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

Bivariate regenerative Markov modulated queueing processes are described. 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 during a regenerative cycle and the asymptotic tail of the stationary random variable, respectively, of the level process are given and shown to be subexponential having the heavier tail.