  Many problems in computing, service, and manufacturing systems can be modeled
via infinite repeating Markov chains with an infinite number of levels and a
finite number of phases. Many such chains are quasi-birth-death processes
(QBDs) with transitions that are skip-free in level, in that one can only
transition between consecutive levels, and unidirectional in phase, in that one
can only transition from lower-numbered phases to higher-numbered phases. We
present a procedure, which we call Clearing Analysis on Phases (CAP), for
determining the limiting probabilities of such Markov chains exactly. The CAP
method yields the limiting probability of each state in the repeating portion
of the chain as a linear combination of scalar bases raised to a power
corresponding to the level of the state. The weights in these linear
combinations can be determined by solving a finite system of linear equations.
