Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow

We investigate queueing systems with regenerative input flow in the presence of heavy tails. Our goal is asymptotics of the probability of the exceeding of the high level by the waiting time process in the stationary regime. As an input flow $X(t)$ we consider the total service time of customers arrived during time-interval $[0,t]$. This allows to consider the case when service times $\{\eta_n\}_{n=1}^\infty$ are dependent random variables, besides they may be dependent on the number of customers arrived in $[0,t]$. We obtain conditions under which the virtual waiting time process in steady state has a subexponential distribution function. This result we apply to a renovated stationary ergodic sequence to find the tail probabilities for the maximum of a random walk with negative drift. We also consider a queue with a doubly stochastic Poisson flow in case when the random intensity is a regenerative process. We show that these results could be transferred to corresponding systems with an unreliable server.