In this paper we give a definition of "algorithm," "finite algorithm," "equivalent algorithms," and what it means for a single algorithm to dominate a set of algorithms. We define a derived algorithm which may have a smaller mean execution time than any of its component algorithms. We give an explicit expression for the mean execution time (when it exists) of the derived algorithm. We give several illustrative examples of derived algorithms with two component algorithms. We include mean execution time solutions for two-algorithm processors whose joint density of execution times are of several general forms. For the case in which the joint density for a two-algorithm processor is a step function, we give a maximum-likelihood estimation scheme with which to analyze empirical processing time data.