Two different threshold-based stochastic drop mechanisms for queuing systems

In this article two different threshold-based stochastic drop mechanisms of requests (either at the moment of arrival or at the moment of the end of service) for the $G|M|1|\infty$ system are presented. The threshold (the control parameter of the drop mechanism) in the queue not only determines the moment when the stochastic dropping of tasks (arriving or accepted into the system) is enabled, but also sets the safe area in the queue from which accepted into the system tasks cannot be dropped. The formulas for the main probability characteristics of the system (such as the stationary distribution of the number of tasks in the system, the probabilities for arriving tasks to be served or to be dropped (lost)) are derived. For the case of a Poisson incoming flow, the obtained probabilistic characteristics are compared