A discrete time queueing system with negative customers and single working vacation

In this paper, we analyze a discrete time queueing system with geometrical arrivals of both positive and negative customers in which the server works at a lower rate during working vacation. Using embedded Markov chain and the matrix analysis solution method, we derive the probability generating function (PGF) of the number of customers waiting in the system and stationary queue length. From the process of the proof and the results, we also obtain the probabilities that the server is idle, busy, in working vacation, and in regular busy period, respectively. Finally, We introduce the application of the proposed model.