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Solution of large sparse fixed-point problems, Mline over x = over x and Mline over x + line over b = over x, may be seen as underpinning many important performance-analysis calculations. These calculations include steady-state, passage-time and transient-time calculations in discrete-time Markov chains, continuous-time Markov chains and semi-Markov chains. In recent years, much work has been done to extend the application of asynchronous iterative fixed-point solution methods to many different contexts. This work has been motivated by the potential for faster solution, more efficient use of the communication channel and/or access to memory, and simplification of task management and programming. In this paper, we present theoretical developments which allow us to extend theapplication of asynchronous iterative solution methods to solve for the key performance metrics mentioned above-such that we may employ the full breadth of Chazan and Miranker's classes of asynchronous iterations.