Equilibrium joining probabilities for an M/G/1 queue

We study the customers' Nash equilibrium behavior in a single server observable queue with Poisson arrivals and general service times. Each customer takes a single decision upon arrival: to join or not to join. Furthermore, future regrets are not allowed. The customers are homogenous with respect to their linear waiting cost and the reward associated with service completion. The cost of joining depends on the behavior of the other customers present, which naturally forms a strategic game. We present a recursive algorithm for computing the (possibly mixed) Nash equilibrium strategy. The algorithm's output is queue-dependent joining probabilities. We demonstrate that depending on the service distribution, this equilibrium is not necessarily unique. Also, we show that depending on the service time distribution, either the 'avoid the crowd' phenomenon or the 'follow the crowd' phenomenon may hold.