Heyman, Daniel P.O'Leary, Dianne P.We present an algorithm for solving linear systems involving the probability or rate matrix for a Markov chain. It is based on a UL factorization but works only with a submatrix of the factor U. We demonstrate its utility on Erlang-B models as well as more complicated models of a telephone multiplexing system. (Also cross-referenced as UMIACS-TR-96-24)en-USTechnical Report