An automated chemical or biological sample analyzer includes a plurality of linear sample carrying racks that are moved to a processing station for conducting an analysis using one or more carousel devices. Under computer control, rotational movements of the carousel or carousels, linear movements of the transfer or shuttle mechanisms, and operation of a pipetting station are performed in a coordinated fashion so as to handle the processing of a large number of samples simultaneously in an optimized fashion that allows random ordering of samples for processing.