A compressed cyclic reduction for QBDs with low rank upper and lower transitions

Abstract

Consider a quasi-birth-and-death (QBD) Markov chain [6], having probability transition matrix P = 2 6 6 6 6 6 6 4 B0 B1 0 B−1 A0 A1 A−1 A0 A1 A−1 A0 . . . 0 . . . . . . 3 7 7 7 7 7 7 5, where B i, Ai, i = −1, 0, 1, are m × m matrices. In the numerical solution of QBD Markov chains a crucial step is the efficient computation of the minimal nonnegative solution R of the quadratic matrix equationTo this purpose, many numerical methods, with different properties, have been designed in the last years (see for instance [1, 2, 3, 4]). However, many of these numerical methods are defined for general block coefficients A−1, A0 and A1, and do not exploit the possible structure of these blocks.Recently, some attention has been addressed to the case where A−1 has only few non-null columns, or A1 has only few non-null rows. These properties are satisfied when the QBD has restricted transitions to higher (or lower) levels.In particular, in [7] the authors exploit these properties of the matrix A−1, or A1, to formulate the QBD in terms of an M/G/1 type Markov chain, where the block matrices have size smaller than m; in particular, when both A−1 and A1 have the desired property, the latter M/G/1 type Markov chain reduces to a QBD. In [5] the structure of A−1 is used in order to reduce the computational cost of some algorithms for computing R.Here we assume that both the matrices A−1 and A1 have small rank with respect to their size m. In particular, if A−1 and A1 have only few non-null columns and rows, respectively, they have small rank. We show that, under this assumption, the matrix R can be computed by using the cyclic reduction algorithm, where the matrices A (k) i , i = −1, 0, 1, generated at the kth step of the algorithm, can be represented by small rank matrices. In particular, if r−1 is the rank of A−1, and if r1 is the rank of A1, then each step of cyclic reduction can be performed by means of O((r−1+r1) 3 ) arithmetic operations. This cost estimate must be compared with the cost of O(m 3 ) arithmetic operations, needed without exploiting the structure of A−1 and A1. Therefore, if r−1 and r1 are much smaller than m, the advantage is evident.It remains an open issue to understand how the structure can be exploited in the case where only one between A−1 and A1 has low rank.