Abstract: The joint capacity and appointment schedule with walkin demands were modeled. The objective function was proved to be multimodular when the service time follows an exponential distribution and the cost function is a monotonically increasing convex function. Based on multimodular function， a neighborhood search algorithm converging to optimal solution was proposed to determine booking limits and schedule with maximizing utility. Numerical experiments were conducted to analyze the effects of system parameters on optimal utility and of walkin demands on the design of appointment system. The results indicate that the optimal schedule overbooks the first slot， and the walkin demand reduces the utility of the appointment system in most scenarios.

Key words: stochastic optimization, appointment schedule, multimodular function, queuing system, preregistration