A single-server G-queue in discrete-time with geometrical arrival and service process

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. This paper analyses a discrete-time single-server queue with geometrical arrivals of both positive and negative customers. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, the two sub-cases in which a customer currently being served can and cannot be killed by a negative customer. Thus, we carry out a complete study of these systems, including the ergodicity condition as well as exact formulae for the associated stationary distribution. The effect of several parameters on the systems is shown numerically.