We consider the Mn/Gn/1 queue with vacations and exhaustive service in which the server takes (repeated) vacations whenever it becomes idle, the service time distribution is queue length dependent, and the arrival rate varies both with the queue length and with the status of the server, being busy or on vacation. Using a rate balance principle, we derive recursive formulas for the conditional distribution of residual service or vacation time given the number of the customers in the system and the status of the server. We also derive a closed-form expression for the steady-state distribution as a function of the probability of an empty system. As an application of the above, we provide a recursive computation method for Nash equilibrium joining strategies to the observable M/G/1 queue with vacations.
Oz, B., Adan, I. J. B. F., & Haviv, M. (2019). The Mn/Gn/1 queue with vacations and exhaustive service. European Journal of Operational Research, 277(3), 945-952. https://doi.org/10.1016/j.ejor.2019.03.016