M. Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, C-30(2):116-125, February 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....spaces of C 1 and C 2 . Process C possesses the transition rate matrix R which is given by the Kronecker sum of R 1 and R 2 : R = R 1 Phi R 2 = R 1 Omega I d2 I d1 Omega R 2 where Omega denotes Kronecker product, Phi denotes Kronecker sum and I d denotes an identity matrix of size d [37]. If, however, C 1 and C 2 are not independent, but perform certain transitions synchronously, the expression for the overall transition rate matrix changes to R = R 1;i Phi R 2;i a Delta R R 2;a where R 1;i and R 2;i contain those transitions which C 1 and C 2 perform independently ....

M. Davio. Kronecker Products and Shuffle Algebra. IEEE Transactions on Computers, C-30(2):116--125, February 1981.

....K k=1 E k , ffl use this decomposition to get an expression of Q as function f(Q 1 ; QK ) where Q k is the generator of the CTMC restricted to E k , ffl compute the solution with :f(Q 1 ; QK ) 0. 2 In our context, the functions f are sums of tensor products of the Q k (see [6] and [14] for details about tensor algebra and its use in the area of stochastic transition systems) The main interest of this method is to allow the computation of the steady state probabilities directly using the tensor expression of Q, without computing the Q matrix. Trying to merge the two ....

....f q (W Gamma (q; t) c) An implicit place (without explicit set P 0 ) is an implicit place w.r.t some P 0 . Appendix B Tensor algebra a short overview We give here some basic results about tensor algebra used in decomposition methods. Demonstrations and further properties may be found in [6, 14]. Multibase indexing is the result of the lexicographic ordering used on the product space. Denition B.1 (tensor product) Let A 2 M n 1 p 1 and B 2 M n 2 p 2 . The tensor product ( N ) of A and B is the matrix C 2 M (n 1 n 2 ) p 1 p 2 ) C = A O B with c ij = a i 1 j 1 b i 2 j 2 where i = ....

M. Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, 30(2):116125, 1981.

....q2P 0 f q (W (q; t) c) An implicit place (without explicit set P 0 ) is an implicit place w.r.t some P 0 . Appendix B Tensor algebra a short overview We give here some basic results about tensor algebra used in decomposition methods. Demonstrations and further properties may be found in [8, 18]. Multibase indexing is the result of the lexicographic ordering used on the product space. De nition B.1 (tensor product) Let A 2 M n 1 p 1 and B 2 M n 2 p 2 . The tensor product ( N ) of A and B is the matrix C 2 M (n1n2 ) p1 p2 ) C = A O B with c ij = a i 1 j 1 b i 2 j 2 where i = i 1 ....

M. Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, 30(2):116125, 1981.

....Structured representations are based on Kronecker algebra to combine incidence matrices R i of reachability graphs of components RG(PN i ) Kronecker algebra is sometimes named tensor algebra as well, however the former term seems to be more usual. We define Kronecker products according to [19], but only for square matrices, since only square matrices are relevant in our context. Definition 3.1 Kronecker product, Kronecker sum Let A 0 ; A N Gamma1 be square matrices of dimension (k i Theta k i ) for i 2 f0; N Gamma 1g, then their Kronecker product A = N N ....

M. Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, C-30(2):116--125, Feb. 1981. 21

....have been described in [1, 3, 10, 22, 24] Most previous approaches are based on the graph and binary representations of these networks. It has been proved that the graph and binary representations for a given class of interconnection networks are equivalent to tensor product representations [2]. However, the tensor product notation is more versatile in the ability to represent both algorithms and architectures and hence is expected to serve as a better representation for algorithm mapping. In this paper, we focus on modeling direct interconnection networks. Using the tensor product and ....

Marc Davio. Kronecker Products and Shuffle Algebra. IEEE Transaction on Computers, C-30(2):116--125, Feb. 1981.

....use bit strings to represent the permutations performed by single and multistage interconnection networks to partition a class of networks. It has been proved that the graph and binary representations for a given class of interconnection networks are equivalent to the tensor product representation [8]. However, the tensor product representation is more versatile in the ability to represent both algorithms and architectures. In this paper, we use an algebraic theory based on the tensor product notation for representing multistage interconnection networks. A class of full access, unique path, ....

Marc Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, C-30(2):116--125, 1981.

....synchronization transitions. By definition of the tensor sum and product, G is a jPSj Theta jPSj matrix, and it is shown in [21, 32] how the non null entries of the vector , solution of the equation Delta G = 0, are the steady state solution of S. Moreover, a solution process may be devised [18, 32] that does not require the explicit computation and storing of G, so that the biggest memory requirement is that of the vector . The technique is extended to transient analysis in [28] The computational cost, under full matrix implementation assumption, is smaller than the classical vector to ....

.... b 12 a 21 b 13 a 22 b 11 a 22 b 12 a 22 b 13 a 21 b 21 a 21 b 22 a 21 b 23 a 22 b 21 a 22 b 22 a 22 b 23 1 C C C A In case of square matrices A and B can be interpreted as the matrices of transition probabilities of two discrete time Markov chains, it is immediate to recognize (see Davio in [18]) that C is the transition probabilities matrix of the process obtained as independent composition of the two original processes. Let us now define the Kronecker (or tensor) sum of two square matrices 30 Definition 15 Let A be a n Theta n matrix, and B be a p Theta p one; D is the tensor ....

M. Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, 30(2):116-- 125, 1981.

...., ffl use this decomposition to get an expression of Q as function f(Q 1 ; QK ) where Q k is the innitesimal generator of the CTMC restricted to E k , ffl compute the solution with :f(Q 1 ; QK ) 0. In our context, the functions f are sums of tensor products of the Q k (see [4] and [12] for details about tensor algebra and its use in the area of stochastic transition systems) In the framework of Petri nets, two composition methods are used: the synchronous composition of subnets by transition merging (corresponding to temporal synchronization of subsystems (as in the ....

M. Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, 30(2):116125, 1981.

....to generate the state space and to find the steady state probabilities of a stochastic extension of DSSP in a net driven, efficient way. Essentially, we give an expresion of an auxiliary matrix, G, which is a supermatrix of the infinitesimal generator of a DSSP. G is a tensor algebra [9] expression of matrices of the size of the components for which it is possible to numerically solve the characteristic equation Delta G = 0, without the need to explicitly compute G. Therefore, we obtain a method that computes the steady state solution of a DSSP without ever explicitly ....

M. Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, 30(2):116-- 125, 1981.

....of normal form. lumping 1 0 B B B 0 1 2 0 0 0 0 0 0 0 0 0 0 X 1 C C C A 0 B 0 1 2 0 0 0 0 1 C A 0 1 2 0 2 recX : X 1 2 1 2 Figure 11: Recursion example that makes further lumping necessary For an introduction to tensor algebra refer to [Dav81]. Assuming that QA is of dimension s, it follows that Q is of dimension s n . The s n states of Q can be numbered using n tuples built from the digits f0; 1; s Gamma 1g in ascending lexicographical ordering: state 0 = 0 ; 0 ; 0 ) state 1 = 0 ; 0 ; 1 ) state ....

M. Davio. Kronecker Products and Shuffle Algebra. IEEE Transactions on Computers, C-30(2):116--125, February 1981.

....Similar approaches are taken in [3, 5] In this paper, we will briefly describe the Stochastic Automata Networks with special emphasis in the continuous time models. Particularly the next section will present the modelling formalism and its representation through a tensor algebra description [4]. In the following section some solution methods are cited in order to present the advantages to compute solutions from a SAN instead of apply standard solutions to huge transition matrices. 2 Tensor Algebra Description Since SAN uses a compact form of tensor algebra description to represent ....

....A (k) Omega I n k 1 Omega Delta Delta Delta Omega I nN ; 4) where n k is the order of the matrix A (k) and I n k is the identity matrix of order n k . The operators Omega and Phi are not commutative. More information concerning the properties of tensor algebra may be found in [4]. 2.2 Generalized tensor operators Let us take the following matrices A and B as example. Assume that these matrices are transition matrices. Namely A is a transition matrix of two states (a 1 and a 2 ) automaton. A = a 11 a 12 a 21 a 22 and B(a) 0 B b 11 (a) b 12 (a) b 13 (a) b ....

M. Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, Vol. C-30, No. 2, Feb. 1981.

....blocks. A model which is composed of subsystems with almost no mutual interference falls into the class of NIS. The state space of the joined model is equal to the Cartesian product of the subsystems state spaces. NIS models have a generator matrix structure which is close to the tensor sum [12] of the subsystems generator matrices. The major difficulty with the decomposition approach is given the joined model to recognize the way in which the model should be decomposed. A model is decomposa2 Structured Model Description NIS Approx. NCD [Courtois] Ciardo] PFQN [BCMP] Exact Model ....

M. Davio. Kronecker Products and Shuffle Algebra. IEEE Transactions on Computers, C-30(2):116--125, February 1981.

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M. Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, C-30(2):116-125, February 1981.

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M. Davio. Kronecker Products and Shu#e Algebra. IEEE Transactions on Computers, C-30(2):116--125, February 1981.

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M. Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, C-30(2):116--125, February 1981.

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M.Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, vol. C-30, no. 2, 1981, pp. 116-125.

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M.Davio. Kronecker products and shue algebra. IEEE Transactions on Computers, vol. C-30, no. 2, 1981, pp. 116-125.

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