Optimal control of premature queueing

Given an M/G/1 queue, a special customer, and an exponential random variable M that is independent of the queueing process, if the special customer reaches the front of the queue before the time specified by M, it cannot be processed and must re-enter the queue at the back. While waiting in the queue, the available decisions for the special customer are either to keep its position in the queue or to leave its position to join the back of the queue. The author´s goal is to find a strategy that minimizes the expected delay until the special customer starts being processed. The move-along policy is to always stay in the queue until the front of the queue is reached, and if it cannot be served, to join the back of the queue. The basic result is that if the queue is stable, the move-along policy is optimal