We consider a service system (QS) that operates according to the FCFS discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially to the system and decide whether or not to join, using decision rules based upon the queue length on arrival to QS. Each customer is interested in selecting a rule that meets a certain optimality criterion with regards to their expected sojourn time in the system; as a consequence, the decision rules of other customers need to be taken into account. Within a particular class of decision rules for an associated infinite player game, the structure of the Nash equilibrium routing policies is characterized. We prove that within this class, there exist a finite number of Nash equilibria, and that at least one of these is non-randomized. Finally, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data with the aid of simulation experiments
