Decomposability. Queueing and Computer System Applications by P. J. Courtois, Robert L. Ashenhurst

By P. J. Courtois, Robert L. Ashenhurst

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X [ t - 1 ] , may have to be added. 45) must be taken into consideration at the corresponding levels of aggregation. 35), for example, would be block stochastic, as well as its first level aggregation matrix P [ 1 ] , if the row sums were equal in each submatrix Rj, Sj, and TSi s 2 s3] Ls 4 s 5 s 6 J s7 s9 s8 Sio Sn Sl2 - A numerical example of a two-level aggregation is given at the end of Appendix III. 5 The lesson to be drawn from this section is simply the following. For an approximation by multilevel aggregation to be satisfactorily accurate, the couplings ει between aggregates must not only be small at each level, but they must also be small compared with the intraaggregate couplings ηι-γ.

3 Block Stochastic Systems It is interesting to examine the specific case where the nearly completely decomposable matrix is at the same time block stochastic; such matrices are indeed particularly well suited for an analysis by aggregation. Two cases must be distinguished: a nearly completely decomposable matrix may be, as we shall call it, either row block stochastic or column block stochastic; in the first case the sum of each row's elements in respective submatrices QJJ are equal; in the second case the sums of each column's elements are equal.

1 which is involved require ε to be small. This explains why the corollary does not specify that Q must be nearly completely decomposable. Let us note that, since the aggregative matrix P is identical to the matrix \WU~\ when Q is row block stochastic, the first part of this corollary is similar to Haynsworth's (1959) theorem, which states that the eigenvalues of \_WU~] are also eigenvalues of Q. 1 is that nearly completely decomposable matrices that are row block stochastic are well suited for an analysis by 38 II ON THE DEGREE OF APPROXIMATION aggregation.

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