Tail asymptotics of the waiting time and the busy period for the $${{\varvec{M/G/1/K}}}$$ queues with subexponential service times

Abstract

We study the asymptotic behavior of the tail probabilities of the waiting time and the busy period for the $$M/G/1/K$$ queues with subexponential service times under three different service disciplines: FCFS, LCFS, and ROS. Under the FCFS discipline, the result on the waiting time is proved for the more general $$GI/G/1/K$$ queue with subexponential service times and lighter interarrival times. Using the well-known Laplace---Stieltjes transform (LST) expressions for the probability distribution of the busy period of the $$M/G/1/K$$ queue, we decompose the busy period into a sum of a random number of independent random variables. The result is used to obtain the tail asymptotics for the waiting time distributions under the LCFS and ROS disciplines.