A maximization of the finishing probability of two jobs processed in a queue

抄録

We deal with a situation where a worker processes two kinds of jobs. Job A (JA) can be processed only at a discrete time queueing system. He/She must join the queue when he decides to process JA and stays there until he completes JA. Job B (JB) is completed after processing several steps and each step needs a

unit time. At the end of each step, he observes the queue length and decides whether he joins the queue. If JB is completed, he joins the queue to process JA. If he decides to join the queue, he processes the rest of JB after JA. There is a time limit to process both jobs and the objective is to maximize the probability to finish the two jobs within the time limit. We prove a monotone property of the optimal policy by a dynamic programming formulation.