We study the market partition between two distinct firms that deliver services to waiting time sensitive customers. In our model, the incoming customers select a firm on the basis of its posted price, the expected waiting time and its location. Using a diffusion process, the market sharing dynamics is described by the time evolution of a boundary point, which exhibits a noise-induced phase transition. Indeed, we are able to analytically characterize a transition between a regime dominated by location considerations and a regime dominated by waiting time considerations.