Some Results for the Problem “Waiting for Godot”

Problem Statement: Consider an M/D/1 queueing system (Poisson arrival process, deterministic service times) and a test customer. The test customer is waiting for a friend whose arrival time is an exponentially distributed random variable. The test customer can either join the queue, if one exists, or wait outside the queue. Once the test customer joins the queue, he must stay in the queue until he reaches the server. If the test customer reaches the server after his friend arrives, he is served. Otherwise, he can either join the back of the queue, or wait outside the queue. What policy should the test customer follow to minimize the mean delay until service?