Stochastic Automata Networks and Near Complete Decomposability

Abstract

Abstract. Stochastic automata networks (SANs) have been developed and used in the last fifteen years as a modeling formalism for large systems that can be decomposed into loosely connected components. In this work, we extend the near complete decomposability concept of Markov chains (MCs) to SANs so that the inherent difficulty associated with solving the underlying MC can be forecasted and solution techniques based on this concept can be investigated. A straightforward approach to finding a nearly completely decomposable (NCD) partitioning of the MC underlying a SAN requires the computation of the nonzero elements of its global generator. This is not feasible for very large systems even in sparse matrix representation due to memory and execution time constraints. We devise an efficient decompositional solution algorithm to this problem that is based on analyzing the NCD structure of each component of a given SAN. Numerical results show that the given algorithm performs much better than the straightforward approach.

Discrete time queueing networks with product form steady state. Availability and performance analysis in an integrated model

“…Their investigation suffers from the curse of combinatorial complexity in the same way as the networks considered here. In [16] it is shown that even for loosely coupled systems of automata one has to revert to approximations for computing the steady state distribution of the underlying Markov chains.…”

Lumpable continuous-time stochastic automata networks

“…We remark that the case in which a synchronizing transition probability matrix has zero rows corresponds to an implicit functional dependency between the master automaton of the synchronizing event and the slave automaton whose synchronizing transition probability matrix has zero rows [20]. The reason behind using the reverse of the topological ordering in Definition 4 is the direction of the arcs we choose in G to represent dependencies between automata.…”

“…We set C 1 ¼ C 2 ¼ C 3 with values given in column C i . Since the generator has transient states, we first run the state classification (SC) algorithm discussed in [20] to classify the states into recurrent and transient subsets. Columns n r and nz r respectively give the number of recurrent states and the number of nonzero elements in the corresponding submatrix of the generator.…”

“…Without loss of generality, we restrict ourselves to the case in which synchronizing transition probability matrices of a SAN have row sums of 1 or 0. In [20], it is shown how a SAN that does not satisfy this condition can be transformed to an equivalent SAN having synchronizing transition probability matrices with row sums of 1 or 0.…”

“…Further improvements in time and space requirements of numerical solution techniques can be obtained by employing reachability analysis and sparse storage schemes [7] possibly with reordering and grouping of automata under the presence of functional transitions [17] in the efficient vector-tensor product multiplication algorithms. It is also possible to use the directed graph induced by the tensor representation to develop efficient analysis methods for SANs [19][20][21].…”