We consider customer service chat systems where customers can receive real time service from agents using an instant messaging application over the Internet. A unique feature of these systems is that agents can serve multiple customers simultaneously. The number of customers that an agent is serving, referred to as the level of that agent, determines the rate at which each customer assigned to that agent receives service. Customers are impatient and may abandon the system during service. Our objective is to minimize the number of agents while providing a certain service level, measured in terms of the proportion of customers who abandons the system in the long run. We propose a framework involving measure-valued processes to model the system dynamics. Deterministic fluid models are developed to provide first-order approximations for system performance. In particular, the equilibrium states of the fluid models provide simple approximations for various performance metrics of the system in steady state. Numerical experiments show that these approximations are fairly accurate. The significance of the approximations is that they reveal how general service and patience time distributions impact the system performance, and that they lead to design of optimal control and staffing policies for these systems.