Stationary characteristics of the finite capacity queueing system with inverse service order and generalized probabilistic priority

Consideration is given to the $M/G/1/(r-1)$ queueing system with LIFO (last in, first out) preemptive generalized probabilistic priority policy. It is assumed that customer's service time becomes known upon its arrival at the system and at any time instant remaining service times of all customers present in the system are available. On arrival of a customer at a nonempty system, its service time is compared to the (remaining) service time of the customer in service and one of the following events occurs: both customers leave the system at once, one of the customers leaves the system (the other occupies the server), or both customers stay in the system (one occupies the server, the other — one place in the queue). Those customers which stay in the system acquire new service time according to a known distribution, which can depend on their initial service times. Arriving customers which find the queue full, leave the system and have no influence on it. Analytical expressions for the computation of the joint stationary distribution of the number of customers in the system and the remaining service time of the customer in the server, of the busy period and the stationary sojourn time (in terms of Laplace–Stieltjes transform) are proposed.