Lumpable continuous-time stochastic automata networks

Abstract

The generator matrix of a continuous-time stochastic automata network (SAN) is a sum of tensor products of smaller matrices, which may have entries that are functions of the global state space. This paper specifies easy to check conditions for a class of ordinarily lumpable partitionings of the generator of a continuous-time SAN in which aggregation is performed automaton by automaton. When there exists a lumpable partitioning induced by the tensor representation of the generator, it is shown that an efficient aggregation-iterative disaggregation algorithm may be employed to compute the steady-state distribution. The results of experiments with two SAN models show that the proposed algorithm performs better than the highly competitive block Gauss-Seidel in terms of both the number of iterations and the time to converge to the solution.

Cycle bases of reduced powers of graphs

“…Redundancy in master Markov chains for interacting stochastic automata can often be eliminated without approximation. Both lumpability at the level of individual automata and model composition have been extensively researched, though the latter reduces the state space in a manner that eliminates Kronecker structure [4,6,13]. To see this, consider k identical and indistinguishable stochastic automata, each with v states, that interact via transition rates that are functions of the global state, that is, Q = k F where F (i, j) :…”

Analyzing Markov Chains using Kronecker Products

“…Note that replication refers to a very specific kind of symmetry in the Kronecker representation, and with ordinary lumpability only performance measures of interest over S lumped can be computed. Lumpability can also be investigated among the state spaces S .h/ by considering dependencies and matrix properties in the Kronecker representation as in [87,88]. There, sufficient conditions that satisfy ordinary lumpability are specified and an iterative steady-state solution method that is able to compute performance measures over S is given for CTMCs and DTMCs in the presence of functional transitions.…”

“…The work identifies lumpable partitionings on the underlying MC induced by the nested block structure of the Kronecker representation in (2.2). Although the particular approach of lumping one or more state spaces S .h/ totally as in [87,88] is a very specific kind of performance equivalence and lumping considered in [21,23], due to its accommodation of functional transitions, it also enables the detection of certain ordinarily lumpable partitionings in which blocks are composed of multiple (nonidentical) state spaces, but the individual state spaces cannot be lumped by themselves. This is not possible with the approaches in [9,21,23].…”

“…We remark that neither of the two approaches in [9] and [87,88] that investigate lumpability among the state spaces S .h/ for h D 1; : : : ; H is completely automated, uses a Kronecker representation for the lumped MC, and possesses a proper complexity analysis. Furthermore, since the Kronecker representation is rich in structure and the three approaches presented in this chapter do not work on the flat representation, there can very well be other symmetries in the Kronecker representation that also lead to lumpability.…”

Lumping Matrix Diagram Representations of Markov Models

“…In other words, we can apply compositional and model-level lumping techniques simultaneously on a compositional model whose underlying CTMC is represented as an MD. Our work is related to [11] in that we argue on the level of a block structured matrix to observe lumpability, but unlike [11], is not limited to Kronecker matrices and stochastic automata networks. It is related to [3] in that we have a local condition but do not separate local and synchronized actions as was done for the automata theoretic approach in [3].…”