$M/G/1$ queue with state-dependent heterogeneous batch arrivals, inverse service order, and probabilistic priority

Consideration is given to the stationary characteristics of single-server queues with the queue of infinite capacity, independent and identically-distributed service times, LCFS (last-come-first-served) service order, and probabilistic priority discipline. Most of the results for such type of queueing systems have been obtained under the assumption of either Poisson arrivals or phase-type arrivals. Another important assumption made was that the arrival process is independent from the system state. The author shows that the latter assumption can be relaxed to some, quite large extent. The author considers an $M/G/1/\infty$ queue with batch Poisson arrival flow in which ($i$) the arrival rate depends on the total number of customers present in the system at the arrival instant; and ($ii$) the size of the arriving batch $k$ and the remaining service times $x_1,\dots,x_k$ of the customers in the batch have the arbitrary continuous joint probability distribution $B_k(x_1,\dots,x_k)$. The author obtains analytic expressions for the computation of the joint stationary distribution of the total number of customers in the system and their remaining service times. Busy period, waiting and sojourn time distributions are also given in terms of the Laplace–Stieltjes transforms.