A single-server queueing system with LIFO service, probabilistic priority, batch Poisson arrivals, and background customers

Consideration is given to the single-server queueing system with two independent flows of customers: a batch Poisson flow of (primary) customers and a saturated flow of background customers. Primary customers have relative priority over background customers, i. e., the service of a background customer cannot be interrupted. A background customer is instantly taken for service every time the buffer for primary customers is empty upon the service completion. The service times of primary and background customers are independent and are allowed to be generally distributed. The implemented service policy is LIFO (last in, first out) with the probabilistic priority. The method and analytic expressions for the computation (in terms of transforms) of the system's main stationary performance characteristics, including the stationary distribution of the waiting and sojourn times of the primary customers, are presented.