The potential demand in the market and customers' perception of service value are crucial factors in pricing strategies, resource allocation, and other operational decisions. However, this information is typically private and not readily accessible. In this paper, we analyze a service system operating across two stations, each with its own customer flow. Customers arriving at the system are informed of the waiting times at both stations and can choose to either join the local station, switch to the other station, or balk. Our objective is to estimate the arrival rates at each station and customers' perceived service value based on the observed workloads at both stations. A significant challenge arises from the inability to observe balking customers and the lack of distinction between local arrivals and customers switching from the other station, as the switching cost is unknown. To address this issue, we employ maximum likelihood estimation and validate the effectiveness of the estimator through a series of simulations.