Tail asymptotics for M/G/1 type queueing processes with subexponential increments

Bivariate regenerative Markov modulated queueing processes {I-n,L-n} are described. {I-n} is the phase process, and {L-n} is the level process. Increments in the level process have subexponential distributions. A general boundary behavior at the level 0 is allowed. The asymptotic tail of the cycle maximum, M-Creg, during a regenerative cycle, C-reg, and the asymptotic tail of the stationary random variable L-infinity, respectively, of the level process are given and shown to be subexponential with L-infinity having the heavier tail.