Efficient descriptor-vector multiplications in stochastic automata networks

Abstract

This paper examines numerical issues in computing solutions to networks of stochastic automata. It is well-known that when the matrices that represent the automata contain only constant values, the cost of performing the operation basic to all iterative solution methods, that of matrix-vector multiply, is given by ρ N = Π N i-1 n i × Σ N i=1 n i , where n i is the number of states in the i th automaton and N is the number of automata in the network. We introduce the concept of a generalized tensor product and prove a number of lemmas concerning this product. The result of these lemmas allows us to show that this relatively small number of operations is sufficient in many practical cases of interest in which the automata contain functional and not simply constant transitions. Furthermore, we show how the automata should be ordered to achieve this.

An Algorithmic Approach to Stochastic Bounds

“…Second, the space required to store the description of components is in general much smaller than the explicit list of transitions, even in a sparse representation. However, using this representation instead of the usual sparse matrix form increases the time required for numerical analysis of the chains [6,15,37,33]. Note that we are interested in performance indices R defined as reward functions on the steady-state distribution (i.e.…”

“…Despite considerable works [7,12,15,37], the numerical analysis of Markov chains, is still a very difficult problem when the state space is too large or the eigenvalues badly distributed. Fortunately enough, while modeling high speed networks, it is often sufficient to satisfy the requirements for the Quality of Service (QoS) we expect.…”

Memory-efficient kronecker algorithms with applications to the modelling of parallel systems

“…In SANs [6,19] and in SGSPNs [18], it has been shown that the transition matrix can be expressed as:…”

“…Therefore, the multiplication of the z in n • k by Q (i) gives z out n • k. A pseudo-code of the E-Sh can be found in [20]. Complexity: The complexity of the algorithm for multiplying a vector and a classic tensor product may be obtained by observing the number of vector-matrix multiplications that are executed [6,19]. If we assume that the matrices Q (i) are stored in a sparse compacted form, and letting nz i denote the number of nonzero elements in the matrix Q (i) , then the complexity of E-Sh is given by…”

“…Within this Kronecker framework, a number of algorithms have been proposed. The first and perhaps best-known, is the shuffle algorithm [5,6,19,20], which computes the product but never needs the matrix explicitly. However, as has been shown previously, this algorithm needs to use vectorsπ the size ofŜ, that we shall call extended.…”