Preemptive Priority Queuing System with Randomized Push-Out Mechanism and Negative Customers

A single-server priority queuing system with limited buffer size, Poisson arrivals, an exponentially distributed service time is considered. The primary customers take preemptive priority over secondary customers. We also consider a randomized push-out mechanism. It allows pushing secondary customers out of the system to free up space that could be taken by primary customers. Studied a new model where in addition to mentioned above two kinds of regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the buffer. The type of customer to be removed is determined in accordance with the following kill strategy. If at the moment of the occurrence of the next negative customer, both types of positive customers were presented in the system, then the primary customer is getting removed with a given probability. If there is only one type of customers in the system, then the customer of the existing type is deleted. Finally, if the system does not contain any positive customers at all, then a negative customer does not affect it. It is shown that such a queuing system can be investigated using the technique developed earlier by the authors for similar systems without negative customers. Using the method of generating functions, loss probabilities for both types of positive customers are obtained. The dependence of these loss probabilities on the basic parameters of the model (such as the probability of pushing out and the probability of crowding out a positive customer by a negative one) is investigated.