Stochastic machine scheduling to minimize waiting time related objectives with emergency jobs

We consider a new scheduling model where emergency jobs appear during the processing of current jobs and must be processed immediately after the present job is completed. All jobs have random processing times and should be completed on a single machine. The most common case of the model is the surgery scheduling problem, where some elective surgeries are to be arranged in an operation room when emergency cases are coming during the operating procedure of the elective surgeries. Two objective functions are proposed to display this practice in machine scheduling problem. One is the weighted sum of the waiting times and the other is the weighted discounted cost function of the waiting times. We address some optimal policies to minimize these objectives.