  The treatment of two-dimensional random walks in the quarter plane leads to
Markov processes which involve semi-infinite matrices having Toeplitz or block
Toeplitz structure plus a low-rank correction. Finding the steady state
probability distribution of the process requires to perform operations
involving these structured matrices. We propose an extension of the framework
of [5] which allows to deal with more general situations such as processes
involving restart events. This is motivated by the need for modeling processes
that can incur in unexpected failures like computer system reboots.
Algebraically, this gives rise to corrections with infinite support that cannot
be treated using the tools currently available in the literature. We present a
theoretical analysis of an enriched Banach algebra that, combined with
appropriate algorithms, enables the numerical treatment of these problems. The
results are applied to the solution of bidimensional Quasi-Birth-Death
processes with infinitely many phases which model random walks in the quarter
plane, relying on the matrix analytic approach. This methodology reduces the
problem to solving a quadratic matrix equation with coefficients of infinite
size. We provide conditions on the transition probabilities which ensure that
the solution of interest of the matrix equation belongs to the enriched
algebra. The reliability of our approach is confirmed by extensive numerical
experimentation on some case studies.
