Times until service completion and abandonment in an M/M/ \begin{document}$ m$\end{document} preemptive-resume LCFS queue with impatient customers

We consider an M/M/ \begin{document}$ m$\end{document} preemptive-resume last-come first-served (PR-LCFS) queue without exogenous priority classes of impatient customers. We focus on analyzing the time interval from the arrival to either service completion or abandonment for an arbitrary customer. We formulate the problem as a one-dimensional birth-and-death process with two absorbing states, and consider the first passage times in this process. We give explicit expressions for the probabilities of service completion and abandonment. Furthermore, we present sets of recursive computational formulas for calculating the mean and second moment of the times until service completion and abandonment. The two special cases of a preemptive-loss system and an ordinary M/M/ \begin{document}$ m$\end{document} queue with patient customers only, both incorporating the preemptive LCFS discipline, are treated separately. We show some numerical examples in order to demonstrate the computation of theoretical formulas.