The classical repairman problem, cf. Feller (1967) is generalized to consider r failure-prone machine types, each type having its own individual failure rate and also repair rate. Each failed machine joins its type queue, and is repaired by a single server. Several dynamic service priority schemes are considered that approximate first-come, first-served, longest-line first, and least-available first. A heavy-traffic asymptotic analysis determines approximations to the time dependent mean and covariance of individual type queue lengths, and shows that the marginal joint distribution of queue lengths is approximately Ornstein-Uhlenbeck. Numerical illustrations of approximation accuracy are provided, as are suggested applications to computer performance and manufacturing systems analysis.