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 where Bi, Ai, i = ?1, 0, 1, are m x 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 equation X = X 2 A?1 + XA 0 + A 1 . (1) To 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 numericalmethods 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 A 1 , 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 A 1 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 r 1 is the rank of A 1 , 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(m3) arithmetic operations, needed without exploiting the structure of A ?1 and A 1 . Therefore, if r 1 and r 1 /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 A 1 has low rank.