A model of queueing for a single server by several types of customers (messages, or jobs), with a simple dynamic priority rule, is considered. The rule is equivalent to selecting the next server occupant type with a probability proportional to the number of that type enqueued. The situation studied here occurs in fields such as computer and communication system performance analysis, in operational analysis of logistics systems, and in the repair of elements of a manufacturing system. It is assumed that the population sizes of the items of different types are large, and that the mean service rates are correspondingly large, in comparison with the service demand rates. Moreover, it is assumed that the system is in heavy traffic. Under these assumptions, asymptotic approximations are derived for the steady-state means and covariances of the number of items of different types either waiting or being served. Numerical comparisons with simulated results show excellent agreement.