Homogeneous customers renege from invisible queues at random times under deteriorating waiting conditions

We consider a memoryless first-come first-served queue in which customers' waiting costs are increasing and convex with time. Hence, customers may opt to renege if service has not commenced after waiting for some time. We assume a homogeneous population of customers and we look for their symmetric Nash equilibrium reneging strategy. Besides the model parameters, customers are aware only, if they are in service or not, and they recall for how long they are have been waiting. They are informed of nothing else. We show that under some assumptions on customers' utility function, Nash equilibrium prescribes reneging after random times. We give a closed form expression for the resulting distribution. In particular, its support is an interval (in which it has a density) and it has at most two atoms (at the edges of the interval). Moreover, this equilibrium is unique. Finally, we indicate a case in which Nash equilibrium prescribes a deterministic reneging time.