C.D. Meyer, `Stochastic Complementation, Uncoupling Markov Chains and the Theory of Nearly Reducible Systems', SIAM Reviews, vol.31, no.2, pp. 240-272, 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....solution for a K server system, when the servers are homogeneous, and for a 2 server system, when the servers are heterogeneous. The authors experience difficulties in extending the Green s function method beyond 2 heterogeneous servers. In [16] exact solutions, using stochastic complementation [17], for the K server homogeneous, heterogeneous, and bulk arrival variations of the multi server threshold queueing system with hysteresis are given; no restrictions are placed on the number of servers or the bulk sizes or the size of the waiting room. Stochastic complementation is a more intuitive ....

....contribution of this paper is an efficient solution of a threshold based queueing system with hysteresis obtained through a computation of tight performance bounds. More specifically, we compute the steady state probabilities of the bounding models using a combination of stochastic complementation [17] and the matrix geometric [21] methods. Given the steady state probabilities, we can compute tight bounds on various performance measures of interest. The ease with which we are able to obtain these bounds demonstrates the extensibility of our method. The remainder of this paper is organized as ....

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C.D. Meyer, "Stochastic Complementation, Uncoupling Markov Chains and the Theory of Nearly Reducible Systems ", SIAM Review, Vol. 31, No. 2, pp. 240-272, 1989.

....this paper can be applied to such modified QAM systems with clustered state values in order to reduce the computationalcomplexity of demodulation. Much work has been carried out into grouping states associated with Markov chains (see for example, 7] Techniques such as stochastic complementation [8] are sophisticated methods of producing reduced complexity representations of Markov chains which have large numbers of states. They have been used mainly to evaluate steady state probability distributions [9] and reduced order controllers for Markov systems [10, 11] When there exist only weak ....

C. D. Meyer, "Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems," SIAM Review, vol. 31, no. 2, pp. 240--272, 1989.

....this block structure to define the problem : 8 : X 1 ; X 2 ) A B C D = 0 (X 1 ; X 2 )e = 1 (2) where e is a column vector whose components are all equal to 1. Note that A and D are square blocks. First, note that if the whole matrix is irreducible, then matrix D is not singular [Meyer 1989]. Let us define the matrix H as H = GammaBD Gamma1 . We now have to solve what is known as a reduced problem in Markov chains [Lal and Bhat 1987] 8 : X 1 (A HC) 0 X 1 (e 1 He 2 ) 1 where (e 1 ; e 2 ) t = e X 2 = X 1 H (3) The computation of the reduced system in X 1 is ....

C.D. Meyer, "Stochastic Complementation, Uncoupling Markov Chains, and the Theory of Nearly Reducible Systems", SIAM review, Vol. 31, No. 2, pp. 240-272, June 1989.

....is no directed cycle in the subgraph of G restricted to vertices included in S 1 . So, this subgraph is a directed acyclic graph (dag) and as D is the matrix associated to this subgraph, D is upper triangular if the states in S 1 are numbered with a topological order. Proof of ii) may be found in [10]. 2 Now, we prove the complexity of algorithms based on reduction on a transversal. Lemma 5 If D is upper triangular, then the i th column of H may be computed in time O(N 0 Theta d Gamma i ) Proof: By definition of H , we have for all i: X 1 [i] P N 0 j=1 X 0 [j]H [j; i] As D is ....

....manner: Q = A B 1 K 1 B 2 : K 2 B n Gamma1 B n C 0 . 0 K n Gamma1 0 0 K n where A is a square matrix of size N 0 Theta N 0 and matrices K i are square matrices of size N i Theta N i . We can now give the complexity of this algorithm. The proof is omitted. Note that some results in [10] prove that the matrices K i are not singular. Theorem 3 This reduction algorithm computes the steady state distribution of Markov chain with a complexity of the order of O N 0 m n X i=0 N 3 i N 0 n X i=1 N 2 i : This technique may be applied to a larger family of SAN. ....

Meyer C.D. "Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems," SIAM review, Vol. 31(2), pp. 240--272, June 1989.

....is no directed cycle in the subgraph of G restricted to vertices included in S 2 . So, this subgraph is a directed acyclic graph (dag) and as D is the matrix associated to this subgraph, D is upper triangular if the states in S 2 are numbered with a topological order. Proof of ii) may be found in [5]. 2 Now, we prove the complexity of algorithms based on reduction on a transversal. Lemma 2 If D is upper triangular, then the i th column of H may be computed in time O(N 1 Theta d Gamma i ) Proof : By definition of H, we have for all i : X 2 [i] P N 1 j=1 X 1 [j]H[j; i] As D is ....

C.D. Meyer "Stochastic Complementation, Uncoupling Markov Chains, and the Theory of Nearly Reducible Systems", SIAM review, Vol. 31, No. 2, pp. 240-272, June 1989.

....chain. The stochastic complementation approach provides the theoretical basis for uncoupling Markov chains and furthermore, if applied on the stochastic matrix Q XS , provides as a by product the matrix Q S as we shall see later in one of our results. Definition 5. stochastic complement) [16] Let Q be a n n irreducible stochastic matrix partitioned as (9) where all diagonal blocks are square. For a given index i, let Q i denote the principal block submatrix of Q obtained by deleting the i th row and i th column of blocks from Q, and let Q i and Q i designate (10) That is, Q i ....

....an irreducible stochastic matrix in which the diagonal blocks are square, the stochastic complement of Q 11 is given by and the stochastic complement of Q 22 is . It can be shown that every stochastic complement in Q is also an irreducible matrix. In addition, the following theorem has been proved [16]: Theorem 2. Let Q be an n n irreducible stochastic matrix as in (9) whose stationary probability vector p can be written as (12) with F i e = 1 for i = 1, 2, p 1 . Then F i is the unique stationary probability vector for the stochastic complement S ii and x = x 1 , x 2 , x p ) is ....

C.D. Meyer, `Stochastic Complementation, Uncoupling Markov Chains and the Theory of Nearly Reducible Systems', SIAM Reviews, vol.31, no.2, pp. 240-272, 1989.

.... e; k 1 e; q e) 10) 4 General Aggregation Step The concept of stochastic complementation provides a theoretical basis for reducing a single large chain into a collection of smaller independent chains whose solutions can be aggregated to solve the original chain, see Meyer [7] and the references contained therein. Definition 4.1 (Meyer [7] The stochastic complement of P kk in P, is defined to be the matrix S [k] P kk P k (I Gamma P k ) Gamma1 P k where P k and P k are the k th row and k th column of blocks in P except that P kk is deleted. The ....

....Step The concept of stochastic complementation provides a theoretical basis for reducing a single large chain into a collection of smaller independent chains whose solutions can be aggregated to solve the original chain, see Meyer [7] and the references contained therein. Definition 4. 1 (Meyer [7]) The stochastic complement of P kk in P, is defined to be the matrix S [k] P kk P k (I Gamma P k ) Gamma1 P k where P k and P k are the k th row and k th column of blocks in P except that P kk is deleted. The stochastic complement S [k] is the transition matrix of the reduced ....

Meyer, C. D.: "Stochastic Complementation, Uncoupling Markov Chains, and the Theory of Nearly Reducible Systems," SIAM Review, Vol. 31, No. 2, pp. 240--272, 1989.

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C.D. Meyer, `Stochastic Complementation, Uncoupling Markov Chains and the Theory of Nearly Reducible Systems', SIAM Reviews, vol.31, no.2, pp. 240-272, 1989.

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C. D. Meyer, "Stochastic Complementation, Uncoupling Markov Chains and the Theory of Nearly Reducible Systems," SIAM Review, vol. 31, pp. 240--272, June 1989.

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