This paper examines prioritization in a service system and analyzes whether, in the presence of heterogeneous customers who have different needs and a costly sorting process, it pays to prioritize. In particular, in our model, sorting is costly because the task of gathering information to prioritize jobs consumes resources. We investigate a stylized model in which there are two classes of jobs - one whose waiting cost is high, called urgent, and the other whose waiting cost is low, called non-urgent. There are two types of employees, sorters, who collect information on a job and then decide whether it is urgent or non-urgent, and processors, who execute the job or provide the service. We begin by assuming that sorters categorize customers perfectly, and we relax this assumption later in the paper. We optimize two performance metrics, waiting costs (under a given budget) and total costs, and find the conditions under which prioritization is beneficial for these two metrics.