discrete-time SAN models. In the next section, we use the SAN model of the mass storage problem as an example to show that it is not difficult to apply Theorem 4 to a continuous-time SAN model.…”

“…Hence, compared with the original IAD algorithm, the aggregation phase of Algorithm 1 is performed once and each subsequent iteration consists only of disaggregation. The implementation of AID for discrete-time SANs can be found in [22].…”

“…In other words, we assume that the descriptor corresponding to the SAN model at hand is a sum of generalized tensor products. Using the basic results in [22], we derive easy to check conditions for a continuous-time SAN on descriptions of its automata and their ordering that enable us to identify a class of lumpable partitionings in which lumping is performed automaton by automaton. We remark that most of the existing work on exact aggregation of tensor based formalisms amounts to defining equivalence relations among states in a component of the modeled system or among the components of the system [2,3,6].…”

“…Continuing our work in [22], we propose an aggregation-iterative disaggregation (AID) algorithm for a class of lumpable continuous-time SANs and compare its performance with that of block Gauss-Seidel (BGS) on two continuous-time SAN models. To the best of our knowledge, this is the first numerical study of AID on lumpable continuous-time SANs.…”