We consider a fluid queue with two modes of service, that represents a production facility, where the processing of the customers (units) is typically carried out at a much faster time-scale than the machine-related processes. We examine the strategic behavior of the customers, regarding the joining/balking dilemma, under two levels of information upon arrival. Specifically, just after arriving and before making the decision, a customer observes the level of the fluid, but may or may not get informed about the state of the server (fast/slow). Assuming that the customers evaluate their utilities based on a natural reward/cost structure, which incorporates their desire for processing and their unwillingness to wait, we derive symmetric equilibrium strategy profiles. Moreover, we illustrate various effects of the information level on the strategic behavior of the customers. The corresponding social optimization problem is also studied and the inefficiency of the equilibrium strategies is quantified via the Price of Anarchy (PoA) measure.