M/M/1/N Queues with Energy Enabled Service and General Vacation Times

We consider the performance analysis of an M/M/1/N queueing system where the server consumes energy from a battery during the service of customers. The energy supply of the battery depends on a randomly changing environment. When the battery gets empty, the server goes on vacation for a random amount of time. We model the energy level of the battery as the fluid in an infinite fluid buffer and the environment as a continuous-time Markov chain (CTMC). The analytical framework resembles the one used in fluid vacation models with exhaustive discipline, but the properties of the considered queueing model require the extension of the available methodology, because the model evolution is different during the service and the vacation period due to the inactivity of the server during vacation. Essentially, new results are derived to cope with the general properties of the considered model. Consequently, the results in this article extend the analysis of fluid vacation models to a more general class than the fluid vacation models with exhaustive discipline. The steady-state vector density of the fluid level, its Laplace transform, and the mean fluid level are derived together with the probability mass function of the queue size. A special case of the model with phase-type (PH)-distributed vacation time is also analyzed.